diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzddbs" "b/data_all_eng_slimpj/shuffled/split2/finalzzddbs" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzddbs" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:sec1}\n\nThe first theoretical prediction of the famous phenomenon known as\nBose-Einstein condensation (BEC) was made in 1924 and 1925 by Bose\n\\cite{Bose24} and Einstein \\cite{Einstein 24-25}, respectively. In\na system of particles obeying Bose-Einstein statistics where the\ntotal number of particles is conserved, there should be a\ntemperature below which a finite fraction of all the particles\ncondense into the same one-particle state\n\\cite{Griffin95,Dalfavo99,Leggett01,Parkins98,Fetter98,Courteille01}.\nSeventy years later, in a remarkable experiment, Anderson \\emph{et\nal.} \\cite{Anderson95} have cooled magnetically trapped $^{87}$Rb\ngas to nanokelvin temperatures, and observed the BEC. This\ndiscovery has generated a huge amount of theoretical\ninvestigations\n\\cite{Baym96,Dalfavo96,Esry97,Fabrocini99,Fabrocini01,DuBois01,Minguzzi00,Naraschewski99,Pitaevski99}.\n\nThe main feature of the trapped alkali-metal and atomic hydrogen\nsystems (which obey the Bose-Einstein statistics) is that they are\ndilute. The crucial parameter defining the condition of diluteness\nis the gas parameter $\\chi=na^3$, where $n$ is the density of the\nsystem and $a$ is the s-wave scattering length \\cite{Fabrocini01}.\nThere are two ways to bring $\\chi$ outside the regime of validity\nof the mean field description. The first one is to increase the\ndensity, while the second one to change the effective size of the\natoms.\n\nThe diluteness of the gas is ensured when the effective atomic\nsize is small compared both to the trap size and to the\ninteratomic distance. However, the effects of inter-particle\ninteractions are of fundamental importance in the study of the BEC\ndilute-gas where the physics should be dominated by two-body\ncollisions described in terms of the s-wave scattering length $a$.\nIn the case of positive $a$, it is equivalent to consider a very\ndilute (atomic) system of hard spheres, whose diameter coincides\nwith the scattering length itself \\cite{Fabrocini99}. The natural\nstarting point for studying the behavior of those systems is the\ntheory of weakly interacting bosons which, for inhomogeneous\nsystems, takes the form of the Gross-Pitaevskii equation\n\\cite{Pitaevski61,Gross61}. This is a mean-field approach for the\norder parameter associated with the condensate \\cite{Dalfavo99}.\n\nIn the present work we study BEC in a phenomenological way where\nthe Bose gas is considered as a many-body system\n\\cite{Ketterle99}. In particular, we study the ground state of a\nsystem of correlated bosonic atoms at zero temperature, trapped by\na harmonic oscillator potential (HO). The key quantities for this\neffort is the one- and two-body density matrices \\cite{Lowdin}. As\nthe mean-field approach (non-interacting atoms) fails to\nincorporate the interparticle interactions which are necessary for\nthe description of the correlated Bose system, we introduce the\nrepulsive interactions among the atoms, through the Jastrow\ncorrelation functions $f\\left(|\\textbf{r}_1-\\textbf{r}_2| \\right)$\n\\cite{Moustakidis02}.\n\nWe focus our efforts to the calculation of the one- and two-body\ndensity and momentum distributions and the calculation of the\nstatic structure factor\n\\cite{Moustakidis02,MoustaChatz05,Moustakidis04,Massen05}.\nOne-body density and momentum distributions are complementary\ndescriptions of the Bose gas and related directly with the\nmean-square radius and mean kinetic energy of the trapped Bose gas\nrespectively. In addition the two-body density distribution is\nrelated with the calculation of the static structure factor, a\nquantity which gives information for the ground and excited states\nof the gas. Special effort has been devoted to the derivation of\nthe natural orbital and natural occupation numbers through the\ndiagonalization of the one-body density matrix.\n\n In recent years information-theoretic methods play an increasing\nrole for the study of quantum mechanical systems. An example is\nthe application of the Maximum Entropy Principle \\cite{Kapur89}\n(MEP) to the calculation of the wave function in a potential\n\\cite{Canosa92} using as constraints expectation values of simple\nobservables and reconstructing a quantum wave function from a\nlimited set of expectation values. The idea behind MEP is to\nchoose the least biased result, compatible with the constraints of\nthe problem. Thus the MEP provides the least biased description\nconsistent with the available relevant information. This is done\nby employing a suitably defined quantum entropy that measures the\nlack of information associated with the distribution of a quantum\nstate over a given known basis.\n\nInformation entropy is important for the study of quantum\nmechanical systems in two cases: first in the clarification of\nfundamental concepts of quantum mechanics and second in the\nsynthesis of probability densities in position and momentum space\n\\cite{Garbaczewski05}.\n\nIn the present work special effort is devoted for the calculation\nof various quantum information properties including Shannon\nentropy, Onicescu energy, Kullback-Leibler relative entropy and\nalso Jensen-Shannon divergence. The information properties are\ncalculated for the interatomic correlations. In addition the\nGross-Piatevskii equation is solved giving the density and\nmomentum distribution which are employed to calculate the above\nquantum information properties of the Bose gas\n\\cite{Massen05,Massen02}. The results are compared with those\ntaken in a phenomenological way in the framework of the Jastrow\ncorrelations.\n\nThe plan of the paper is the following: In Sec. \\ref{sec:sec2} the\ngeneral definitions related to the density matrices of a Bose\nsystem are considered. Details of the lowest-order cluster\nexpansion, analytical expressions and numerical results are\nreported in Sec. \\ref{sec:sec3}. In Sec. \\ref{sec:sec4} formulas\nfor the quantum information properties (both for the one-and\ntwo-body density matrices) are reviewed and analytical results are\npresented. Quantum information properties based on\nGross-Piatevskii equation are presented in Sec. \\ref{sec:sec5}\nwhile the summary of the work is given in Sec. \\ref{sec:sec6}.\n\n\\section{Definition of Density Matrices}\\label{sec:sec2}\n\nLet $\\Psi({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)$ be the wave\nfunction describing the trapped Bose gases. In the case where this\nsystem is composed of non-interacting bosonic atoms at zero\ntemperature, all atoms occupy the same single-particle ground\nstate. The many body ground state wave function $\\Psi_0({\\bf\nr}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)$ is then a product of $A$\nidentical single particle ground state wave functions. This ground\nstate wave function is therefore called the condensate wave\nfunction or macroscopic wave function and has the form\n\\cite{Ketterle99}\n\\begin{equation}\n\\Psi_0({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)= \\psi_0({\\bf\nr}_1)\\psi_0({\\bf r}_2) \\cdots \\psi_0({\\bf r}_A), \\label{WF-1}\n\\end{equation}\n\n\\noindent where $\\psi_0({\\bf r})$ is the normalized to one\nground-state single-particle wave function describing a bosonic\natom. It is worth to indicate that Eq. (\\ref{WF-1}) is valid even\nwhen\n weak interactions are included. In this case the wave function\n$\\Psi_0({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)$ is still, to a very\ngood approximation, a product of $A$ single particle wave\nfunctions obtained now from the solution of a non-linear\nSchr\\\"odinger equation, the well known Gross-Pitaevskii equation.\nHowever, in the general case where interactions between atoms are\nincluded, the ground state wave function $\\Psi({\\bf r}_1,{\\bf\nr}_2,\\ldots,{\\bf r}_A)$ is modified from the simple form of Eq.\n(\\ref{WF-1}). In that case a percentage of atoms is moving from\nthe condensate orbit $\\psi_0$ to higher orbits.\n\n\nIn the present work we adopt the following normalization of the\nwave function $\\Psi({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)$,\n\n\\begin{equation}\n \\int \\Psi^{*}({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)\n \\Psi({\\bf r}_1,{\\bf r}_2,\\ldots,{\\bf r}_A)\n d\\textbf{r}_1 d\\textbf{r}_2 \\cdots d\\textbf{r}_A=1,\n\\end{equation}\n\n\\noindent where the integration is carried out over the radius\nvectors $\\textbf{r}_1$, $\\textbf{r}_2$, $\\ldots$, $\\textbf{r}_A$.\n\nA quantity characterizing very important aspects of a Bose gas (as\nwell a variety of quantum many-body systems) is the one-body\ndensity matrix defined as in \\cite{Lowdin}\n\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_1')=\\int \\Psi^{*}({\\bf r}_1,{\\bf\nr}_2,\\ldots,{\\bf r}_A) \\Psi({\\bf r}_1',{\\bf r}_2,\\ldots,{\\bf r}_A)\nd{\\bf r}_2 \\cdots d{\\bf r}_A. \\label{OBDM-1}\n\\end{equation}\n\nThe one-body density matrix is connected to the position- and\nmomentum-space properties of the Bose gas and in addition it is\nthe quantity which gives the percentage of the condensate of the\nsystem.\n\nThe two-body density matrix is a generalization of the one-body\ndensity matrix and is defined as\n\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_2;{\\bf r}_1',{\\bf r}_2')= \\int\n\\Psi^{*}({\\bf r}_1,{\\bf r}_2,{\\bf r}_3,\\cdots,{\\bf r}_A) \\Psi({\\bf\nr}_1',{\\bf r}_2',{\\bf r}_3,\\cdots,{\\bf r}_A) d{\\bf r}_3 \\cdots\nd{\\bf r}_A. \\label{TBDM-1}\n\\end{equation}\nThe above density matrices are related by the following equation\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_1')=\\int \\rho({\\bf r}_1,{\\bf r}_2;{\\bf\nr}_1',{\\bf r}_2) d{\\bf r}_2. \\label{O-T-1}\n\\end{equation}\n\nThe two-body density matrix is related directly to the interatomic\ninteraction and its diagonal part provides the two-body density\ndistribution $\\rho(\\textbf{r}_1,\\textbf{r}_2)$ (expresses the\njoint probability of finding two atoms at the positions ${\\bf\nr}_1$ and ${\\bf r}_2$, respectively), a key quantity of the\npresent work\n\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_2)= \\rho({\\bf r}_1,{\\bf r}_2;{\\bf\nr}_1',{\\bf r}_2')\\mid_{{\\bf r}_1'={\\bf r}_1, {\\bf r}_2'={\\bf\nr}_2}. \\label{TBDD-1}\n\\end{equation}\n\nOn the other hand the diagonal part of the one-body density matrix\nis just the density distribution of the Bose gas and expresses the\nprobability of finding an atom at position $\\textbf{r}_1$\n\n\\begin{equation}\n\\rho({\\bf r}_1)=\\rho({\\bf r}_1,{\\bf r}_1')|_{{\\bf r}_1={\\bf\nr}_1'}. \\label{DD-1}\n\\end{equation}\n\nThe quantities $\\rho(\\textbf{r}_1)$ and\n$\\rho(\\textbf{r}_1,\\textbf{r}_2)$ are also related by the\nfollowing integral\n\n\\begin{equation}\n\\rho({\\bf r}_1)=\\int \\rho({\\bf r}_1,{\\bf r}_2)\n d{\\bf r}_2. \\label{DD-2}\n\\end{equation}\n\nVery interesting is also the description of the Bose gas in\nmomentum-space via the quantities of the one- and two-body\nmomentum distributions. The two-body momentum distribution\n$n(\\textbf{k}_1,\\textbf{k}_2)$ expresses the joint probability of\nfinding two atoms with momentum $\\textbf{k}_1$ and $\\textbf{k}_2$\nrespectively and is given by a particular Fourier transform of the\ncorresponding two-body density matrix $\\rho({\\bf r}_1,{\\bf\nr}_2;{\\bf r}_1',{\\bf r}_2')$\n\n\\begin{equation}\nn({\\bf k}_1,{\\bf k}_2)=\\frac{1}{(2\\pi)^6} \\int \\rho({\\bf r}_1,{\\bf\nr}_2;{\\bf r}_1',{\\bf r}_2') \\exp[i{\\bf k}_1({\\bf r}_1-{\\bf r}_1')]\n\\exp[i{\\bf k}_2({\\bf r}_2-{\\bf r}_2')] d {\\bf r}_1 d {\\bf r}_1' d\n{\\bf r}_2 d {\\bf r}_2'. \\label{TBMD-1}\n\\end{equation}\n\nThe one-body momentum distribution (or simply momentum\ndistribution) $n(\\textbf{k})$, expresses the probability of\nfinding an atom with momentum $\\textbf{k}$, and it is given by a\nparticular Fourier transform of the one-body density matrix\n$\\rho({\\bf r}_1,{\\bf r}_1')$\n\n\\begin{equation}\nn({\\bf k})=\\frac{1}{(2\\pi)^3} \\int \\rho({\\bf r}_1,{\\bf r}_1') \\exp\n\\left[i{\\bf k}({\\bf r}_1-{\\bf r}_1')\\right]\n d {\\bf r}_1 d {\\bf r}_1'. \\label{mom}\n\\end{equation}\n\nIt can be shown easily that in the case where the Bose gas is\ndescribed by the wave function of Eq. (\\ref{WF-1}) the two-body\ndensity matrix is given by\n\\begin{equation}\n\\rho_{0}({\\bf r}_1,{\\bf r}_2;{\\bf r}_1',{\\bf r}_2')= \\rho_{0}({\\bf\nr}_1,{\\bf r}_1')\\rho_{0}({\\bf r}_2,{\\bf r}_2'), \\label{TBDM-BG}\n\\end{equation}\nwhere\n\\begin{equation}\n\\rho_{0}({\\bf r}_1,{\\bf r}_1')=\\psi_{0}^{*}({\\bf r}_1)\n\\psi_{0}({\\bf r}_1'). \\label{eq:eq12new}\n\\end{equation}\n\nFrom Eqs. (\\ref{TBMD-1}), (\\ref{mom}) and (\\ref{TBDM-BG}) we get\n\\begin{eqnarray}\nn_0(\\textbf{k}_1,\\textbf{k}_2) &=& \\displaystyle{\n\\frac{1}{(2\\pi)^3}} \\int \\rho_0(\\textbf{r}_1,\\textbf{r}_1')\\,\n\\textrm{exp}\\left[ i \\textbf{k}_1 (\\textbf{r}_1-\\textbf{r}_1')\n\\right]\\,d\\textbf{r}_1\nd\\textbf{r}_1' \\nonumber \\\\\n& & \\times \\displaystyle{ \\frac{1}{(2\\pi)^3}} \\int\n\\rho_0(\\textbf{r}_2,\\textbf{r}_2')\\, \\textrm{exp}\\left[ i\n\\textbf{k}_2 (\\textbf{r}_2-\\textbf{r}_2') \\right]\\,d\\textbf{r}_2\nd\\textbf{r}_2' \\\\\n&=& n_0(\\textbf{k}_1)n_0(\\textbf{k}_2). \\nonumber\n\\end{eqnarray}\n\nFinally form Eqs. (\\ref{mom}) and (\\ref{eq:eq12new}) we get\n\n\\begin{eqnarray}\nn_0(\\textbf{k})&=& \\displaystyle{ \\frac{1}{(2\\pi)^{3\/2}}} \\int\n\\psi_{0}^{*}({\\bf r}_1) \\textrm{exp}\\left[ i \\textbf{k}\n\\textbf{r}_1 \\right] d\\textbf{r}_1 \\nonumber \\\\\n& & \\times \\displaystyle{ \\frac{1}{(2\\pi)^{3\/2}}} \\int\n\\psi_{0}^{*}({\\bf r}_1') \\textrm{exp}\\left[ i \\textbf{k}\n\\textbf{r}_1' \\right] d\\textbf{r}_1' \\label{eq:eq14new} \\\\\n&=& \\tilde{\\psi}_{0}^{*}({\\bf k})\\tilde{\\psi}_{0}({\\bf k}).\n\\nonumber\n\\end{eqnarray}\n\nFrom Eq. (\\ref{eq:eq14new}) it is obvious that\n$\\tilde{\\psi}_{0}^{*}({\\bf k})$ is the particular Fourier\ntransform of the single particle wave function $\\psi_{0}({\\bf\nr})$.\n\n\\subsection{Static Structure Factor}\n\nSpectroscopic studies have been used to assemble a complete\nunderstanding of the structure of atoms and simple molecules\n\\cite{Stamper-Kurn}. The static structure factor $S(k)$ is a\nfundamental quantity, connected with the atomic structure, and is\nthe Fourier transform of the radial distribution function $g(r)$.\n$S(k)$ gives the magnitude of the density fluctuation in the\nsystem (atomic, molecular, electronic or nuclear) at wavelength\n$2\\pi\/k$, where $k$ is the momentum transfer. In recent papers,\nthe Bragg spectroscopic method was used to measure $S(k)$ either\nin the phonon regime \\cite{Stamper-Kurn} or\/and in the\nsingle-particle regime \\cite{Steinhauer}.\n\nThe static structure factor in a finite system is defined as\n\\cite{Zambelli}\n\n\\begin{equation}\nS({\\bf k})=1+ \\frac{1}{N} \\int e^{i{\\bf k}({\\bf r}_1-{\\bf r}_2)}\n\\left[ \\rho({\\bf r}_1,{\\bf r}_2) - \\rho({\\bf r}_1) \\rho({\\bf r}_2\n) \\right] d {\\bf r}_1 d {\\bf r}_2. \\label{eq:eq18}\n\\end{equation}\n\nIn the most general case the two-body density distribution\n$\\rho(\\textbf{r}_1,\\textbf{r}_2)$ and the one-body density\ndistribution $\\rho(\\textbf{r})$ are connected via the following\nrelation\n\n\\begin{equation}\n \\rho(\\textbf{r}_1,\\textbf{r}_2)=C\\rho(\\textbf{r}_1)\\rho(\\textbf{r}_2)f^2(r_{12})\n =C\\rho(\\textbf{r}_1)\\rho(\\textbf{r}_2)g(r_{12}), \\label{eq:eq19}\n\\end{equation}\nwhere $g(r_{12})$ is the radial distribution function and $C$ is\nthe normalization factor which ensures that\n\n\\begin{equation}\n \\int \\rho(\\textbf{r}_1,\\textbf{r}_2) d\\textbf{r}_1\n d\\textbf{r}_2=N(N-1). \\label{eq:eq20}\n\\end{equation}\n\nWe also consider that\n\\begin{equation}\n \\int \\rho(\\textbf{r}_1) d\\textbf{r}_1=N, \\label{eq:eq21}\n\\end{equation}\nwhere $N$ is the number of the atoms of the Bose condensate.\n\nIn the uncorrelated case (non-interacting gas) the radial\ndistribution function is $g(r_{12})=1$ (absence of correlations),\nand the two-body density distribution becomes\n\n\\begin{equation}\n \\rho(\\textbf{r}_1,\\textbf{r}_2)=\n \\frac{N-1}{N}\\rho(\\textbf{r}_1)\\rho(\\textbf{r}_2).\n\\end{equation}\n\nUsing Eq. (\\ref{eq:eq19}), Eq. (\\ref{eq:eq18}) is written as\n\\begin{equation}\nS({\\bf k})=1+ \\frac{1}{N} \\int \\textrm{exp} \\left[i{\\bf k}({\\bf\nr}_1-{\\bf r}_2)\\right] \\rho({\\bf r}_1) \\rho({\\bf r}_2 ) [C\ng(r_{12})-1] d {\\bf r}_1 d {\\bf r}_2. \\label{str-fin2}\n\\end{equation}\n\nConditions (\\ref{eq:eq20}) and (\\ref{eq:eq21}) ensure that\n$S(0)=0$.\n\nThe integration in Eq. (\\ref{str-fin2}) can be performed if the\nfunction $g(r)$ is known. $g(r)$ must obey the rules $g(r=0)=0$\nand $\\displaystyle{\\lim_{r \\rightarrow \\infty} g(r) \\rightarrow\n1}$. The first rule introduces the repulsive correlations between\nthe atoms and the second the absence of such correlations in long\ndistances. In general the form of $S(k)$ is affected appreciably\nfrom the form of $g(r)$. More specifically, the long range\nbehavior of $g(r)$ affects $S(k)$ for small values of $k$ while\nits short range behavior affects $S(k)$ for large values of $k$ as\na direct consequence of the Fourier transform theory.\n\n\n\\subsection{Natural Orbitals and Natural Occupation Numbers}\n\nIn the case of the inclusion of the inter-particle interactions\nbetween the atoms, which give rise to the depletion of the\ncondensate, the one-body density matrix is written\n\\cite{Stringari01}\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_1')= n_0 \\psi_{0}^{*}({\\bf r}_1)\n\\psi_{0}({\\bf r}_1')+ \\sum_{i\\neq 0}n_i \\psi_{i}^{*}({\\bf r}_1)\n\\psi_{i}({\\bf r}_1'),\n\\end{equation}\nwhere $\\displaystyle{\\sum_{i}n_i=1}$. The sum\n$\\displaystyle{\\sum_{i\\neq 0}n_i \\psi_{i}^{*}({\\bf r}_1)\n\\psi_{i}({\\bf r}_1')}$ is the contribution arising from the atoms\nout of the condensate. The eigenfunctions $\\psi_i({\\bf r}),$ which\nare called natural orbitals (NO's), and the eigenvalues $n_i$,\ncalled natural occupation numbers (NON's), are obtained by\ndiagonalizing the one-body density matrix through the eigenvalue\nequation\n\\begin{equation}\n\\int \\rho({\\bf r}_1,{\\bf r}_1') \\psi_i({\\bf r}_1') d {\\bf r}_1'\n=n_i\\psi_i({\\bf r}_1). \\label{diag-rho}\n\\end{equation}\nThe condition, generally adopted, for the existence of\ncondensation is that there should be one eigenvalue $n_i$ which is\nof the order of the number of the particles in the trap.\n\nThe NO's $\\psi_{i}({\\bf r}_1)$ and the NON's $n_i$ are obtained by\ndiagonalizing the one-body density matrix through the eigenvalue\nequation (\\ref{diag-rho}) by expanding first the one-body density\nmatrix in a series of Legendre polynomials $P_l(x)$\n\\begin{equation}\n\\rho({\\bf r},{\\bf r}')=\\rho(r, r',\\cos\\omega_{rr'})=\n\\sum_{l=0}^{\\infty}\\rho_l(r,r') P_l(\\cos\\omega_{rr'}),\n\\label{ch4-NO4}\n\\end{equation}\nwhere $\\rho_l(r,r')$ are the coefficients of the expansion\n\\begin{equation}\n\\rho_l(r,r')=\\frac{2l+1}{2}\\int_{-1}^{1} \\rho(r,\nr',\\cos\\omega_{rr'}) \\ P_l(\\cos\\omega_{rr'}) \\ {\\rm d}\n(\\cos\\omega_{rr'}). \\label{ch4-NO5}\n\\end{equation}\nFrom the Eqs. (\\ref{diag-rho}), (\\ref{ch4-NO4}) and\n(\\ref{ch4-NO5}) the eigenvalue equation is written\n\\begin{equation}\n4\\pi \\int_{0}^{\\infty}\\rho_{l}(r,r')\\varphi_{nl}^{NO}(r')\n{r'}^2{\\rm d} r'=n_{nl}^{NO}\\varphi_{nl}^{NO}(r), \\label{NO-3}\n\\end{equation}\nwhere $\\varphi_{nl}^{NO}(r)$ is the radial part of\n $\\psi_{i}({\\bf r})$\n($\\psi_{i}({\\bf r})=\\varphi_{nl}^{NO}(r)Y_{lm}(\\Omega_r) $).\n\n\n\n\\section{Jastrow type Correlated Properties of a Trapped Bose\nGas}\\label{sec:sec3}\n\n\n\\subsection{Correlated Density Matrices}\\label{sub:sub3-1}\n\nA dilute trapped Bose gas can be studied using the lowest-order\napproximation \\cite{Fabrocini99}. In this approximation the\ntwo-body density matrix has the form\n\\cite{Moustakidis02,MoustaChatz05}\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_2;{\\bf r}_1',{\\bf r}_2')= N_0 \\rho_0({\\bf\nr}_1,{\\bf r}_1' ) \\rho_0({\\bf r}_2,{\\bf r}_2') f(|{\\bf r}_1' -\n{\\bf r}_2'|) f(|{\\bf r}_1-{\\bf r}_2|), \\label{TBDM-1}\n\\end{equation}\nwhere $f(|{\\bf r}_1-{\\bf r}_2|)$ is the Jastrow correlation\nfunction, which depends on the inter-particle distance and $N_0$\nis the normalization factor which ensures that\n\\[\\int \\rho({\\bf\nr}_1,{\\bf r}_2;{\\bf r}_1',{\\bf\nr}_2')|_{(\\textbf{r}_1=\\textbf{r}_1', \\textbf{r}_2=\\textbf{r}_2')}\n\\, d\\textbf{r}_1 d\\textbf{r}_2=1.\n\\]\n\nThe diagonal part of $\\rho({\\bf r}_1,{\\bf r}_2;{\\bf r}_1',{\\bf\nr}_2')$ that is the two body density distribution takes the form\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_2)= N_0 \\rho_0({\\bf r}_1)\\rho_0({\\bf r}_2)\nf^2(r_{12}), \\label{TBDD-12}\n\\end{equation}\nwhile the one-body density matrix is given by the integral\n\\begin{equation}\n\\rho({\\bf r}_1,{\\bf r}_1')= N_0 \\rho_0({\\bf r}_1,{\\bf r}_1') \\int\n\\rho_0({\\bf r}_2,{\\bf r}_2) f(|{\\bf r}_1' - {\\bf r}_2|)f(|{\\bf\nr}_1 - {\\bf r}_2|) d\\textbf{r}_2. \\label{OBDM-1}\n\\end{equation}\n\nThe density distribution, which is the diagonal part of $\\rho({\\bf\nr}_1,{\\bf r}_1')$, can also be obtained from the integral\n\\begin{eqnarray}\n\\rho({\\bf r})= \\int \\rho({\\bf r},{\\bf r}_2) d{\\bf r}_2 =N_0\n\\rho_0({\\bf r}) \\int \\rho_0({\\bf r}_2) f^2(|{\\bf r}-{\\bf r}_2|) d\n{\\bf r}_2. \\label{cor-dd1}\n\\end{eqnarray}\n\nIn the present work we consider that the atoms are confined in an\nisotropic HO well where the normalized to $1$ ground state single\nparticle wave function $\\psi_0(r)$ has the form of a Gaussian\ngiven by the formula\n\\[\n \\psi_0(r)=\\left( \\frac{1}{\\pi b^2} \\right)^{3\/4}\n \\textrm{exp}\\left[-\\frac{r^2}{2 b^2}\\right], \\quad\n \\mbox{\\textrm{where the width}}\\,\\, b=\\left( \\frac{\\hbar}{m\\omega}\n \\right)^{1\/2},\n\\]\nwhile the density distribution has the form $\\rho_0({\\bf r})=|\n\\psi_0({\\bf r}) |^2$. The correlation function $f(r_{12})$ is\ntaken to be of the form\n\\begin{equation}\nf(r)=1-\\exp\\left[-\\frac{y r^{2}}{b^2}\\right], \\label{case-1}\n\\end{equation}\nwhere $r=|{\\bf r}_1-{\\bf r}_2|$. The correlation function $f(r)$\ngoes to $1$ for large values of $r$ and goes to $0$ for $r\n\\rightarrow 0$. It is obvious that the effect of the correlations\nintroduced by the function $f(r)$, becomes large when the\ncorrelation parameter $y$ becomes small and vice versa.\n\nThe above defined correlation function was used in\n\\cite{Moustakidis02,MoustaChatz05} to find analytical expressions\nof the one-body density matrices in position and momentum spaces\nand static structure factor, while the NO's and NON's are\ncalculated numerically employing Eq. (\\ref{NO-3}).\n\n\nThe analytical expression of the two-body density matrix obtained\nfrom Eq. (\\ref{TBDM-1}) has the form\n\n\\begin{eqnarray}\n \\rho(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)= &\n \\displaystyle{\\frac{N_0}{\\pi^3 b^6}} \\, \\Big(\n O_1(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)-\n O_{21}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)- \\nonumber \\\\\n & -O_{22}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)+\n O_{23}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)\n \\Big),\n\\end{eqnarray}\nwhere $N_0$ is the normalization factor of the form\n\\cite{MoustaChatz05}\n\\begin{equation}\nN_0=\\left[1-\\frac{2}{(1+2y)^{3\/2}}+\\frac{1}{(1+4y)^{3\/2}}\\right]^{-1}\n\\label{norm-1b}\n\\end{equation}\nand\n\\[\n \\begin{array}{lcl}\n O_1(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)&=&\n \\textrm{exp}\\left[\\displaystyle{-\\frac{r_{1b}^2+r_{1b}^{'2}+r_{2b}^2+r_{2b}^{'2}}{2}}\\right], \\\\\n O_{21}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)&=&\n \\textrm{exp}\\left[\\displaystyle{-\\frac{r_{1b}^2+r_{2b}^2}{2}}\\right]\\,\n \\textrm{exp}\\left[\\displaystyle{-\\frac{(1+2y)(r^{'2}_{1b}+r^{'2}_{2b})}{2}}\\right]\\,\n \\textrm{exp}\\left[2y \\textbf{r}'_{1b} \\textbf{r}'_{2b}\\right], \\\\\n O_{22}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)&=&\n \\textrm{exp}\\left[\\displaystyle{-\\frac{r^{'2}_{1b}+r^{'2}_{2b}}{2}}\\right]\\,\n \\textrm{exp}\\left[\\displaystyle{-\\frac{(1+2y)(r_{1b}^2+r_{2b}^2)}{2}}\\right]\\,\n \\textrm{exp}\\left[2y \\textbf{r}_{1b} \\textbf{r}_{2b}\\right], \\\\\n O_{23}(\\textbf{r}_1,\\textbf{r}_2,\\textbf{r}'_1,\\textbf{r}'_2)&=&\n \\textrm{exp}\\left[\\displaystyle{-\\frac{(1+2y)(r_{1b}^2+r_{1b}^{'2}+r_{2b}^2+r_{2b}^{'2})}{2}}\\right]\\,\n \\textrm{exp}\\left[2y(\\textbf{r}_{1b}\\textbf{r}_{2b}+\\textbf{r}'_{1b}\\textbf{r}'_{2b})\\right].\n\\end{array}\n\\]\nwhere $r_b=r\/b$.\n\nThe two-body density distribution in accordance with Eq.\n(\\ref{TBDD-1}) is given by\n\n\\begin{equation}\n \\rho(\\textbf{r}_1,\\textbf{r}_2)=\\frac{N_0}{\\pi^3 b^6}\n \\,\\textrm{exp}[-r_{1b}^2]\\,\\textrm{exp}[-r_{2b}^2]\\,\n \\Big(1-\\textrm{exp}[-y({\\bf r}_{1b}-{\\bf r}_{2b})^2]\\Big)^2.\n\\end{equation}\n\nThe analytical expressions of the one-body density matrix obtained\nfrom Eq. (\\ref{OBDM-1}) has the form \\cite{Moustakidis02}\n\\begin{equation}\n\\rho({\\bf r},{\\bf r}')= \\frac{N_0}{\\pi^{3\/2}b^{3}}\\,\\left[O_1({\\bf\nr},{\\bf r}')-O_{21}({\\bf r},{\\bf r}')- O_{22}({\\bf r},{\\bf\nr}')+O_{23}({\\bf r},{\\bf r}')\\right], \\label{cluster-11}\n\\end{equation}\n\n\\noindent where the one- and the two-body terms of the expansion\nin the low order approximation have the forms\n\\begin{eqnarray}\n O_1({\\bf r},{\\bf r}')&=&\n\\exp\\left[-\\frac{r_b^2+{r_b'}^2}{2}\\right], \\\\\n& & \\nonumber\\\\\nO_{21}({\\bf r},{\\bf r}')&=& \\frac{1}{(1+y)^{3\/2}}\n\\exp\\left[-\\frac{1+3y}{1+y}\\frac{r_b^2}{2}-\\frac{{r_b'}^2}{2}\\right], \\\\\n& & \\nonumber\\\\\nO_{22}({\\bf r},{\\bf r}')&=&O_{21}({\\bf r}',{\\bf r}), \\\\\n& & \\nonumber\\\\\nO_{23}({\\bf r},{\\bf r}')&=& \\frac{1}{(1+2y)^{3\/2}}\n\\exp\\left[-(1+2y)\\frac{r_b^2+{r_b'}^2}{2}\\right] \\nonumber\\\\\n&& \\times \\exp\\left[\\frac{y^2}{1+2y}({\\bf r}_b+{\\bf\nr}_b')^2\\right], \\label{case1-o1r}\n\\end{eqnarray}\n\n\nThe analytical expression of the density distribution can be found\nfrom Eq. (\\ref{cluster-11}), putting ${\\bf r'}={\\bf r}$\n\\begin{eqnarray}\n\\rho(r)&=& \\frac{N_0}{\\pi^{3\/2} b^3} \\left(\n\\exp\\left[-r_b^2\\right] - \\frac{2}{(1+y)^{3\/2}}\n\\exp\\left[-\\frac{1+2y}{1+y}r_b^2\\right] \\right.\n\\nonumber\\\\\n&& \\left. + \\frac{1}{(1+2y)^{3\/2}}\n\\exp\\left[-\\frac{1+4y}{1+2y}r_b^2\\right] \\right).\n\\label{cluster-nr}\n\\end{eqnarray}\n\nThe two-body momentum distribution is calculated form the integral\nof Eq. (\\ref{TBMD-1}) and has the form\n\n\\begin{eqnarray}\n n(\\textbf{k}_1,\\textbf{k}_2)&=&\\frac{b^6}{\\pi^3} N_0 \\,\n \\textrm{exp}[-k_{1b}^2]\\,\\textrm{exp}[-k_{2b}^2] \\nonumber \\\\\n & & \\times \\left( 1-\\frac{1}{(1+4y)^{3\/2}}\\,\n \\textrm{exp}\\left[-\\frac{y}{1+4y}(\\textbf{k}_{1b}-\\textbf{k}_{2b})^2\\right]\n \\right)^2,\n\\end{eqnarray}\n\n\\noindent while the momentum distribution can be found\nanalytically using Eq. (\\ref{mom}) and has the form\n\\begin{eqnarray}\nn(k)&=& \\frac{N_0 b^3}{\\pi^{3\/2}} \\, \\left(\n\\exp\\left[-k_b^2\\right] - \\frac{2}{(1+3y)^{3\/2}}\n\\exp\\left[-\\frac{1+2y}{1+3y}k_b^2\\right]\n\\nonumber \\right.\\\\\n&& \\left. + \\frac{1}{(1+2y)^{3\/2}(1+4y)^{3\/2}}\n\\exp\\left[-\\frac{1}{1+2y}k_b^2\\right] \\right),\n\\label{cluster-nk}\n\\end{eqnarray}\nwhere $k_b =kb$.\n\nThe above analytical expressions of $\\rho(r)$ and $n(k)$ have been\nused to find the analytical expressions of the mean square radius\nand kinetic energy of the trapped gas. The expressions we found,\nfor $\\langle r^2 \\rangle$ and $\\langle T \\rangle$, are\n\\begin{equation}\n\\langle r^2 \\rangle=N_0 b^2 \\left[ \\frac{3}{2} -\n3\\frac{1+y}{(1+2y)^{5\/2}} +\\frac{3}{2}\\frac{1+2y}{(1+4y)^{5\/2}}\n\\right] \\label{eq:rad-1}\n\\end{equation}\nand\n\\begin{equation}\n\\langle T \\rangle= N_0 \\hbar \\omega \\left[ \\frac{3}{4} -\n\\frac{3}{2}\\frac{1+3y}{(1+2y)^{5\/2}} +\n\\frac{3}{4}\\frac{1+2y}{(1+4y)^{3\/2}}\\right]. \\label{eq:kinetic-1}\n\\end{equation}\n\n\nThese expressions, which for a given HO trap are functions of the\ncorrelation parameter $y$, could be used to find the value of $y$\nfrom Eq. (\\ref{eq:rad-1}), if the rms radius of the trapped atoms\nis known and then to define $\\langle T \\rangle$ from Eq.\n(\\ref{eq:kinetic-1}) and vice versa. For very large values of $y$\nEqs. (\\ref{eq:rad-1}) and (\\ref{eq:kinetic-1}) give the HO\nexpressions of $\\langle r^2 \\rangle$ and $\\langle T \\rangle$, i.e.\n$\\langle r^2 \\rangle = \\frac{3}{2} b^2$ and $\\langle T \\rangle =\n\\frac{3}{4}\\hbar\\omega$, respectively.\n\n\n\nThe calculation of the density distribution of a trapped Bose gas,\nconfined in an isotropic HO potential with length $b=10^4$ \\AA,\nhas been carried out on the basis of Eq. (\\ref{cluster-nr})\n\\cite{Moustakidis02}. The dependence of the density distribution\non the parameter $y$, including also the uncorrelated case\n($y=\\infty$), has been plotted in Fig. \\ref{fig:fig1}(a). It is\nseen that, the large values of $y$ ($y>10$) correspond to the\nGaussian distribution (HO case), while when $y$ becomes small\nenough ($y<1$) the density distribution spreads out as in\nGross-Pitaevskii's theory \\cite{Moustakidis02}. For $y>10$ the\neffect of correlations is small, while for very large correlations\n($y \\lesssim 0.1$) the density distribution is modified entirely\ncompared to the Gaussian form originating from the HO trap.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=5.0cm,width=4.0cm]{fig1a.eps}\n\\hspace{0.5cm}\n\\includegraphics[height=5.0cm,width=4.cm]{fig1b.eps} \\caption{(a) The\ndensity distribution $\\rho(r)$ versus $r_b=r\/b$ ($b=10^4$ \\AA)\n for various\nvalues of the parameter $y$. The normalization is $\\int \\rho({\\bf\nr}) d {\\bf r}=1$. (b) The rms radius, $\\sqrt{\\langle r_b ^{2}\n\\rangle}$, versus the parameter $y$.}\\label{fig:fig1}\n\\end{figure}\n\n\n\nThe rms radius of the Bose gas has also been calculated\nanalytically from Eq. (\\ref{eq:rad-1}) for various values of the\nparameter $y$. From Fig. \\ref{fig:fig1}(b), it is seen that\n$\\sqrt{\\langle r_{b}^2\\rangle}$ ($r_b=r\/b$) is a decreasing\nfunction of the parameter $y$ and for $y > 10$ approaches the rms\nradius of the uncorrelated case.\n\n\\subsection{Natural Orbitals and Natural Occupations\nNumbers}\\label{sub:3-1}\n\nThe NO's and the NON's were calculated \\cite{Moustakidis02}, by\ndiagonalizing the one-body density matrix through Eq.\n(\\ref{NO-3}). The NON $n_{1s}$, gives directly the condensation\nfraction $n_0$ as a result of the repulsive interaction between\nthe atoms of the Bose gas at zero temperature. The NON's for the\n$1s$, $1p$, $1d$ and $1f$ states are given in Table\n\\ref{tbl:table1}. It seems that, for strong correlations, a\nfraction of atoms spread out into many states. The condensation\nfraction $n_{1s}$, versus the parameter $\\frac{1}{y}$ is plotted\nin Fig. \\ref{fig:fig2}. From that figure and from Table\n\\ref{tbl:table1} it is seen that the effect of the correlations on\n$n_{1s}$ is small and all the atoms occupy the 1s ground state,\nwhen $y>10$. The effect of the correlations is prominent when\n$y<10$, while the decrease of the parameter $y$ (strong\ncorrelations) induces a significant depletion of the condensated\natoms spreading them into many states.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=5.0cm,width=4.0cm]{fig2.eps}\n\\caption{The condensate fraction $n_{1s}$, at zero temperature,\nfor interacting atoms versus $1\/y$.}\\label{fig:fig2}\n\\end{figure}\n\n\n\\begin{table}[pt]\n\\centering\n\\begin{tabular}{@{}c c c c c c@{}}\n\\hline $y$ & $n_{1s}$ & $n_{1p}$ & $n_{1d}$ & $n_{1f}$ & Sum \\\\\n\\hline\n100.00 & 0.99988 & - & - & - & 0.99988 \\\\\n\\hphantom{0}10.00 & 0.99634 & 0.00063 & 0.00042 &0.00042 & 0.99781 \\\\\n\\hphantom{00}5.00 & 0.99055 & 0.00273 & 0.00108 &0.00108 & 0.99544 \\\\\n\\hphantom{00}2.50 & 0.97771 & 0.00960 & 0.00186 &0.00186 & 0.99103 \\\\\n\\hphantom{00}1.00 & 0.94422 & 0.03462 & 0.00172 &0.00172 & 0.98228 \\\\\n\\hphantom{00}0.50 & 0.90815 & 0.06830 & 0.00082 &0.00082 & 0.97809 \\\\\n\\hphantom{00}0.10 & 0.83097 & 0.15185 & 0.00001 & 0.00001 & 0.98284 \\\\\n\\hphantom{00}0.01 & 0.79273 & 0.19414 & - & - & 0.98687\\\\\n\\hline\n\\end{tabular}\n\\caption{The natural occupation numbers for various values of the\ncorrelation parameter $y$ \\cite{Moustakidis02}.}\\label{tbl:table1}\n\\end{table}\n\n\n\nThe NO's of the states $1s$, $1p$ and $1d$ for $y=1$ are shown in\nFig. \\ref{fig:fig3}. It is seen that the interatomic correlations\nin the $1s$-state spread out the ground state wave function and\nconsequently the condensation appears in the outer region of the\ntrap. From the same figure it is obvious that the NO's of the $1p$\nand $1d$ states are much more localized in coordinate space than\nthe equivalent HO orbitals.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=5.0cm,width=3.5cm]{fig3a.eps}\n\\hspace{0.3cm}\n\\includegraphics[height=5.0cm,width=3.5cm]{fig3b.eps}\n\\hspace{0.3cm}\n\\includegraphics[height=5.0cm,width=3.5cm]{fig3c.eps}\n \\caption{The NO's (dot lines) versus $r_b$ ($r_b=r\/b$) of the\nstates (a) 1s, (b) 1p, and (c) 1d obtained by diagonalization of\nthe one-body density matrix (for $y$=1). The solid lines\ncorrespond to the HO wave-function with the trap length $b=10^4$\n\\AA.}\\label{fig:fig3}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=5.0cm,width=3.5cm]{fig4a.eps}\n\\hspace{0.3cm}\n\\includegraphics[height=5.0cm,width=3.5cm]{fig4b.eps}\n\\hspace{0.3cm}\n\\includegraphics[height=5.0cm,width=3.5cm]{fig4c.eps}\n \\caption{(a) The normalized to $1$ momentum distribution versus\n$k_b$ ($k_b=kb$) for various values of the correlation parameter\n$y$. (b) The momentum distribution and the contribution to it from\nthe NO's of the $1s$ NO state and of the rest of the NO states\n(for $y=0.1$). (c) The mean kinetic energy per atom, $\\langle T_b\n\\rangle$ ($T_b=T\/(\\hbar\\omega)$), versus $y$. The solid curve\ncorresponds to the total values of $\\langle T_b \\rangle$, while\ndashed and dotted lines are the contributions to the total\n$\\langle T_b \\rangle$ of the NO's of the $1s$ NO state and of the\nrest of the NO states, respectively.}\\label{fig:fig4}\n\\end{figure}\n\nThe momentum distribution can be calculated analytically from Eq.\n(\\ref{cluster-nk}) or by Fourier transform of the NO's. The\nmomentum distribution calculated analytically for various values\nof the parameter $y$ has been plotted in Fig. \\ref{fig:fig4}(a).\nIt is seen that the large values of $y$ ($y>10$) correspond to the\nGaussian distribution, while when $y$ becomes small enough ($y<1$)\nthe momentum distribution has a sharp maximum for $k=0$. The\nmomentum distribution of the $1s$ NO state as well as of the rest\nof the NO states for $y=0.1$ are shown and compared with the total\nmomentum distribution in Fig. \\ref{fig:fig4}(b). It is obvious\nthat although the $1s$ NO state gives the main contribution to the\nmomentum distribution, the additional NO states contribute to the\nmomentum distribution mainly in the large values of the momentum\n$k$.\n\nThe dependence of the mean kinetic energy $\\langle T \\rangle$ on\nthe parameter $y$ calculated analytically, using Eq.\n(\\ref{eq:kinetic-1}), is presented in Fig. \\ref{fig:fig4}(c). It\nis seen that $\\langle T \\rangle$ has a maximum for $y\\simeq2.5$.\nIt is interesting to note that for the same value of the parameter\n$y$ the NON's of the states $1d$ and $1f$ have the same value as\ncan be seen from Table \\ref{tbl:table1}. The contribution of the\n$1s$ NO state and of the rest of the NO states to $\\langle T\n\\rangle$ are shown in the same figure. It is seen that, for large\nvalues of the parameter $y$ (weak correlations) the main\ncontribution to $\\langle T \\rangle$ comes from the $1s$ NO state,\nwhile for strong correlations there is a significant contribution\ncoming from the NO's of the additional states.\n\n\n\nA few comments are appropriate. In this section we study the\nbehavior of various condensate quantities treated in the Jastrow\nmanner, which introduces one parameter. The determination of that\nparameter could be made by fit of the theoretically calculated\nquantities (density distribution, momentum distribution,\n$\\sqrt{\\langle r^2\\rangle}$, and $\\langle T \\rangle$) to the\nexperimental ones as we mentioned in the end of subsection\n\\ref{sub:sub3-1}, provided that there are experimental data for\nthe corresponding quantities. It could be determined also by using\nthe density distribution or the two-body density matrix as a trial\none and applying the variation principle to the ground state\nenergy of the system. The present approach is quite frequent in\nthe study of the quantum many body problem when the solution of\nthe Schr\\\"odinger equation is very difficult. It should be noted\nalso that in the present work there is not a direct dependence\nbetween the condensation and the number of the atoms. The\ninter-particle correlations are incorporated in the mean field\nonly by the correlation function which, in some way, depends on\nthe effective size of the atoms. That dependence can be found from\nthe information entropy $S$ using the linear dependence of $S$ on\n$\\ln (Na_b)$ and the linear dependence of $S$ on\n$\\ln(\\frac{1}{y})$ (see Sec. \\ref{sub:sub5-1}).\n\n\n\\subsection{Static Structure Factor}\n\nIn order to calculate the static structure factor in the framework\nof the atomic calculations we choose two trial forms for $g(r)$\n\\cite{Moustakidis04}. The first one is a gaussian type which has\nbeen extensively and successfully used for the study of similar\nproblems in atomic physics (Bose gas, liquid helium) as well in\nnuclear physics. The relevant $g(r)$ and the entailed $S(k)$ (Case\n1) are\n\\begin{eqnarray}\ng(r)&=& 1- \\exp[-\\beta r^2], \\nonumber \\\\\nS(k)&=&1+N(C_1-1)\\exp\\left[-\\frac{k_b^2}{2}\\right] \\nonumber \\\\\n& &-\\frac{N\nC_1}{(1+2y^2)^{3\/2}}\\exp\\left[-\\frac{k_b^2}{2}(1+2y)\\right],\n\\label{fin-cs-1}\n\\end{eqnarray}\nwhere $k_b=kb$, $y=\\beta b^2 $, $\\beta$ is the correlation\nparameter and $C_1$ is the normalization factor.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=5.0cm,width=3.5cm]{fig5a.eps}\n\\hspace{0.3cm}\n\\includegraphics[height=5.0cm,width=3.5cm]{fig5b.eps}\n \\caption{The static structure factor $S(k)$ of the trapped Bose gas\n in various cases versus the momentum $k$,\n(a) in Case 1 for various values of the correlation parameter\n$\\beta$ as well as for the uncorrelated case (harmonic\noscillator), (b) in Case 2 for the least squares best fit value of\nthe parameter $a$. The experimental points are from reference\n\\cite{Steinhauer}. For the various cases see\ntext.}\\label{fig:fig5}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=5.0cm,width=3.5cm]{fig6.eps}\n \\caption{The radial distribution function $g(r)$ for Case 1 and\n2 (corresponding to inhomogeneous Bose gas) with the best fit\nvalues of the correlation parameters.}\\label{fig:fig6}\n\\end{figure}\n\nThe second trial function $g(r)$ and the relevant $S(k)$ (Case 2)\nare of the form\n\\begin{eqnarray}\ng(r)&=& 1- \\frac{\\sin^4ar}{(ar)^4}, \\nonumber \\\\\nS(k)&=&1+N(C_2-1)\\exp\\left[-\\frac{k_b^2}{2}\\right] \\nonumber\\\\\n& &- \\frac{N C_2}{2^{15\/2} ab \\pi k_b} \\sum_{i=1}^{5}\n\\alpha_i\\left[\\beta_i\\exp\\left[-\\frac{\\beta_i^2}{4}\\right]+\n\\sqrt{\\pi}\\left(1+\\frac{\\beta_i^2}{2}\\right) {\\rm\nerf}\\left(\\frac{\\beta_i}{2}\\right) \\right], \\label{fin-cs-2}\n\\end{eqnarray}\nwhere $a$ is the correlation parameter, $\\alpha_i$ are known\ncoefficients, $\\beta_i=\\beta_i(a,b,k_b)$, ${\\rm\nerf}(z)=\\displaystyle{\\frac{2}{\\sqrt{\\pi}}} \\int_{0}^{z} e^{-t^2}\ndt$ and $C_2$ is the normalization factor \\cite{Moustakidis04}.\n\n\nThe behavior of S(k), in Case 1, for various values of the\ncorrelation parameter $\\beta$ is shown in Fig. \\ref{fig:fig5}(a).\nIt is obvious that the effect of correlations, induced by the\nfunction $g(r)$, becomes large when the parameter $\\beta$ becomes\nsmall and vice versa. The case where $\\beta\\rightarrow \\infty$,\ncorresponds to the uncorrelated case (HO). For the values of $k$\nemployed in the experiment of Ref. \\cite{Steinhauer} (hereafter\nEXP) the prediction of the HO model is always close to 1 for\n$S(k)$. When the correlation parameter $\\beta$ decreases\nconsiderably (strong correlations) the theoretical prediction of\n$S(k)$ is in good agreement with the experimental data. The value\n$\\beta$=5.3 ${\\rm \\mu m}^{-2}$ gives the best least squares fit in\nthat case. In general the gaussian form of $g(r)$, in spite of its\nsimplicity, reproduces fairly well the experimental data of EXP,\nboth in low and high values of the momentum $k$. Within our\ntheoretical model, the gaussian type of $g(r)$ is flexible enough\nto obtain values for $S(k)$ in agreement with the experimental\ndata.\n\nFig. \\ref{fig:fig5}(b) displays the results in Case 2, which are\ncompared with those of the data of EXP. The model reproduces well\nthe experimental data in the range 1.5-3 ${\\rm \\mu m}^{-1}$ (with\nbest least squares fit value $a=1.34 \\ \\mu m^{-1}$), but fails in\nthe range $k>3$ ${\\rm \\mu m}^{-1}$. The main drawback of this\nmodel is the predicted negative values of $S(k)$ in the range\nclose to $k=0$ when the correlation parameter $a$ decreases\nconsiderably (strong correlation case).\n\nThe correlation function $g(r)$ corresponding to Cases 1 and 2 for\nthe correlation parameters $\\beta$=5.3 ${\\rm \\mu m}^{-2}$ and\n$a=1.34 \\ \\mu m^{-1} $ respectively is sketched in Fig.\n\\ref{fig:fig6}. Those values of the parameters $\\beta$ and $a$\ngive the best $x^2$ value in the fit of the theoretical\nexpressions of $S(k)$ to the data of EXP. The most striking\nfeature in Case 2 is the existence of strong correlations,\nintroduced by $g(r)$, in order to reproduce the experimental data\nof $S(k)$. It is worthwhile to point out that $g(r)$, in Case 2,\nexhibits fluctuations in the range $r>2$ ${\\rm \\mu m}$ but this is\nnot visible in Fig. \\ref{fig:fig6}.\n\nThe possibility of a linear dependence of $S(k)$ on $k$ for small\nvalues of $k$, as predicted from other works \\cite{Zambelli}, is\nprohibitive, on the basis of Eq. (\\ref{eq:eq18}) at least in the\ncase where the trap is an harmonic oscillator one. That can be\nseen considering the ground state wave function to be the\nharmonic oscillator one and transforming ${\\bf r}_1$ and ${\\bf\nr}_2$ in Eq. (\\ref{eq:eq18}) into the coordinates of the relative\nmotion (${\\bf r}={\\bf r}_1-{\\bf r}_2$) and the center of mass\nmotion (${\\bf R}=({\\bf r}_1+{\\bf r}_2)\/2 $). After some algebra\n$S({\\bf k})$ takes the form\n\\begin{equation}\nS({\\bf k}) \\sim \\int \\textrm{exp} \\left[ i{\\bf k}{\\bf r} \\right]\n \\textrm{exp}\\left[ -r_b^2\\right] [Cg(r)-1] d {\\bf r}. \\label{sk-rel}\n\\end{equation}\n\nFor finite systems, as a trapped Bose gas, we can expand the\nexponential $\\textrm{exp} \\left[ i{\\bf k}{\\bf r} \\right]$, since\n${\\bf r}$ is bounded. Thus:\n\\begin{equation}\n\\textrm{exp} \\left[ i{\\bf k}{\\bf r} \\right]=1+i{\\bf k}{\\bf r}\n+\\frac{(i{\\bf k}{\\bf r})^2}{2 !}+ \\frac{(i{\\bf k}{\\bf r})^3}{3\n!}+\\cdots .\\label{expand}\n\\end{equation}\n\nSubstituting Eq. (\\ref{expand}) into Eq. (\\ref{sk-rel}) and\ntaking into account that the terms with odd powers of $k$ do not\ncontribute to the integral, $S(k)$ takes the form\n\\begin{equation}\nS(k) \\sim a_1k^2+a_2k^4+\\cdots .\\label{sk-rel-2}\n\\end{equation}\n\nHence, for small values of $k$, $S(k)$ depends linearly on $k^2$.\nThe gaussian factor $\\textrm{exp}\\left[ -r_b^2\\right]$,\noriginating from the harmonic oscillator wave function of the\ntrapped Bose gas, ensures the convergence of the integrals $a_i$\ncorresponding to the even powers of the expansion\n\\cite{Moustakidis04}.\n\n\n\n\\section{Quantum-Information properties of trapped\nBose gas}\\label{sec:sec4}\n\n\\subsection{Shannon Information Entropy}\n\nThe Boltzmann-Gibbs-Shannon information entropy\n\\cite{Shannon48,Halliwell93} of a finite probability distribution\n($p_1$,$p_2$,$\\cdot$,$p_k$) is defined as the quantity\n\\begin{equation}\nS=-\\sum_{i=1}^{k}p_i \\ln p_i, \\label{BGS}\n\\end{equation}\nwith the constraint: $\\displaystyle{\\sum_{i=1}^{k}p_i=1}$. $S$ is\nmeasured in bits if the base of the logarithm is 2 and nats\n(natural units of information) if the logarithm is natural.\n\n$S$ appears in different areas: information theory, ergodic theory\nand statistical mechanics. It is closely related to the entropy\nand disorder in thermodynamics. The maximum value of $S$ is\nobtained if $p_1=p_2=\\cdots=p_k=\\frac{1}{k}$ i.e. $S_{max}=\\ln k$.\nThe minimum value of $S$ is found when one of the $p_i$'s is equal\n$1$ and all the others are equal to $0$. Then $S_{min}=0$. The\nabove definition holds for discrete probability distributions\n\\cite{Chatzisavvas05}. In quantum mechanics we are often\ninterested in a continuous probability distribution $p(x)$. In\nthis case the obvious generalization of Eq. (\\ref{BGS}) is the\ninformation entropy\n\\begin{equation}\nS=-\\int p(x) \\ln p(x) dx, \\label{S-con}\n\\end{equation}\nwhere $\\int p(x) dx=1$. Now $p(x)$ is a quantum mechanical\nprobability distribution and $S$ may be called the quantum entropy\n\\cite{Ohya93}. $S$ indicates the amount of disorder or randomness\n(uncertainty) in a physical system. Shannon considered this\nuncertainty attached to the system as the amount of information\ncarried by the system. If a physical system has a large\nuncertainty and one obtains information on the system by some\nprocedure, as a measurement, then the information is more valuable\nthan that received from a system having less uncertainty. Thus,\nbefore a measurement, the uncertainty of the position of a\nparticle is small for a localized probability distribution, while\nfor a diffuse distribution is large. The same holds for the\nmissing information due to a limited knowledge of the system via a\nprobability distribution. After the measurement the gain in\ninformation for a localized distribution is smaller than the\ncorresponding gain for a diffuse distribution.\n\nAn important step is the discovery of an entropic uncertainty\nrelation (EUR) \\cite{Bialynicki75}, which for a three-dimensional\nsystem has the form\n\\begin{equation}\\label{eq:equ2}\n S=S_r+S_k\\geq 3\\,(1+\\ln{\\pi})\\simeq 6.434,\n\\end{equation}\nwhere $S_r$ is the information entropy in position-space of the\ndensity distribution $\\rho(\\textbf{r})$ of a quantum system\n\\begin{equation}\\label{eq:equ3}\n S_r=-\\int\n \\rho(\\textbf{r})\\,\\ln{\\rho(\\textbf{r})}\\,d\\textbf{r},\n\\end{equation}\nand $S_k$ is the information entropy in momentum-space of the\ncorresponding momentum distribution $n(\\textbf{k})$\n\\begin{equation}\\label{eq:equ4}\n S_k=-\\int n(\\textbf{k})\\,\\ln{n(\\textbf{k})}\\,d\\textbf{k}.\n\\end{equation}\n\nThe total information entropy is given by\n\\begin{equation}\\label{eq:stot}\n S=S_r+S_k.\n\\end{equation}\n\nThe density distributions $\\rho(\\textbf{r})$ and $n(\\textbf{k})$\nare normalized to one. Inequality (\\ref{eq:equ2}), for the\ninformation entropy sum in conjugate spaces, is a joint measure of\nuncertainty of a quantum mechanical distribution, since a highly\nlocalized $\\rho(\\textbf{r})$ is associated with a diffuse\n$n(\\textbf{k})$, leading to low $S_r$ and high $S_k$ and\nvice-versa. Expression (\\ref{eq:equ2}) is an\ninformation-theoretical relation stronger than Heisenberg's.\n\nIn previous work we proposed a universal property of $S$ for the\ndensity distributions of nuclei, electrons in atoms and valence\nelectrons in atomic clusters \\cite{Massen98}. This property has\nthe form\n\\begin{equation}\\label{eq:equ5}\n S=a+b \\ln{N},\n\\end{equation}\nwhere $N$ is the number of particles of the system and the\nparameters $a, b$ depend on the system under consideration. It is\nnoted that recently we have obtained the same form for systems of\ncorrelated bosons in a trap \\cite{Massen02}. This concept was also\nfound to be useful in a different context. Using the formalism in\nphase-space of Ghosh, Berkowitz and Parr \\cite{Ghosh84}, we found\nthat the larger the information entropy, the better the quality of\nthe nuclear density distribution \\cite{Lalazissis98}. Recently the\nShannon information entropy has been applied successfully to the\nstudy of the free expansion of impenetrable bosons on the\none-dimensional optical lattices \\cite{Rigol05}.\n\n\\subsection{Onicescu's Information Entropy}\n\nOnicescu tried to define a finer measure of dispersion\ndistributions than that of Shannon's information entropy\n\\cite{Onicescu96}. Thus, he introduced the concept of information\nenergy $E$. For a discrete probability distribution\n$(p_1,p_2,\\ldots,p_k)$ the information energy $E$ is defined by\n\\begin{equation}\\label{eq:equ9}\n E=\\sum_i^k p_i^2,\n\\end{equation}\nwhich is extended for a continuous density distribution $\\rho(x)$\nas\n\\begin{equation}\\label{eq:equ10}\n E=\\int \\rho^2(x)\\,dx.\n\\end{equation}\nThe meaning of (\\ref{eq:equ10}) can be seen by the following\nsimple argument: For a Gaussian distribution of mean value $\\mu$,\nstandard deviation $\\sigma$ and normalized density\n\\begin{equation}\\label{eq:equ11}\n \\rho(x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}\\, \\textrm{exp}\n \\left[-\\frac{(x-\\mu)^2}{2\\sigma^2} \\right],\n\\end{equation}\nrelation (\\ref{eq:equ10}) gives\n\\begin{equation}\\label{eq:equ12}\n E=\\frac{1}{2\\pi\\sigma^2} \\int_{-\\infty}^{\\infty}\n \\textrm{exp}\n \\left[-\\frac{(x-\\mu)^2}{\\sigma^2}\n \\right]\\,dx=\\frac{1}{2\\sigma\\sqrt{\\pi}}.\n\\end{equation}\n$E$ is maximum if one of the $p_i$'s equals 1 and all the others\nare equal to zero i.e. $E_{max}=1$, while $E$ is minimum when\n$p_1=p_2=\\ldots=p_k=\\frac{1}{k}$, hence $E_{min}=\\frac{1}{k}$\n(total disorder). $E$ has been called information energy, although\nit does not have the dimension of energy \\cite{Lepadatu03}. This\nis due to the fact that $E$ becomes minimum for equal\nprobabilities (total disorder), by analogy with thermodynamics.\n\nIt is seen from (\\ref{eq:equ12}) that the greater the information\nenergy, the more concentrated is the probability distribution,\nwhile the information content decreases. $E$ and information\ncontent are reciprocal, hence one can define the quantity\n\\cite{MoustaChatz05}\n\\begin{equation}\\label{eq:equ13}\n O=\\frac{1}{E},\n\\end{equation}\nas a measure of the information content of a quantum system\ncorresponding to Onicescu's information energy.\n\nRelation (\\ref{eq:equ10}) is extended for a 3-dimensional\nspherically symmetric density and momentum distribution as follow\n$\\rho(\\textbf{r})$\n\\begin{eqnarray}\\label{eq:equ14}\n E_r=\\int \\rho^2(\\textbf{r})\\,d\\textbf{r} \\nonumber \\\\\n E_k=\\int n^2(\\textbf{k})\\,d\\textbf{k}.\n\\end{eqnarray}\n\n\n$E_r$ has dimension of inverse volume, while $E_k$ of volume. Thus\nthe product $E_r E_k$ is dimensionless and can serve as a measure\nof concentration (or information content) of a quantum system. It\nis also seen from (\\ref{eq:equ12}),(\\ref{eq:equ13}) that $E$\nincreases as $\\sigma$ decreases (or concentration increases) and\nthe information (or uncertainty) decreases. Thus $O$ and $E$ are\nreciprocal. In order to be able to compare $O$ with Shannon's\nentropy $S$, we redifine $O$ as\n\\begin{equation}\\label{eq:equ15}\n O=\\frac{1}{E_r E_k},\n\\end{equation}\nas a measure of the information content of a quantum system in\nboth position and momentum spaces, inspired by Onicescu's\ndefinition.\n\n\\subsection{Landsberg's Order Parameter}\n\nLandsberg \\cite{Landsberg84} defined the order parameter $\\Omega$\n(or disorder $\\Delta$) as\n\\begin{equation}\n \\Omega = 1-\\Delta = 1- \\frac{S}{S({\\rm max})},\n \\label{omega}\n \\end{equation}\n %\nwhere $S$ is the information entropy (actual) of the system and\n$S({\\rm max})$ the maximum entropy accessible to the system. Thus\nthe concepts of entropy and disorder are decoupled and it is\npossible for the entropy and order to increase simultaneously. It\nis noted that $\\Omega =1$ corresponds to perfect order and\npredictability, while $\\Omega =0$ means complete disorder and\nrandomness.\n\n\\subsection{Two-body information entropies}\n\nThe two-body Shannon information entropy both in position- and\nmomentum-space and in total are defined respectively\n\\cite{MoustaChatz05,Amovilli04,Cover91}\n\\begin{equation}\nS_{2r}=-\\int \\rho({\\bf r}_1,{\\bf r}_2) \\ln \\rho({\\bf r}_1,{\\bf\nr}_2) d {\\bf r}_1 d{\\bf r}_2 \\label{S2r-1}\n\\end{equation}\n\\begin{equation}\nS_{2k}=-\\int n({\\bf k}_1,{\\bf k}_2) \\ln n({\\bf k}_1,{\\bf k}_2)\nd{\\bf k}_1 d{\\bf k}_2, \\label{S2k-1}\n\\end{equation}\n\\begin{equation}\nS_2=S_{2r}+S_{2k}. \\label{S2-1}\n\\end{equation}\n\nThe one-body Onicescu's information entropy is already defined in\n(\\ref{eq:equ14}) and (\\ref{eq:equ15}), where the generalization to\nthe two-body information entropy is straightforward and is given\nby\n\\begin{equation}\nO_2= \\frac{1} {E_{2r} E_{2k}}, \\label{O1-2}\n\\end{equation}\nwhere\n\\begin{eqnarray}\nE_{2r}&=&\\int \\rho^2({\\bf r}_1,{\\bf r}_2) d{\\bf r}_1 d{\\bf r}_2 \\nonumber \\\\\nE_{2k}&=&\\int n^2({\\bf k}_1,{\\bf k}_2) d{\\bf k}_1 d {\\bf k}_2.\n\\label{E2r-2k}\n\\end{eqnarray}\n\n\\subsection{Kullback-Leibler relative entropy and Jensen-Shannon\ndivergence}\n\nA well known measure of distance of two discrete probability\ndistributions $p_i^{(1)}, p_i^{(2)}$ is the Kullback-Leibler\nrelative entropy \\cite{Kullback59}\n\\begin{equation}\n K(p_i^{(1)},p_i^{(2)})=\\sum_i\n p_i^{(1)}\\,\\ln{\\frac{p_i^{(1)}}{p_i^{(2)}}}, \\label{eq:equ6}\n\\end{equation}\nwhich for continuous probability distributions $\\rho^{(1)},\n\\rho^{(2)}$ is defined as\n\\begin{equation}\n K=\\int\n \\rho^{(1)}(x)\\,\\ln{\\frac{\\rho^{(1)}(x)}{\\rho^{(2)}(x)}}\\,dx, \\label{eq:equ7}\n\\end{equation}\nwhich can be easily extended for 3-dimensional systems.\n\nOur aim is to calculate the relative entropy (distance) between\n$p^{(1)}$ (correlated) and $p^{(2)}$ (uncorrelated) densities both\nat the one- and the two-body levels in order to assess the\ninfluence of short range correlations (SRC) through the\ncorrelation parameter $y$, on the distance $K$\n\\cite{MoustaChatz05}. It is noted that this is done for both\nsystems under consideration: nuclei and trapped Bose gases. An\nalternative definition of distance of two probability\ndistributions was introduced by Rao and Lin \\cite{Rao87,Lin91},\ni.e. a symmetrized version of $K$, the Jensen-Shannon divergence\n$J$ \\cite{Majtey05}\n\\begin{equation}\n J(p^{(1)},p^{(2)})=H\\left(\\frac{p^{(1)}+p^{(2)}}{2}\\right)-\\frac{1}{2}H\\left(p^{(1)}\\right)\n -\\frac{1}{2}H\\left(p^{(2)}\\right), \\label{eq:equ8}\n\\end{equation}\nwhere $H(p)=\\displaystyle{-\\sum_i p_i \\ln{p_i}}$ stands for\nShannon's entropy. We expect for strong SRC the amount of\ndistinguishability of the correlated from the uncorrelated\ndistributions is larger than the corresponding one with small SRC.\nWe may also see the effect of SRC on the number of trials $L$\nneeded to distinguish $p^{(1)}$ and $p^{(2)}$ (in the sense\ndescribed in \\cite{Majtey05}).\n\nIn addition to the above considerations, we connect $S_r$ and\n$S_k$ with fundamental quantities i.e. the root mean square radius\nand kinetic energy respectively. We also argue on the effect of\nSRC on EUR and we propose a universal relation for $S$, by\nextending our formalism from the one- and two-body level to the\n$N$-body level, which holds exactly for uncorrelated densities in\ntrapped Bose gas and it is conjectured to hold approximately for\ncorrelated densities in Bose gases (see Sec. \\ref{sub:sec4-7}).\n\nThe Kullback-Leibler relative information entropy $K$ for\ncontinuous distributions $\\rho_i^{(1)}$ and $\\rho_i^{(2)}$ is\ndefined by relation (\\ref{eq:equ7}). It measures the difference of\n$\\rho_i^{(1)}$ form the reference (or apriori) distribution\n$\\rho_i^{(2)}$. It satisfies: $K\\geq 0$ for any distributions\n$\\rho_i^{(1)}$ and $\\rho_i^{(2)}$. It is a measure which\nquantifies the distinguishability (or distance) of $\\rho_i^{(1)}$\nfrom $\\rho_i^{(2)}$, employing a well-known concept in standard\ninformation theory. In other words it describes how close\n$\\rho_i^{(1)}$ is to $\\rho_i^{(2)}$ by carrying out observations\nor coin tossing, namely $L$ trials (in the sense described in\n\\cite{Majtey05}). We expect for strong SRC the amount of\ndistinguishability of the correlated $\\rho_i^{(1)}$ and the\nuncorrelated distributions $\\rho_i^{(2)}$ is larger than the\ncorresponding one with small SRC.\n\nHowever, the distance $K$ does not satisfy the triangle inequality\nand in addition is i) not symmetric ii) unbounded and iii) not\nalways well defined \\cite{Majtey05}. To avoid these difficulties\nRao and Lin \\cite{Rao87,Lin91} introduced a symmetrized version of\n$K$ (recently discussed in \\cite{Majtey05}), the Jensen-Shannon\ndivergence $J$ defined by relation (\\ref{eq:equ8}). $J$ is minimum\nfor $\\rho^{(1)}=\\rho^{(2)}$ and maximum when $\\rho^{(1)}$ and\n$\\rho^{(2)}$ are two distinct distributions, when $J=\\ln{2}$. In\nour case $J$ can be easily generalized for continuous density\ndistributions. For $J$ minimum the two states represented by\n$\\rho^{(1)}$ and $\\rho^{(2)}$ are completely indistinguishable,\nwhile for $J$ maximum they are completely distinguishable. It is\nexpected that for strong SRC the amount of distinguishability can\nbe further examined by using Wooter's criterion \\cite{Majtey05}.\nTwo probability distributions $\\rho^{(1)}$ and $\\rho^{(2)}$ are\ndistinguishable after $L$ trials $(L\\rightarrow \\infty)$ if and\nonly if $\\left( J(\\rho^{(1)},\\rho^{(2)})\n\\right)^{\\frac{1}{2}}>\\frac{1}{\\sqrt{2L}}$.\n\n\n\nThe relative entropy is a measure of distinguishability or\ndistance of two states. It is defined \\cite{MoustaChatz05},\ngeneralizing (\\ref{eq:equ7}), by\n\\begin{equation}\n K=\\int \\psi^2(\\textbf{r})\n \\ln{\\frac{\\psi^2(\\textbf{r})}{\\phi^2(\\textbf{r})}}\\,d\\textbf{r}. \\label{eq:equ16}\n\\end{equation}\nIn our case $\\psi(\\textbf{r})$ is the correlated case and\n$\\phi(\\textbf{r})$ the uncorrelated one. Thus\n\\begin{equation}\n K_{1r}=\\int\n \\rho(\\textbf{r})\\,\\ln{\\frac{\\rho(\\textbf{r})}{\\rho'(\\textbf{r})}} \\label{eq:equ17}\n \\,d\\textbf{r},\n\\end{equation}\nwhere $\\rho(\\textbf{r})$ is the correlated one-body density and\n$\\rho'(\\textbf{r})$ is the uncorrelated one-body density.\n\nA corresponding formula holds in momentum-space\n\\begin{equation}\n K_{1k}=\\int\n n(\\textbf{k})\\,\\ln{\\frac{n(\\textbf{k})}{n'(\\textbf{k})}}\n \\,d\\textbf{k},\n \\label{eq:equ18}\n\\end{equation}\nwhere $n(\\textbf{k})$ is the correlated one-body density and\n$n'(\\textbf{k})$ is the uncorrelated one.\n\nFor the two-body case we have\n\\begin{equation}\n K_{2r}=\\int \\rho(\\textbf{r}_1,\\textbf{r}_2)\\,\n \\ln{\\frac{\\rho(\\textbf{r}_1,\\textbf{r}_2)}{\\rho'(\\textbf{r}_1,\\textbf{r}_2)}}\\,\n d\\textbf{r}_1 d\\textbf{r}_2, \\label{eq:equ19}\n\\end{equation}\nwhere $\\rho(\\textbf{r}_1,\\textbf{r}_2)$ is the correlated two-body\ndensity in position-space and $\\rho'(\\textbf{r}_1,\\textbf{r}_2)$\nis the uncorrelated one.\n\nThe generalization to momentum-space is straightforward\n\\begin{equation}\n K_{2k}=\\int n(\\textbf{k}_1,\\textbf{k}_2)\n \\ln{\\frac{n(\\textbf{k}_1,\\textbf{k}_2)}{n'(\\textbf{k}_1,\\textbf{k}_2)}}\\,\n d\\textbf{k}_1 d\\textbf{k}_2, \\label{eq:equ20}\n\\end{equation}\nwhere $n(\\textbf{k}_1,\\textbf{k}_2)$ is the correlated two-body\ndensity in momentum-space and $n'(\\textbf{k}_1,\\textbf{k}_2)$ is\nthe uncorrelated one.\n\nFor the Jensen-Shannon divergence $J$ we may write formulas for\n$J_1$ (one-body) and $J_2$ (two-body), employing definition\n(\\ref{eq:equ8}) and putting the corresponding correlated\n$\\rho^{(1)}$ and uncorrelated $\\rho^{(2)}$ distributions in\nposition- and momentum-spaces. We calculate $K$ and $J$ in\nposition- and momentum-spaces, for nuclei and bosons.\n\n\\subsection{Numerical Results and Discussion}\n\nFor the sake of symmetry and simplicity we put the width of the HO\npotential $b=1$. Actually for $b=1$ in the case of uncorrelated\ncase it is easy to see that $S_{1r}=S_{1k}$ and also\n$S_{2r}=S_{2k}$ (the same holds for Onicescu entropy), while when\n$b\\neq 1$ there is a shift of the values of $S_{1r}$ and $S_{1k}$\nby an additive factor $\\ln{b^3}$. However, the value of $b$ does\nnot affect directly the total information entropy $S$ (and also\n$O$). $S$ and $O$ are just functions of the correlation parameter\n$y$ \\cite{MoustaChatz05}.\n\n\nIn Fig. \\ref{fig:fig7} we present the Shannon information entropy\n$S_1$ using relation (\\ref{eq:stot}) and $S_2$ using relation\n(\\ref{S2-1}) in trapped Bose gas as functions of the correlation\nparameter $\\ln{(\\frac{1}{y})}$. It is seen that $S_1$ and $S_2$\nincrease almost linearly with the strength of SRC i.e.\n$\\ln{(\\frac{1}{y})}$ in both systems. The relations $S_2=2 S_1$\nand $O_2=O_1^2$ hold exactly for the uncorrelated densities while\nthe above relations are almost exact for the uncorrelated\ndensities. For the sake of comparison we also present the\ndecomposition of $S$ in coordinate and momentum spaces i.e.\n$S_{1r}$, $S_{1k}$, $S_{2r}$, $S_{2k}$ employing (\\ref{eq:equ3}),\n(\\ref{eq:equ4}), (\\ref{S2r-1}), (\\ref{S2k-1}). The most striking\nfeature concluded from the above Figures is the similar behavior\nbetween $S_{1r}$ and $S_{2r}$ and also $S_{1k}$ and $S_{2k}$\nrespectively.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=5.0cm,width=3.5cm]{fig7a.eps}\n \\hspace{0.3cm}\n \\includegraphics[height=5.0cm,width=3.5cm]{fig7b.eps}\n \\caption{ (a) Shannon information entropy (one- and\n two-body) (b) Shannon information entropy (one- and two-body) both in\ncoordinate- and momentum-space, in a trapped Bose\ngas.}\\label{fig:fig7}\n\\end{figure}\n\n\nIn Fig. \\ref{fig:fig8} we plot the Onicescu information entropy\nboth one-body $(O_1)$ and two-body $(O_2)$ (relations\n(\\ref{eq:equ15}), (\\ref{O1-2})). We conclude by noting once again\nthe strong similarities of the behavior between one- and two-body\nOnicescu entropy.\n\n\\begin{figure}\n \\centering\n \\includegraphics[height=5.0cm,width=3.5cm]{fig8.eps}\n \\caption{The Onicescu information entropy (both one- and two-body) in a trapped Bose gas.}\\label{fig:fig8}\n\\end{figure}\n\nIt is interesting to observe the relation of the rms radii\n$\\sqrt{\\langle r^2 \\rangle}$ with $S_r$ as well as the\ncorresponding relation of the mean kinetic energy $\\langle T\n\\rangle$ with $S_k$, as functions of the strength of SRC,\n$\\ln{(\\frac{1}{y})}$. This is done in Fig. \\ref{fig:fig9} for\n$\\sqrt{\\langle r^2 \\rangle}$ and $\\langle T \\rangle$ after\napplying the suitable rescaling. The corresponding curves are\nsimilar for nuclei and trapped Bose gas.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[height=5.0cm,width=3.5cm]{fig9a.eps}\n \\hspace{0.3cm}\n \\includegraphics[height=5.0cm,width=3.5cm]{fig9b.eps}\n \\caption{ (a) Mean-square radius and the Shannon information entropy\n$S_{1r}$ (b) Mean kinetic energy $\\langle T \\rangle$ (in $\\hbar\n\\omega$ units) and the Shannon information entropy $S_{1k}$, as\nfunctions of the correlation parameter $\\ln{(\\frac{1}{y})}$ in a\ntrapped Bose gas.}\\label{fig:fig9}\n\\end{figure}\n\n\n\nA well-known concept in information theory is the distance between\nthe probability distributions $\\rho_i^{(1)}$ and $\\rho^{(2)}$, in\nour case the correlated and the uncorrelated distributions\nrespectively. A measure of distance is the Kullback-Leibler\nrelative entropy $K$ defined previously. The correlated and\nuncorrelated cases are compared for the one-body case $(K_1)$ and\nfor the two-body case $(K_2)$ in Fig. \\ref{fig:fig10}, decomposing\nin position- and momentum-spaces according to\n(\\ref{eq:equ17})-(\\ref{eq:equ20}). It is seen that $K_{1r}$,\n$K_{2r}$ increase as the strength of SRC increases, while\n$K_{1k}$, $K_{2k}$ have a maximum at a certain value of\n$\\ln{(\\frac{1}{y})}$ depending on the system under consideration.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[height=5.0cm,width=3.5cm]{fig10a.eps}\n \\hspace{0.3cm}\n \\includegraphics[height=5.0cm,width=3.5cm]{fig10b.eps}\n \\caption{(a) One- body Kullback-Leibler relative entropy both in\ncoordinate- and momentum-space (b) Two-body Kullback-Leibler\nrelative entropy both in coordinate- and momentum-space, in a\ntrapped Bose gas.}\\label{fig:fig10}\n\\end{figure}\n\nCalculations are also carried out for the Jensen-Shannon\ndivergence for one-body density distribution ($J_1$ entropy) as\nfunction of $\\ln{(\\frac{1}{y})}$, decomposed in position- and\nmomentum- spaces (Fig. \\ref{fig:fig11}). We observe again that\n$J_1$ increases with the strength of SRC in position-space, while\nin momentum-space there is a maximum for a certain value of\n$\\ln{(\\frac{1}{y})}$. It is verified that $0 10$) correspond to a\nGaussian distribution, while for $y\\lesssim 0.05$ (very strong\ncorrelations) higher order terms must be included in the expansion\nof the density.\n\nIt should be noted that in the present approach there is not a\ndirect dependence between the condensation and the number of the\natoms. The inter-particle correlations are incorporated in the\nmean field only by the correlation function which, in some way,\ndepends on the effective size of the atoms. We could find that\ndependence making the assumption that the correlated parameter $y$\ndepends on $N$ and $a_b$ through the relation\n\\begin{equation}\n\\frac{1}{y}= \\left(\\lambda_1 Na_b \\right)^{\\lambda_2}\n\\label{inv-y}\n\\end{equation}\n and try to find $\\lambda_1$ and $\\lambda_2$ equating the rhs of\n Eqs. (\\ref{SrSk-3}) and (\\ref{SrSk-4}). In this way $\\lambda_1$\n and $\\lambda_2$ can be found as functions of the parameters $a$\n and $b$ of Eqs. (\\ref{SrSk-3}) and (\\ref{SrSk-4}) having the\n forms\n \\begin{equation}\n \\lambda_1 ={\\rm e}^{(a_1-a_2)\/b_1},\\quad\n \\lambda_2 =b_1\/b_2\n \\label{a12-b12}\n \\end{equation}\nwhere $a_1$ and $b_1$ are the values of the parameters $a$ and $b$\nof Eq. (\\ref{ab-SrSk-3b})\n and $a_2$ and $b_2$ the parameters of Eq. (\\ref{ab-srsk-4}).\n The numerical values of\n$\\lambda_1$ and $\\lambda_2$ are: $\\lambda_1=0.0188$ and\n$\\lambda_2=0.7330$.\n\nThe values of $S$ for the bosonic systems ${}^{87}$Rb and\n$^{133}$Cs found in Jastrow correlation method (JCM), versus $\\ln\n(Na_b)$ (calculated from Eqs. (\\ref{inv-y}) and (\\ref{a12-b12})),\nare shown in Fig. \\ref{fig:fig14}(b) with open squares. It is seen\nthat the two bosonic systems studied with the GP theory and the\nbosonic system studied with the JCM give very similar results for\n$S$. It seems that the information entropy $S$ for the bosonic\nsystems depends only on $\\ln(Na_b)$.\n\n\\section{Summary}\\label{sec:sec6}\nThe effect of the interparticle correlations between Bose atoms at\nzero temperature is examined using a phenomenological way to\nincorporate the atomic correlations. This is made by introducing\nthe Jastrow correlation function in the two-body density matrix.\nAnalytical expressions are found for the one- and the two-body\ndensity and momentum distribution, mean-square radius, kinetic\nenergy and static structure factor. The introduction of\ncorrelations changes the shape of the density and momentum\ndistributions compared with the Gaussian one, corresponding to the\nharmonic oscillator model. There is a decrease of the density\ndistribution in the central region of the atomic system while the\nmomentum distribution increases in the region of small $k$ and\nthus there is a decrease of the mean kinetic energy of the system.\nIn addition the natural orbitals and the natural occupation\nnumbers have been calculated and consequently the condensate\nfraction has been obtained for different values of the parameter\n$y$. A theoretical calculation of the static structure factor is\nreported also by applying two trial forms for the radial\ndistribution function. Our results are compared with recent\nexperimental data concerning trapped Bose gas. By applying\nsuitable parametrization the experimental data are reproduced\nquite well.\n\nVarious kinds of quantum information properties of the trapped\nBose gas are calculated i.e. the Shannon and Onicescu information\nmeasures for the correlated and uncorrelated cases which are\ncompared as functions of the strength of the short range\ncorrelations. It can be seen that increasing the short range\ncorrelations the information entropies $S$ and $O$ increase. There\nis a relation between $\\sqrt{\\langle r^2 \\rangle}$ and $S_r$ and\nbetween $\\langle T \\rangle$ and $S_k$. It is also conjectured\nthat the relation $S_N=N(a+b\\,\\ln{N})$ holds approximately for the\ncorrelated system. The Gross-Pitavskii equation is solved in order\nto calculate the information properties of the Bose gas from\nanother point of view. It is concluded that the Shannon\ninformation entropy obeys the functional form $S=a+b\\,\\ln{N}$.\nFinally it is shown that Landsberg's order parameter $\\Omega$ is\nan increasing function of the number of Bose atoms $N$.\n\n\n\n\n\n\\section*{Acknowledgments}\n\nThe work of K.~Ch.~Chatzisavvas and C.~P.~Panos was supported by\nHerakleitos Research Scholarships (21866) of\n$\\textrm{E}\\Pi\\textrm{EAEK}$ and the European Union while the work\nof S.~E.~Massen, Ch.~C.~Moustakidis and C.~P.~Panos was supported\nby the Pythagoras II Research project (80861) of\n$\\textrm{E}\\Pi\\textrm{EAEK}$ and the European Union.\n\n\\clearpage\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nThe idealized tasks on which machine learning models are benchmarked commonly involve a single data source and readily available labels. Many datasets, however,\nare composed of multiple sources, e.g., different MRI modalities~\\cite{calhoun2016multimodal, fedorov2020self} in medical imaging, LiDAR, and video for self-driving cars~\\cite{xiao2020multimodal}, and confounder-influenced data~\\cite{scholkopf2016modeling}. Additionally, labels can be scarce, incorrect, or too definitive, which leads to the need for self-supervised or at least semi-supervised learning. In this work, we will be addressing both these constraints simultaneously by proposing an analysis of self-supervised learning approaches to multimodal data and comparing them to more classical methods. The self-supervised learning methods we consider are based on contrastive Deep InfoMax (DIM)~\\cite{dim}.\n\nClassical approaches to multimodal data include \\emph{canonical correlation analysis} (CCA)~\\cite{hotelling1992relations}, which finds maximally correlated linear projections of two data sources. More recently, CCA has been extended to allow for representations obtained using neural networks in works such as \\emph{deep canonical correlation analysis} (DCCA)~\\cite{andrew2013deep} and \\emph{deep\ncanonically correlated autoencoders} (DCCAE)~\\cite{wang2015deep}. Another family of methods is based on variational autoencoders, such as the \\emph{multimodal mixture of experts VAE} (MMVAE)~\\cite{shi2019variational}, which has resulted in considerable performance improvements.\n\n\\begin{figure}[t]\n \\center\n \\includegraphics[width=\\linewidth]{dataset_scheme.png}\n \\caption{Sample images from Two-View MNIST, MNIST-SVHN, OASIS-3, and a general scheme of all the possible ways to maximize the mutual information between two modalities. The arrows represent all possible combinations as coordinated connections between vectors of locations (along convolutional channels) or whole latent representations. The red arrow is Convolution-to-Representation (CR), the pink --- Convolution-Convolution (CC), the burgundy --- Cross-Convolution-to-Representation (XX), and the purple --- Representation-to-Representation (RR).}\n \\label{fig:dataset_scheme}\n\\end{figure}\n\n\nContrastive objectives have in recent years become essential components for a large number of self-supervised learning methods. Mutual information estimation~\\cite{belghazi2018mine} has inspired a number of successful uses for single-view data, such as \\emph{deep infomax} (DIM)~\\cite{hjelm2018learning}, \\emph{contrastive predictive coding} (CPC)~\\cite{oord2018representation}) and for multi-view data, such as \\emph{augmented multiscale DIM} (AMDIM)~\\cite{bachman2019amdim}, \\emph{contrastive multiview coding} (CMC)~\\cite{tian2019contrastive}, SimCLR~\\cite{chen2020simple}) image classification, reinforcement learning (\\emph{spatio-temporal DIM} (ST-DIM)~\\cite{anand2019unsupervised}) and zero-shot learning (\\emph{class matching DIM} (CM-DIM)~\\cite{sylvain2019locality,sylvain2020zeroshot}). Such methods have resulted in large representation learning improvements by considering different views of the same instance. These and other related methods mostly operate in a self-supervised fashion, where the goal is to encourage similarity between transformed representations of a single instance. These objectives can also be readily applied to a multimodal context, where different sources can be understood as different views of the same instance.\nIn addition to the relative scarcity of the literature on self-supervised multimodal representation learning, we also note that there are no studies that consider explicit combinations of unsupervised and multimodal objectives. This work aims to contribute to both issues. Further, we propose a taxonomy for self-supervised learning that is readily applicable to newer models: contrasting with momentum encoders MoCo~\\cite{he2020momentum}, prototypical\/clustering approaches SwAV~\\cite{caron2020unsupervised}, PCL~\\cite{li2020prototypical}, and non-contrastive approaches, such as BYOL~\\cite{grill2020bootstrap} and SimSiam~\\cite{chen2020exploring}.\n\nOur contributions are as follows:\n\\begin{itemize}\n \\item We unify contrastive learning methods under a systematic framework and perform a comparison of their merits. By analyzing the effect of adding different contrastive objectives, we show that correctly combining such objectives significantly impacts performance.\n \\item We demonstrate empirically that multimodal contrastive learning has significant benefits over its unimodal counterpart.\n \\item We propose novel models that emerge naturally from our systematic framework and taxonomy.\n \\item We show that simultaneously maximizing the inter- and intra-modality mutual information is essential because intra-modality similarity maximization alone may lead to a collapse of the representations, which could, in turn, be caused by underspecification~\\cite{d2020underspecification}.\n \\item We propose a new analysis with \\emph{centered kernel alignment} (CKA)~\\cite{kornblith2019similarity} to test the alignment of representations between modalities as a measure of their joint shared subspace.\n \\item We propose solutions to the possible detrimental effects of a similarity metric on representations.\n \\item We applied modern contrastive techniques to a complex multimodal medical setting and found a promising solution to uncover connections between modalities through a jointly shared subspace.\n\\end{itemize}\n\n\n\n\n\n\n\n\\begin{figure}[t]\n \\center\n \\includegraphics[width=1\\linewidth]{variants.pdf}\n \\caption{Possible variants of connected pairs to maximize mutual information. The 4 main objectives are CR, XX, CC, RR. CR (Convolution-to-Representation) is a unimodal objective capturing the relation between the location in the Convolutional feature map and the Representation. XX corresponds to a relation between a location in the convolutional feature map of one modality and the representation of the other. CC (Convolution-to-Convolution) relates locations in the Convolutional feature map across modalities. RR captures a multimodal Representation-to-Representation relation. The other objectives are combinations of the first four (CR, XX, RR, and CC).}\n \\label{fig:variants}\n\\end{figure}\n\n\\begin{figure}[t]\n \\center\n \\includegraphics[width=\\linewidth]{variants2.pdf}\n \\caption{Scheme for Supervised, AE, DCCAE, L-CCA, RR-AE and MMVAE.}\n \\label{fig:others}\n\\end{figure}\n\n\\section{Methods}\n\n\\subsection{Problem setting}\n\nLet $\\{D_i = \\{x_1,...,x_N\\}\\}_{i=1,...,n}$ be a set of datasets $D_i$ with $N$ samples and $n$ modalities. For each $i$th modality $D_i$, we define a sampled image $x_i$, a CNN encoder $E_i$, a convolutional feature $c_i$ from a fixed layer $l$ in the encoder as $c_i = E^l_i(x_i)$, which is needed to define DIM~\\cite{dim}-based objective, and a latent representation $z_i$ defined as $z_i = E_i(x_i)$. The AE-based approaches also produce a reconstruction $x'_i$ of the original sample $x_i$.\n\nTo learn the set of encoders $\\{E_i\\}_{i=1,...,n}$ we want to maximize the following objective $L$:\n\\begin{equation*}\n \\mathcal{L} = \\sum_{i=1}^n \\sum_{j=1}^n \\ell(D_i,D_j),\n\\end{equation*}\nwhere $\\ell(D_i,D_j)$ is a loss function between datasets $D_i$ and $D_j$.\nIn this study, we are specifically exploring the self-supervised contrastive objectives based on the maximization of mutual information, $\\ell(D_i, D_j)$ is therefore a loss function that maximizes the mutual information between the datasets of the two modalities.\n\n\\subsection{Contrastive mutual information maximization}\n\nTo maximize the lower bound of the mutual information, we utilize the InfoNCE~\\cite{oord2018representation} estimator $I$. We chose InfoNCE, because it is the common choice in the literature for self-supervised learning in natural images. There are, however, multiple estimators available (e.g., JSD and NWJ)~\\cite{dim, tschannen2019mutual, poole2019variational}. We define $I$ by adopting implementation from AMDIM~\\cite{bachman2019amdim}:\n\\begin{align*}\n &\\ell(D_i,D_j) = I(D_i; D_j) \\\\\n &\\ge \\frac{1}{N} \\sum_{l=1}^N \\log \\frac{e^{f(u^l_i,v^l_j)}}{\\sum_{k=1, k\\ne l}^N e^{f(u^l_i,v^k_j)} + e^{f^{c}(u^l_i,v^l_j)}},\n\\end{align*}\nwhere $f$ is a critic function, $u_i, v_j$ are two embeddings. The embeddings $u_i,v_j$ are obtained by projecting the location of a convolutional feature $c$ or a latent $z$ (e.g. $u_i = \\phi(c_i)$ and $v_j = \\psi(z_j)$). The neural networks that parametrize these projections $\\phi$ and $\\psi$ are also known as projection heads~\\cite{chen2020simple}.\n\nTo describe a critic function, we define positive and negative pairs. A pair $(u_i, v_j)$ is positive if it is sampled from a joint distribution $(u^l_i, v^l_j) \\sim p(D_i, D_j)$ and negative if it is sampled from a product of marginals $(u^l_i, v^k_j) \\sim p(D_i)p(D_j)$.\nA single entity could be represented differently in dataset $D_i$ than in dataset $D_j$. More specifically, the digit \"1\" can be represented by an image in multiple domains, for example, as a handwritten digit in MNIST and a house number in the SVHN dataset. The same number $(1_{\\text{MNIST}}, 1_{\\text{SVHN})}$ chosen from MNIST and SVHN will be a positive pair, whereas a digit in MNIST paired with a different digit in SVHN is a negative pair (such as $(1_{\\text{MNIST}}, 2_{\\text{SVHN}})$).\n\nThe idea behind a critic function $f(u,v)$ is to assign higher values to positive pairs and lower values to negative pairs. The critic used in this study is a separable critic $f(u,v) = \\frac{u^{\\intercal} v}{\\sqrt{d}}$, which is also used in the AMDIM~\\cite{bachman2019amdim} implementation (e.g. there are other possible choices such as bilinear, concatenated critics~\\cite{tschannen2019mutual}). Such a critic is equivalent to the scaled-dot product used in transformers~\\cite{vaswani2017attention}.\n\nAdditionally, we clip scores from critic function $f(x,y)$ to interval $[-c, c]$ by $c\\tanh(\\frac{f(x,y)}{c})$ with $c=20$, like in AMDIM~\\cite{bachman2019amdim}. Thus we need an additional term for a positive pair $e^{f^{c}(u^l_i,v^l_j)} = e^{-c}$. Lastly, we penalize $I(D_i; D_j)$ with the squared matching scores as $\\lambda f(x,y)^2$ with $\\lambda = 4\\mathrm{e}{-2}$.\n\n\n\\subsection{Objectives}\n\nMost contrastive self-supervised methods incorporate an estimator of mutual information. There are multiple ways of doing this, which are schematically shown in Figure~\\ref{fig:dataset_scheme}. The combinations of objectives shown in Figure~\\ref{fig:dataset_scheme} can be formulated as objectives based on Local DIM. In our taxonomy, we refer to the original Local DIM as CR since it captures a unimodal \\emph{Convolution-to-Representation} relation. This method maximizes the mutual information between the location in the convolutional feature map $c_i$ (where the representation is considered to be along the channels) and the latent representation $z_i$. Further, AMDIM~\\cite{bachman2019amdim}, ST-DIM~\\cite{anand2019unsupervised}, and CM-DIM~\\cite{sylvain2019locality} are referred to as XX, which captures a \\emph{Cross-Convolution-to-Representation} relation and as CC, which captures a \\emph{Convolution-to-Convolution} relation. The XX objective pairs the latent $z_i$ with $c_j$, where $i \\ne j$. The CC objective pairs $c_i$ with $c_j$, where $i \\ne j$. The last type of pairing, RR, represents a \\emph{Representation-to-Representation} relation and was introduced with CMC~\\cite{tian2019contrastive} and SimCLR~\\cite{chen2020simple}. These two methods pair $z_i$ and $z_j$, where $i \\ne j$. The CR and XX objectives are more closely related to the Deep InfoMax principle, while the pairing of RR and CC is more closely related to CCA.\n\nResearchers can combine these four basic contrastive objectives as shown in Figure~\\ref{fig:variants} to create new objectives. Each combination of edges is a type of objective. The objective is equal to the total sum of all the edges. To create a full picture, we extend these objectives to other approaches such as autoencoders and DCCA~\\cite{andrew2013deep} by adding the contrastive terms to their respective objective functions. We call these models the \\emph{RR autoencoder} (RR-AE) and the \\emph{CR canonical correlation analysis} (CR-CCA). We show them schematically in Figure~\\ref{fig:others} alongside other baselines: a Supervised uni-modal model, a normal AE, a multimodal DCCAE~\\cite{wang2015deep}, and a MMVAE~\\cite{shi2019variational} with a loose \\emph{ importance weighted autoencoder} (IWAE) estimator. We introduced the supervised model baseline specifically to serve as a discriminative bound on the multimodal dataset.\n\n\n\n\n\\section{Experiments}\n\n\\subsection{Datasets}\nFor our experiments, we incorporate diverse datasets with two input sources (shown in Figure~\\ref{fig:dataset_scheme}). The first experiments are the most straightforward multi-view dataset, Two-View MNIST, and the more challenging multi-domain MNIST-SVHN dataset. These datasets allow us to validate our approach because the datasets gradually increase in multimodal dataset complexity. These datasets have the additional advantage of being tasks DCCAE, and MMVAE were directly evaluated in the original articles, allowing for a direct comparison to the author's reported results. The last dataset we evaluate is an Alzheimer's disease dataset, where we use a functional and structural view of the brain to learn representations that allow us to differentiate between healthy controls and patients with Alzheimer's disease.\n\n\\subsubsection{Multi-view dataset}\nTwo-View MNIST is inspired by~\\cite{wang2015deep}, where each view represents a corrupted version of an original MNIST digit. First, the intensities of the original images are rescaled to the unit interval. The images are then resized to $32 \\times 32$ to fit the DCGAN architecture. Lastly, to generate the first view, we rotate the image by a random angle within the $[-\\pi\/4, \\pi\/4]$ interval. For the second view, we add unit uniform noise and rescale the intensity to a unit interval again.\n\n\\subsubsection{Multi-domain dataset}\nThe multi-domain dataset MNIST-SVHN was used by the authors of the MMVAE~\\cite{shi2019variational}, where the first view is a grayscale MNIST digit and the second view is an RGB street view house number sampled from the SVHN dataset. The MNIST digits are modified by resizing the images to $32 \\times 32$, which is also the default SVHN image size, to use them with the DCGAN encoder. All intensities are scaled to a unit interval. This dataset is more complicated than two-View MNIST because the digit is represented in different underlying domains. It is also more similar to the neuroimaging dataset because the views occur more naturally than the Two-View MNIST, where the views are augmentations of the original MNIST dataset.\n\n\\subsubsection{Multi-modal dataset}\n\nThe multimodal MRI dataset that we use is OASIS3~\\cite{LaMontagne2019.12.13.19014902}. We use it to evaluate different representations for Alzheimer's disease (AD) classification. We use T1-weighted images to account for the anatomy of the brain. The T1-weighted image is brain masked using FSL~\\cite{fsl} (v 6.0.2). T1 is the first modality and captures the structural aspects of the brain. The second modality captures the functional aspects of the brain. The second modality we use is resting-state fMRI (rs-fMRI), which captures the brain's metabolic function. We preprocess rs-fMRI into fALFF (relative low-frequency power in the 0.01 to 0.1 Hz power band) using REST~\\cite{rest}.\nAll images are linearly converted to MNI space and resampled to 3mm isotropic voxel resolution. The final input volume is $64\\times64\\times64$.\nCareful selection (removing poor images and trying to limit race as a confounder) resulted in a final subset of the OASIS-3 dataset with 826 non-Hispanic Caucasian subjects. For each subject, we combined their sMRI and fALFF scans to create 4021 multimodal pairs. We left 100 ($66\/22\/12$) subjects for hold-out and used others in a stratified (about $70\/15\/15\\%$) 5-folds for training and validation. We defined three groups: healthy cohort (HC), AD, and others (subjects with other brain problems). During pretraining, we employ all groups and pairs, whereas, during the linear evaluation, we only take one pair for each subject only use the HC or AD subjects. An additional preprocessing we applied is histogram standardization and z-normalization. During pretraining, we also use simple data augmentations, such as random crops and flips. These data augmentations were done using the TorchIO library~\\cite{perez_garcia_torchio_2020}. The classes in the dataset are highly unbalanced, so we utilize a class balanced data sampler~\\cite{balanced_data_sampler}.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\linewidth]{two_view_mnist.pdf}\n \\caption{Test downstream perfomance with linear evaluation on Two-View MNIST. Cross-modal losses have a strong positive impact on performance.}\n \\label{fig:two_view_mnist}\n\\end{figure}\n\n\\subsection{Evaluation}\n\\subsubsection{Linear evaluation on downstream task}\nTo evaluate representations on natural images, we employ the linear evaluation protocol, which is common for self-supervised approaches~\\cite{bachman2019amdim,chen2020simple}. It trains a linear mapping from the latent representations to the number of classes. The weights in the encoder that produces these representations are kept frozen. In this study, we evaluate the encoder for each modality separately.\n\n\\subsubsection{Measuring similarity between representations}\nTo better understand the underlying inductive bias of the specific objectives, we measure the similarity between the representations of the different modalities it produces. The measure of similarity we use is \\emph{canonical correlation analysis} CCA. CCA measures the average correlation of the aligned directions between the representations. Additionally, we propose to use linear \\emph{centered kernel alignment} (CKA)~\\cite{kornblith2019similarity}, which has been shown to identify the relationship between representations of networks reliably.\n\nSpecifically, in our case, the CCA measure for a pair modalities $i$ and $j$ can be written as :\n\\begin{equation}\n CCA(Z^i, Z^j) = \\frac{1}{d} ||Q^{\\mathrm{T}}_{Z^j}Q_{Z^i}||_*,\n\\end{equation}\nwhere $d$ is a dimension of latent representation, $Z$ is a $n \\times d$ matrix of $d$-dimensional representation for $n$ samples, $||\\cdot||_*$ is the nuclear norm, and $Q_{Z^i}$ is an orthonormal basis for $Z^i$.\n\nThe linear CKA measure is defined as:\n\\begin{equation}\nCKA(Z^i, Z^j) = \\left\\|Z^{j\\mathrm{T}} Z^i\\right\\|_{\\mathrm{F}}^{2} \/\\left(\\left\\|Z^{i\\mathrm{T}} Z^i\\right\\|_{\\mathrm{F}}\\left\\|Z^{j\\mathrm{T}} Z^j\\right\\|_{\\mathrm{F}}\\right),\n\\end{equation}\nwhere $||\\cdot||_F$ is the Frobenius norm.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\linewidth]{mnist_svhn.pdf}\n \\caption{Downstream test performance with linear evaluation on MNIST-SVHN. We can see that overall, multimodal contrastive losses fare better than unimodal contrastive losses. }\n \\label{fig:mnist_svhn}\n\\end{figure}\n\nAdditionally, we considered \\emph{singular value CCA} (SVCCA)~\\cite{NIPS2017_7188} and \\emph{projected weighted CCA} (PWCCA)~\\cite{morcos2018insights}. SVCCA is equivalent to CCA but performs an additional SVD-based dimensionality reduction. PWCCA is a weighted sum of the CCA vectors, where the weights are found through projection weighting. However, their results are similar to CCA.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\linewidth]{oasis_le.pdf}\n \\caption{Holdout test downstream performance with linear evaluation on OASIS-3. The error lines show $\\pm$ standard deviation over models trained on 5 different folds.}\n \\label{fig:oasis}\n\\end{figure}\n\n\\begin{figure*}[ht!]\n \\centering\n \\includegraphics[width=\\linewidth]{march14_dslw_oasis_results.pdf}\n \\caption{CCA, SVCCA, PWCCA and CKA measures on OASIS-3 for the training and holdout test sets.}\n \\label{fig:oasis_svcca_cka}\n\\end{figure*}\n\n\n\\subsection{Implementation details}\n\nThe architecture and the hyper-parameters for each encoder and decoder are entirely based on DCGAN~\\cite{radford2015unsupervised}. However, we removed one layer for experiments with natural images to be able to use an input size of $32x32$. The encoder produces a latent vector $z$, which in our case is $64$-dimensional. All the layers in the models are initialized with a uniform Xavier.\n\nThe projection head for the latent representation $z$ is identity. The convolutional features are taken from the layer with a feature side size of $8$ and passed through the convolutional projection head. The convolutional projection head is a ResNet like a block where one direction consists of $2$ convolutional layers (kernel size $1$, number of output and hidden features $64$, Xavier uniform initialization), and the second direction consists of one convolutional layer which (kernel size $1$, number of output features $64$, with initialization as identity). After summation of features from two directions, we apply a batch normalization layer to get the convolutional embeddings. The weights of the convolutional projection heads are shared across all contrastive objectives but separate for each modality.\n\nThe optimizer we use is RAdam~\\cite{liu2019variance}($ \\text{lr=}4\\mathrm{e}{-4}$) with a OneCycleLR scheduler~\\cite{smith2019super}($ \\text{max\\_lr=}0.01$). We pretrain on the natural images for $50$ epochs and on the volumes in OASIS-3 for $200$ epochs. The linear classification projection for natural images and OASIS-3 is trained for $50$ and $500$ epochs, respectively. All experiments were performed with a batch size of $64$. In some runs, we noticed that the CCA objective is unstable. The MMVAE did not converge for three folds out of 5 or completely collapsed due to the model's greater capacity. The greater capacity leads to GPU memory issues. Thus we had to decrease the batch size to 4. Thus we currently removed MMVAE from the OASIS-3 result.\n\nThe code was developed using PyTorch~\\cite{NEURIPS2019_9015} and the Catalyst framework~\\cite{catalyst}.\nFor data transforms of the brain images we utilized TorchIO~\\cite{perez_garcia_torchio_2020}, for CKA analysis of the representations we used code by anatome~\\cite{hataya2020anatome}, SVCCA~\\cite{NIPS2017_7188}, for AMDIM~\\cite{bachman2019amdim}, for DCCAE~\\cite{perry2020mvlearn}, for MMVAE and MNIST-SVHN~\\cite{shi2019variational}. The experiments were performed on an NVIDIA DGX-1 V100.\n\n\\section{Results}\n\n\\subsection{Two-View MNIST and MNIST-SVHN}\nWe can see in Figures~\\ref{fig:two_view_mnist} and~\\ref{fig:mnist_svhn} that cross-modal contrastive losses have a strong positive impact on downstream test performance across different architectural and model choices. We also note that the formulations have different performances across settings and datasets, leading to the conclusion that applying them in practice requires careful adaptation to a given problem.\n\nAlthough contrastive methods will result in the best performance for a simple multi-view case, reconstruction-based models, such as MMVAE and RR-AA, stand out in multi-domain experiments. The performance for most of the contrastive multimodal models is within $3\\%$ of RR-AE's performance. Thus one can choose decoder-free self-supervised approaches to reduce computational cost. It should also be noted that Uni-source AE, CR, multimodal DCCAE, and CR-CCA are not able to perform well on SVHN.\n\n\n\n\\subsection{OASIS-3 }\n\nFigure~\\ref{fig:oasis} shows the results on OASIS-3. Multimodal approaches exhibit strong performance, but the performance gain is less noticeable than with the previous tasks on natural images. Overall, accounting for robust performance on two modalities, the absolute leader in terms of the self-supervised methods is CR-XX-CC. The supervised baseline trained only on T1 modalities does still outperform this self-supervised method, however. In the case of the fALFF modality, the model that performs the best is XX-CC; notably, it is better than the Supervised model. This suggests that multimodal objectives can improve training for certain modalities that may be hard to extract discriminative representations from. Utilizing such an objective as some form of regularization may be beneficial.\n\nWe also want to note that the maximization of similarity between modalities by itself is not enough. The performance of the RR method shows that it might lead to a collapse of the representations. It might indicate that the T1 representation dominated over the fALFF representation during training because the RR objective model can learn meaningful T1 representations. By adding the reconstruction loss to the contrastive objective, we get RR-AE. The reconstruction loss improves the model. Analogous to the multi-domain experiment, the autoencoder is vital to learning the modality. Adding a reconstruction loss is not only the choice. The objective CR-XX-CC can, for example, be used, and it comes with reduced computational requirements because it does not require a decoder. The RR-AE model is also highly related to the idea and structure of the DCCAE. However, the typical implementation of CCA objectives in DCCAE is less numerically stable and requires the computation of eigenvalues and eigenvectors. Additionally, RR-AE is more robust and has a higher downstream performance.\n\nThe downstream performance might not be the main criteria to select a model, however. The multimodal models can be used to analyze the connection between modalities in neuroimaging~\\cite{calhoun2016multimodal}. Thus one will want to have a model with a representation that has a shared subspace. Based on our experiments on downstream performance in Figure~\\ref{fig:oasis} and similarity measures Figure~\\ref{fig:oasis_svcca_cka} we would advise CR-XX-CC and RR-AE as models that perform well and should be investigated further.\n\n\n\\subsection{Understanding similarity measures}\nMost self-supervised methods and the supervised model in Figure~\\ref{fig:oasis_svcca_cka} are noticeably worse than DCCAE, RR, RR-AE, and AE in terms of the CCA and SVCCA metrics. The lower performance is, however, not found to be significant for the PWCCA metric.\n\nVisually CCA, SVCCA, PWCCA measures of similarity behave comparably with a noticeable difference between training and testing sets. Interestingly, the CKA measure has very close values between training and testing sets. We hypothesize that the CKA measure shows the inductive bias hidden in representations through optimization, architecture, and the learned weights. The authors of the original CKA manuscript~\\cite{kornblith2019similarity} argue that methods with higher CKA have a higher similarity between representations and that more of the subspace is shared. The reason why CCA, SVCCA, PWCCA behave poorly might be related to their sensitivity to noise.\n\n\\section{Conclusion}\nIn this work, we proposed a unifying view on contrastive methods and conducted a detailed study of their performance on multimodal datasets. We\nbelieve that this unifying view will contribute to understanding how to learn powerful representations from multiple\nmodalities. Hopefully, instead of combining the similarities in various\nways and publishing the winning combinations as individual methods, the\nthe field will consider a broader perspective on the problem.\n\nWe empirically demonstrate that multimodal contrastive approaches result in performance improvements over methods that rely on a single modality for contrastive learning. We also show that downstream performance is highly dependent on the composition of such objectives. We argue that maximizing information similarity might not guarantee higher downstream performance. In some cases, it may weaken the representation or have a regularizing effect on the objective. However, high similarities between representations can be significant for other applications, i.e., multimodal analysis~\\cite{calhoun2016multimodal}.\n\nDIM-based methods have a smaller computational cost than autoencoder-based methods because they do not require a decoder to be trained. The smaller computational cost lowers the hardware requirements. While DIM-based methods do have comparable downstream performance, the lower hardware requirements can help democratize medical imaging. DIM-based methods can also be helpful in cases where labels do not exist or are inaccurate, a scenario that is quite common in neurological and mental disorder nosology.\n\nFor future work, we are interested in considering how the conclusions we draw here hold in different learning settings with scarcer data or annotations such as few-shot or zero-shot learning cases. Another goal is to study the joint shared subspace projects in brain space for visualization. The proposed interpretable joint learning approach can help advance work in our search for neuroimaging biomarkers.\n\n\\section{Acknowledgments}\n\\label{sec:acknowledgments}\nThis work is supported by NIH R01 EB006841.\n\nData were provided in part by OASIS-3: Principal Investigators: T. Benzinger, D. Marcus, J. Morris; NIH P50 AG00561, P30 NS09857781, P01 AG026276, P01 AG003991, R01 AG043434, UL1 TR000448, R01 EB009352. AV-45 doses were provided by Avid Radiopharmaceuticals, a wholly-owned subsidiary of Eli Lilly.\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nMirror symmetry is a fascinating relation between a pair of Calabi-Yau manifolds originating from physics. Even though two mirror manifolds may look very different geometrically, they are predicted to give rise to equivalent quantum field theories and thus equivalent notions of physics.\\\\\n\n\nIt took mathematicians decades to understand this relation. For a pair of mirror Calabi-Yau manifolds $X$ and $X^\\vee$, the first thing to notice is that their Hodge diamonds are reflections of each other, i.e. $h^{p,q}(X)=h^{q,\\,p}(X^\\vee)$. In 1990, physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that the enumerative invariants and period integrals of a mirror pair were related to each other, which led to the calculations of Gromov-Witten invariants on quintic 3-folds. \\\\\n\nIn 1994, Maxim Kontsevich proposed the celebrated Homological Mirror Symmetry conjecture which provides a mathematically rigorous explanation for the mysterious mirror symmetry phenomenon. It asserts that the Fukaya category of a symplectic manifold is derived equivalent to the category of coherent sheaves of its mirror dual (usually a complex manifold). \\\\\n\nOn the other hand, Strominger-Yau-Zaslow (SYZ)'s conjecture\\cite{SYZ} gives a geometric intuition for mirror symmetry. It conjectures that mirror symmetry is a torus duality between symplectic geometry (A-side) and complex geometry (B-side), which can be\ninterchanged by implementing a Fourier-transform on torus fibers. Within this geometric framework (following Fukaya\\cite{F} and Abouzaid\\cite{AbF}), one could construct a mirror functor from the Fukaya category of the A-side to the category of coherent sheaves of the B-side which induces the derived equivalence.\n\\subsection{Homological mirror symmetry and coisotropic branes}\nThe Fukaya category consists of Lagrangian branes (Lagrangian submanifolds equipped with flat connections) as objects and Lagrangian intersections as morphisms, and the Floer products (compositions of morphisms in the Fukaya category) are determined by counting holomorphic triangles with vertices at Lagrangian intersections and with edges in Lagrangian submanifolds.\nIn general, Lagrangian submanifolds do not generate the version of Fukaya category for which homological mirror symmetry holds as stated, certain objects are missing. For example, Let $E_\\tau$ be a elliptic curve with a Teichmuller parameter $\\tau$. Suppose $E_\\tau$ admits a complex multiplication, for example, $\\tau =i$, then multiplication by $i$ is an automorphism of $E_\\tau$ which is called a complex multiplication. Let $E=E_\\tau^2$ be the product abelian surface, then the mirror of $E$ is a 4 dimensional symplectic torus $(T,\\,\\omega)$. Considering the Grothendieck groups of the categories involved in homological mirror symmetry and their images under the Chern character map, we expect a commutative diagram.\n$$\n\\begin{tikzcd}\nK_0(D^bCoh(E)) \\arrow[rr] \\arrow[d, \"Ch\"] & & K_0(Fuk(T,\\omega)) \\arrow[d, \"Ch\"] \\\\\nH^*(E) \\arrow[rr] & & H^*(T).\n\\end{tikzcd}\n$$\nBy comparing the images of the Chern character maps, one finds that $Im(Ch)\\otimes \\mathbb{Q}$ for $K_0(D^bCoh(E))$ is 6 dimensional, yet $Im(Ch)\\otimes \\mathbb{Q}$ for $K_0(Fuk(T))$ is 5 dimensional, taking values in the kernel of \n$$ \\omega\\wedge:H^2(T;\\mathbb{Q})\\rightarrow H^4(T;\\mathbb{Q}),$$\nthus the two categories cannot be equivalent. This indicates that Lagrangians do not generate a sufficiently large category. Kapustin and Orlov \\cite{KO} suggest that the missing objects are coisotropic submanifolds equiped with a $U(1)$ connection satisfying certain conditions.\nThe difficulty of incorporating coisotropic submanifolds is to define the morphisms involving coisotropic objects and the product operations involving these morphisms. It is a long standing problem to define the appropriate Fukaya category including coisotropic branes proposed by Kapustin-Orlov \\cite{KO} and further studied by Aldi-Zaslow \\cite{AZ}, Chan-Leung-Zhang \\cite{CLZ}, Herbst \\cite{H}.\n\n\\begin{remark}\n\tAn alternative way to enlarge the Fukaya category is to take the split closure of the derived Fukaya category, i.e. enlarging the Fukaya category by adding formal direct summands of objects\\cite{S} \\cite{AS}.\n\\end{remark}\n\n\n\n\n\\subsection{Main result}\n\n\n\n\nOn a linear symplectic torus $(T,\\,\\omega)$, a new method is proposed to extend the Fukaya category of the torus to include coisotropic submanifolds alongside linear Lagrangian submanifolds as objects of the category. The approach is by considering a twisted doubling torus $T\\times T^\\vee$ (see definition \\ref{Double}) of $T$ and lifting (possibly coisotropic) objects into Lagrangians of $T\\times T^\\vee$.\n\\begin{theorem}\\label{LiftisLag}\\quad\n\t\\begin{enumerate}\n\t\t\\item The lift of a (possibly coisotropic) linear object on $T$ is a Lagrangian submanifold of $T\\times T^\\vee$.\n\t\t\\item The lift is a complex submanifold with respect to a canonical complex structure on $T\\times T^\\vee$.\n\t\\end{enumerate}\n\n\\end{theorem}\nThe Lagrangian Floer theory of $T$ is naturally related to the Lagrangian Floer theory of the twisted doubling torus $T\\times T^\\vee$ of $T$. However, the morphism spaces that we want to consider are only a certain subspace of the Floer cohomology in doubling torus, which we call the ``u-part\".\nThe main result, informally, is that the Floer cohomology of two objects in $T$ is isomorphic to the ``u-part\" Floer cohomology of the lifts in $T\\times T^\\vee$ and respects the Floer products.\n\\begin{theorem}\n\tFor a pair of Lagrangian branes $L,\\,L'$ which are mirror to a pair of line bundles $\\mathcal{L},\\,\\mathcal{L'}$, let $\\boldsymbol{L},\\,\\boldsymbol{L}'$ be the lifts in double torus. Suppose $\\mathcal{L'}\\otimes \\mathcal{L}^{-1}$ is ample. Then the ``u-part\" Floer cohomology $HF_u^*(\\boldsymbol{L},\\boldsymbol{L}')$ is isomorphic to $HF^*(L,\\,L')$. And for two such pair $L,L'$ and $L',L''$, the following diagram commutes\n\t$$\\begin{tikzcd}\n\t{HF^*(L',\\,L'')\\otimes HF^*(L,\\,L')} \\arrow[rr] \\arrow[d, \"\\cong\"] & & {HF^*(L,\\,L'')} \\arrow[d, \"\\cong\"] \\\\\n\tHF^*_u(\\boldsymbol{L}',\\,\\boldsymbol{L}'')\\otimes HF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}') \\arrow[rr] & & HF_u(\\boldsymbol{L},\\boldsymbol{L}'').\n\t\\end{tikzcd}$$\n\\end{theorem}\n By Theorem \\ref{LiftisLag}, a coisotropic submanifold of $T$ lifts to a Lagrangian submanifold of the doubling torus which could be studied using Lagrangian Floer theory. Thus, the enlarged Fukaya category including coisotropic branes is realized as a non-full subcategory of the Fukaya category of the doubling torus. As a corollary of this doubling construction, the Fukaya category of $T$ is equivalent to the Fukaya category of the dual torus $T^\\vee$.\n\n\n\n\\section{HMS for tori}\n\nHomological mirror symmetry for tori was extensively studied by Polishchuk-Zaslow\\cite{PZ}, Fukaya\\cite{F}, Kontsevich and Soibelman\\cite{KS}. Here we will review the construction of the mirror manifold of a symplectic torus, and how Lagrangian branes correspond to coherent sheaves on the mirror.\n\n\n\\subsection{SYZ fibrations and construction of mirrors}\n\n\n\n$T$ be a torus equipped with a complex valued closed 2-form $B+i\\omega$, where real part is called B-field, and where imaginary part is a (non-degenerate) symplectic form.\n\\begin{definition}\n\tA SYZ fibration for $(T,\\,B+i\\omega)$ is a torus fibration such that each fiber is a Lagrangian submanifold.\n\\end{definition}\n\nGiven a SYZ fibration of $(T,\\,B+i\\omega)$ with base $Q$ and fibers $F_q$, $T_qQ$ is naturally identified with $H^1(F_q;\\, \\mathbb{R})$ by $v\\mapsto [\\iota_v\\omega]$, where we lift $v\\in T_qQ$ to a normal vector field along $F_q$, also denoted $v$. Let $T^{\\mathbb{Z}}Q\\subset TQ$ be the lattice corresponding to $H^1(F_q,\\, \\mathbb{Z})$, and $T^*_\\mathbb{Z}Q\\subset T^*Q$ the dual lattice, which is natrually isomorphic to $H_1(F;\\,\\mathbb{Z})$. The key property of this lattice $T^*_\\mathbb{Z}Q$ is that its sections are locally exact 1-forms on $Q$. Indeed, for a class $\\beta\\in H_1(F,\\,\\mathbb{Z})$, near a point $q_0\\in Q$, let $x_\\beta(q)=\\int_{(q-q_0)\\times \\beta}\\omega$, then $dx_\\beta(v)=\\int_{\\beta}\\iota_v\\omega$ is the corresponding section of $T_\\mathbb{Z}^*Q$. If $\\beta_1,...,\\,\\beta_n$ form a basis of $H_1(F;\\,\\mathbb{Z})$, then the local coordinates $x_{\\beta_1},...,x_{\\beta_n}:\\,Q\\rightarrow \\mathbb{R}^n$ induce an integral affine structure on $Q$, and this structure doesn't depend on the choice of basis of $H_1(F;\\,\\mathbb{Z})$.\n\\begin{assumption}\n\tThe restriction of the B-field $B$ to the SYZ fibers vanishes.\n\\end{assumption}\nThis assumption eliminates the case with noncommutative mirrors, and allows one to construct the mirror manifold $Y$ of $(T,\\,B+i\\omega)$ as the moduli space of SYZ fibers equipped with flat $U(1)$ connections. $Y$ naturally admits a complex structure, specified by local coordinates\n\\begin{equation}\\label{coord}\nz_\\beta(F_q,\\,\\nabla)=e^{2\\pi i \\int_{(q-q_0)\\times \\beta}(B+i\\omega)}hol_{\\nabla}(\\beta)\n\\end{equation}\nwhere $F_{q_0}$ is a fixed fiber, $\\nabla$ is a flat $U(1)$ connection on $F_q$, and $(q-q_0)\\times \\beta $ is a relative homology class in $H_2(T,\\,F_q\\cup F_{q_0};\\,\\mathbb{Z})$ with boundary $\\beta$ in $H_1(F_q;\\,\\mathbb{Z})$ and $-\\beta$ in $H_1(F_{q_0};\\,\\mathbb{Z})$.\nThe tangent space of $Y$ at $(F,\\,\\nabla)$ is the quotient of the set of all pairs $(v,\\,\\alpha)\\in C^{\\infty}(NF)\\oplus \\Omega^1(F,\\,\\mathbb{R})$ such that $v$ is an infinitesimal Lagrangian deformation, and $\\alpha$ is a closed $1$-form, viewed as an infinitesimal deformation of the flat connection, by the subspace consisting of Hamiltonian vector fields and exact 1-forms (which correspond to trivial deformations).\n\n\\begin{lemma}\n\t$T_{(F_q,\\,\\nabla)}Y$ is identified with $H^1(F_q,\\,\\mathbb{C})$ via the map\n\t\\begin{equation}\n\t\\phi :(v,\\,\\alpha)\\mapsto \\iota_v (\\omega-iB)+i\\alpha.\n\t\\end{equation}\n\tAnd the map is complex linear.\n\\end{lemma} \n\n\\begin{proof}\n\tThe Arnold-Liouville theorem implies that there are canonical identifications \n\t\\begin{equation}\n\tT_qQ \\cong H^1(F_q;\\,\\mathbb{R}),\\, v\\mapsto [\\iota_v \\omega]\n\t\\end{equation}\n\twhere $v$ is lifted from $T_qQ$ to $C^{\\infty}(NF)$, so $v\\mapsto [\\iota_v \\omega]$ maps bijectively to $H^1(F_q;\\,\\mathbb{R})$. And an exact $1$-form $\\alpha$ is simply a gauge transformation which do not contribute to the deformation of connections, so the deformations of connections are classified by cohomology class of the 1-form $\\alpha$. So $\\phi$ is a bijection.\n\tTo verify $\\phi$ is complex linear, we see that\n\t\\begin{equation}\n\tdlog(z_\\beta)(v,\\,\\alpha)=-2\\pi \\int_{\\beta}(\\iota_v (\\omega-iB)+i\\alpha)\n\t\\end{equation}\n\tis complex linear for every holomorphic coordinate function $z_\\beta$. So $\\phi$ is complex linear. \n\\end{proof}\n\nIn fact, using the coordinates from \\eqref{coord} and observing that interior product with $B+i\\omega$ defines an (injective) linear map $H_1(Q) \\rightarrow H^1(F;\\,\\mathbb{C})$, we can identify the mirror manifold $Y$ with\n\\begin{equation}\nH^1(F;\\,\\mathbb{C})\/H^1(F;\\,\\mathbb{Z})+(B+i\\omega) H_1(Q;\\,\\mathbb{Z}).\n\\end{equation} \n\\begin{example}\\label{stdex}\n\tFor $T=(\\mathbb{R}\/\\mathbb{Z})^{2n}$, $B+i\\omega=\\tau dr\\wedge d\\theta=\\sum \\tau_{jk}dr_j\\wedge d\\theta_k (\\tau \\in M_{n\\times n}(\\mathbb{C}))$, with SYZ fibers $F=\\{r\\}\\times T_\\theta ^n$, the mirror complex torus is $E=\\mathbb{C}^n\/(\\mathbb{Z}^n+\\tau^{T} (\\mathbb{Z}^n))$.\n\\end{example}\n\n\n\\begin{comment\n\\begin{example}\nThe dual torus $T^{\\vee}=(\\mathbb{R}\/\\mathbb{Z})^{2n}, -\\omega^{-1}=-a^{-1}d\\hat{\\theta}\\wedge d\\hat{r} $, with Lagrangian fibers $F^\\vee =T_{\\hat{r}}^n\\times\\{\\hat{\\theta}\\}$, the mirror abelian variety is $E=\\mathbb{C}^n\/(\\mathbb{Z}^n-\\tau^{-1} (\\mathbb{Z}^n))$. \n\\end{example}\n\n\n\\begin{example}\nThe double torus $\\mathbb{T}=(\\mathbb{R}\/\\mathbb{Z})^{4n},\\mathbf{B}+i\\mathbb{\\omega}=\\frac{1}{2}(\\sigma_0+iadr\\wedge d\\theta+ia^{-1}d\\hat{r}\\wedge d\\hat{\\theta}),$\nwith Lagrangian torus fibers \n$\\mathbf{F}=\\{r\\}\\times T_\\theta^n\\times T_{\\hat{r}}^n\\times\\{\\hat{\\theta}\\}$. \nThe mirror is \n\n\\begin{equation*}\n\\mathbb{E}=\\mathbb{C}^{4n}\/(\\mathbb{Z}^{2n}+\\boldsymbol{\\tau}(\\mathbb{Z}^{2n})), \\boldsymbol{\\tau} =\\frac{1}{2}\n\\left(\\begin{array}{cc}\n&\\tau \\ \\ \\ \\mathbbm{1} \\\\\n&-\\mathbbm{1} \\ -\\tau^{-1}\n\\end{array}\\right)\n\\end{equation*}\nUsing coordinates $u=z+\\tau \\hat{z}, v=z-\\tau \\hat{z}$, we have: $\\mathbf{E}\\simeq E\\times E.$\n\\end{example}\n\\end{comment}\n\n\n\\subsection{Floer cohomology and sheaf cohomology}\n\n\\begin{figure}\n\t\\centering \n\t\\includegraphics[scale=0.8]{fig0506.png}\n\t\\caption{The trapezoid bound by $L_0$, $L_z$, $L$ and $L_{z'}$. }\\label{holo}\n\\end{figure}\n\n\\begin{definition}\n\tA linear Lagrangian brane in $(T,\\, B+i\\omega)$ is a linear Lagrangian submanifold $L\\subset (T,\\, \\omega)$ (with a choice of grading and spin structure) together with a complex line bundle over $L$ with a unitary connection $\\nabla$ whose curvature satisfies $F=-B|_L$.\n\\end{definition}\n\\begin{assumption}\n\tAssume there exists a linear Lagrangian brane $L_0$ with trivial local system such that $L_0\\cap F_q$ at only 1 point for each $q\\in Q$.\n\\end{assumption}\nThis $L_0$ will serve as the mirror of structure sheaf on $Y$. In the case of Example \\ref{stdex}, we simply choose $L_0=\\{\\theta=0\\}$. The points of the mirror torus parametrize Lagrangian branes $L_z=(F_q,\\, \\nabla)$ supported on SYZ fibers, and the generators $e_0(z)\\in CF(L_0,\\, L_z)$ are chosen to serve as evaluation map of the structure sheaf at $z$.\nGiven a linear Lagrangian brane $(L,\\, \\nabla)$ in $(T,\\, B+i\\omega)$, assume $L$ is transversal to $F_q$ for each $q\\in Q$. Now we construct its mirror sheaf to be a vector bundle\n\\begin{equation*}\n\\mathcal{L}=\\cup_{z\\in Y}CF^*(L_z,\\, L)\\rightarrow Y.\n\\end{equation*}\nWhen we move $L_z$, let $x(z)$ be a local continuous section of the intersection $L_z\\cap L$, and let $b(z)\\in CF(L_z,\\, L)$ be a rescaling of $x(z)\\in L_z\\cap L$. Then $b(z)$ is a local section of $\\mathcal{L}$.\n\\begin{definition}\n\t$\\mathcal{L}$ admits a natural holomorphic structure such that, if locally for any $z,z'\\in Y$, and N is a trapezoid bound by $L_0$, $L_z$, $L$, $L_{z'}$, see figure \\ref{holo},\n\t\\begin{equation*}\n\te^{2\\pi i\\int_{N}B+i\\omega}hol(\\partial N)\\equiv 1,\n\t\\end{equation*}\n\tthen $b(z)$ is a holomorphic section.\n\tHere $hol(\\partial N)$ is the composition of $e_0$, parallel transport along $L_z$, $b(z)$,parallel transport along $L$, $b(z')^{-1}$, parallel transport along $L_{z'}$, $e_0(z')^{-1}$ and parallel transport along $L_0$. \n\\end{definition}\n\n\\begin{remark}\n\tThese holomorphic sections uniquely determine the holomorphic structure of $\\mathcal{L}$. Here we only consider linear Lagrangians, in which case the Floer differentials are automatically zero, so the corresponding mirror sheaves are holomorphic vector bundles. In general, with Floer differential, one obtains chain complexes of locally free sheaves \\cite{F} \\cite{AbF}. \n\\end{remark}\n\nUnder homological mirror symmetry, we have \n\n\\begin{theorem}\\cite{F} \\cite{KS}\n\t\\begin{equation*}\n\tHF^*(L_1,L_2)\\cong Ext^*(\\mathcal{L}_1,\\mathcal{L}_2)\n\t\\end{equation*}\n\tAnd the following diagram commutes: \\\\\n\t\n\t\\begin{equation*}\n\t\\begin{tikzcd}\n\t{HF^*(L_2,\\,L_3)\\otimes HF^*(L_1,\\,L_2)} \\arrow[rr, \"\\mu_2\"] \\arrow[d, \"\\cong\"] & & {HF^*(L_1,L_3)} \\arrow[d, \"\\cong\"] \\\\\n\t{Ext^*(\\mathcal{L}_2,\\,\\mathcal{L}_3)\\otimes Ext^*(\\mathcal{L}_1,\\,\\mathcal{L}_2)} \\arrow[rr] & & {Ext^*(\\mathcal{L}_1,\\,\\mathcal{L}_3)}\n\t\\end{tikzcd}\n\t\\end{equation*}\n\n\\end{theorem}\n\nThe above theorem is best illustrated by the example of theta functions on the elliptic curve.\n\\begin{figure}\t\\label{line1}\n\t\\centering \n\t\\includegraphics[scale=1]{fig3.pdf}\n\t\t\\caption{Holomorphic triangles computing the Floer product $\\mu^2(e_1,s)\\in HF^*(L_0,L_x)$. }\\label{holo1}\n\\end{figure}\n\n\\begin{example}\n\tLet $T=\\mathbb{R}^{2}\/\\mathbb{Z}^{2}$, $B+i\\omega=\\tau dr\\wedge d\\theta$ be a symplectic two torus with coordinates $r,\\,\\theta$. Let $L_0$ be a horizontal Lagrangian $\\{\\theta=0\\}$ (mirror to the structure sheaf), $L_d$ be a slope $-d$ Lagrangian $\\{\\theta=-dr\\}$ (mirror to a degree d line bundle $\\mathcal{L}^d$), $L_z$ be a vertical Lagrangian with position $x$ and a connection with holonomy $e^{-2\\pi iy}$, $(\\{r=x\\},\\,\\nabla=d+2\\pi iyd\\theta)$, where $z=x+iy$. See figure \\ref{holo1} for the case $d=1$.\n\tThe generators $s_k=(\\frac{k}{d},\\,0)\\in L_0\\cap L_d$ of \n\t$HF(L_0,\\,L_d)$ correspond to the $\\vartheta$-basis of $H^0(E,\\,\\mathcal{L}^d)$\n\t\n\t$$\\vartheta_{d,\\,k} =\\sum_{n\\in \\mathbb{Z}} e^{\\pi\\tau id (n-\\frac{k}{d})^2}e^{2\\pi id(n-\\frac{k}{d}) z}.$$\n\\end{example}\n\n\n\n\\subsection{Coisotropic branes}\n\nKapustin and Orlov introduce the following notion of coisotropic brane, motivated by a string theoretical calculation.\n\n\\begin{definition}\n\tGiven a symplectic manifold $(X^{2n},\\,B+i\\omega)$, a coisotropic brane is a coisotropic submanifold $C^{n+k}$ equiped with a complex line bundle $(\\mathcal{L},\\,\\nabla)$ such that\n\t\\begin{enumerate}\n\t\t\\item Let $-2\\pi iF$ be the curvature of $(\\mathcal{L},\\,\\nabla)$, then $F+B|_C$ vanishes on $TC_{iso}= ker\\ \\omega|_C$, where $F+B|_C$ is viewed as a bundle morphism $TC\\rightarrow TC^*$. In particular, $F=-B|_C$ along the isotropic leaves (foliated by $ker\\, \\omega|_C=TC_{iso}$).\n\t\t\\item $\\omega^{-1}(F+B|_C)$ defines a transverse almost complex structure on $C$, i.e. an almost complex structure on $TC_{red}=TC\/TC_{iso}$. Equivalently, $\\omega+(F+B)\\omega^{-1}(F+B)=0$ on $TC$\n\t\\end{enumerate}\n\t\n\\end{definition}\n\\begin{remark}\n\tThe second condition implies that $F+B+i\\omega$ is a holomorphic symplectic form on the space of isotropic leaves, hence forces $k$ to be even. In the case $n=2,\\,k=2$ and $B=0$, we have a space filling coisotropic brane, and this condition is equivalent to $$\\omega\\wedge F=0,\\, \\omega\\wedge\\omega=F\\wedge F.$$\n\tSince $F\\wedge F$ represents an integral cohomology class, coisotropic branes can only arise for some special $\\omega$.\n\\end{remark}\n\\begin{remark}\n\tThe transverse almost complex structure arising from the geometry of the coisotropic brane is always integrable \\cite{KO}.\n\\end{remark}\n\n\\begin{example}\n\tLet $(T=\\mathbf{R}^4\/\\mathbf{Z}^4,\\, \\omega=dr_1\\wedge d\\theta_1 + dr_2\\wedge d\\theta_2)$ be the standard symplectic four torus. \n\tThen $(C=T,\\,\\nabla=d+2\\pi ir_1d\\theta_2-2\\pi ir_2d\\theta_1)$ is a coisotropic brane. And the induced complex stucture has complex coordinates $r_1-ir_2$, $\\theta_1+i\\theta_2$.\n\\end{example}\n\n\\noindent\nKapustin and Orlov have made a proposal for the endomorphisms of a coisotropic brane, namely $End(C)\\simeq H^{0,\\,*}(C)$, where Dolbeault cohomology is considered with respect to the transverse complex structure, but until now it was not understood how to define morphisms between different branes. In the next sections we propose a definition based on a ``doubling\" construction.\n\n\n\n\n\\section{Doubling and lifting}\n\n\\subsection{Construction of the twisted double torus and lift of coisotropic branes }\n\\subsubsection{Construction for symplectic torus without B-field}\n\nLet $(T=V\/\\Lambda, \\,\\omega)$ be a linear symplectic torus. Let $(T^\\vee=V^\\vee \/\\Lambda^\\vee, \\,-\\omega^{-1})$ be the dual torus with the inverse symplectic form $-\\omega^{-1}(\\alpha, \\,\\beta):=\\alpha(\\omega^{-1}\\beta)$.\n\nWe introduce the following doubling procedure:\n\n\\begin{definition}\\label{Double}\n\tThe twisted doubling torus of $(T, \\,\\omega)$ is a symplectic torus with a B-field\n$$(T\\times T^\\vee, \\ \\frac{1}{2}\\omega\\oplus-\\frac{1}{2}\\omega^{-1}, \\ \\sigma_0=\\sum \\frac{1}{2} dx_j\\wedge d\\hat{x}_j)$$\n\twhere $x_j$ are coordinates on $T$ and $\\hat{x_j}$ are the dual coordinates on $T^\\vee$. $\\sigma_0=\\sum \\frac{1}{2} dx_j\\wedge d\\hat{x}_j$ is the B-field which does not depend on the choice of coordinates.\n\\end{definition}\n\n\\begin{remark}\\label{AS}\n\tThe twisted doubling torus naturally comes with a complex structure $J$ which sends a tangent vector ${v}$ in $T$ to its symplectic dual $v \\lrcorner \\omega$ as a tangent vector of $T^\\vee$. In matrix notation,\n\t\n\t$$ J=\\left( \\begin{array}{ccc}\n\t0 & \\omega^{-1} \\\\\n\t-\\omega & 0\n\t\\end{array}\\right).$$\n\t\n\\end{remark}\n\n\\begin{example}\n\tLet $T=\\mathbb{R}^4\/\\mathbb{Z}^4$, $\\omega=dr_1\\wedge d\\theta_1+dr_2\\wedge d\\theta_2$. Its dual torus is $T^\\vee=\\mathbb{R}^4\/\\mathbb{Z}^4$, $-\\omega^{-1}=d\\hat{r}_1\\wedge d\\hat{\\theta}_1+d\\hat{r}_2\\wedge d\\hat{\\theta}_2$. The twisted double torus is $T\\times T^\\vee=\\mathbb{R}^4\/\\mathbb{Z}^4\\times \\mathbb{R}^4\/\\mathbb{Z}^4$, $ \\frac{1}{2}\\omega\\oplus-\\frac{1}{2}\\omega^{-1}=\\frac{1}{2}(dr_1\\wedge d\\theta_1+dr_2\\wedge d\\theta_2+d\\hat{r}_1\\wedge d\\hat{\\theta}_1+d\\hat{r}_2\\wedge d\\hat{\\theta}_2)$ and $B$ as above.\n\t\n\\end{example}\n\nOne can lift a linear Lagrangian brane or a coisotropic brane of $(T, \\,\\omega)$ to a Lagrangian brane in the doubling torus.\\\\\n\nFor a Lagrangian $(L, \\,\\nabla)$, its lift is $\\boldsymbol{L}=(L\\times L^\\perp, \\,\\nabla\\otimes \\mathds{1})$, where $L^\\perp$ is the conormal of $L$ translated by the holonomy of $\\nabla$.\\\\\n\nFor a coisotropic brane $(C, \\,\\nabla)$, its lift $\\boldsymbol{C}$ is a graph over $T$ in the doubling torus determined by the holonomy of $\\nabla$.\n\n\\begin{definition}\n\tThe lift of a coisotropic (posibly Lagrangian) brane $(C, \\,\\nabla)$ is defined to be\n\t$$\\{(x, \\,\\hat{x})\\in T\\times T^{\\vee}|x\\in C\\ \\text{and} \\\n\t\\langle\\hat{x}, \\,\\gamma_x\\rangle =(-1)^{\\xi(\\gamma_x)}hol_\\nabla(\\gamma_x),\\ \\forall\\ \\gamma_x \\in \\pi_1(C,x)\\},\\quad \\pi_T^*\\nabla$$\n\twhere $\\gamma_x$ is any linear circle passing through $x$ within $C$. And $\\xi:\\,H_1(C)\\rightarrow \\mathbb{Z}\/2$ is such that\n\t\\begin{equation*}\n\t\\xi(\\gamma+\\gamma')-\\xi(\\gamma)-\\xi(\\gamma')=c_1(\\nabla)(\\gamma\\wedge \\gamma') \\ mod \\ 2\n\t\\end{equation*}\n\\end{definition}\n\n\\begin{remark}\n\t$\\xi$ is introduced to make sure $(-1)^{\\xi(\\cdot)}hol_\\nabla(\\cdot)$ is a homomorphism: $\\pi_1(C, \\,x)\\rightarrow U(1)$.\n\t$\\xi$ has a similar role to the spin structure.\n\tThe space of different choices of $\\xi$ is an affine space over $H^1(C; \\,\\mathbb{Z}\/2)$.\n\\end{remark}\n\n\\begin{proposition}\\label{LiftisLag1}\n\tThe lift of a coisotropic (possibly Lagrangian) brane is a Lagrangian brane in the doubling torus. And it is also a $J$-complex (see remark \\ref{AS}) submanifold of $T\\times T^\\vee$.\n\\end{proposition}\n\\begin{proof}\n\tLet $(C, \\,\\nabla)$ be a coisotropic brane, $\\boldsymbol{C}$ be its lift. By linearizing the definition of the lift, we have\n\t$$T\\boldsymbol{C}=\\{u+Fu+\\omega v| u\\in TC, v \\in TC_{iso}\\}\n\t=\\left(\\begin{array}{ll}\n\t1\\ &0 \\\\\n\tF\\ &1\n\t\\end{array}\\right)\n\t\\left(\\begin{array}{ll}\n\t& TC \\\\\n\t& \\omega\\ TC_{iso}\n\t\\end{array}\\right), $$\n\twhere $F$ denote the curvature of $\\nabla$.\n\tGiven two tangent vectors $u_1+Fu_1+\\omega v_1$, $u_2+Fu_2+\\omega v_2$ of $T\\boldsymbol{C}$,\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t&\\omega\\oplus-\\omega^{-1}(u_1+Fu_1+\\omega v_1,\\ u_2+Fu_2+\\omega v_2) \\\\\n\t=&\\omega(u_1,\\,u_2)-\\omega^{-1}(Fu_1+\\omega v_1,\\,Fu_2+\\omega v_2) \\\\\t=&\\omega(u_1,\\,u_2)-\\omega^{-1}(Fu_1,\\,Fu_2)-\\omega^{-1}(\\omega v_1,\\,Fu_2)-\\omega^{-1}(Fu_1,\\,\\omega v_2)-\\omega^{-1}(\\omega v_1,\\,\\omega v_2)\\\\\n\t=&(\\omega+F\\omega^{-1}F)(u_1,\\,u_2)+F(v_1,\\,u_2)+F(u_1,\\,v_2)+\\omega(v_1,\\,v_2)\\\\\n\t=&0\n\t\\end{aligned}\n\t\\end{equation*}\n\tThe last line equals $0$ because $\\omega+F\\omega^{-1}F=0$ and $F$ is zero on $TC_{iso}$. So the lift $\\boldsymbol{C}$ is Lagrangian. \\\\\n\t\t\n\tTo prove that $\\boldsymbol{C}$ is a $J$-complex submanifold of $T\\times T^\\vee$,\n\twe observe that, for $u \\in TC$,\n\t$\\omega u +F\\omega^{-1}Fu $ vanishes on $TC$,\n\thence equals $\\omega v'$ for some $v'\\in TC_{iso}$. Therefore,\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t&J(u+Fu+\\omega v)=\\omega^{-1}Fu+v-\\omega u \\\\\n\t=&\\omega^{-1}Fu+v+F\\omega^{-1}F u -\\omega v'\\\\\n\t=&(\\omega^{-1}Fu+v)+F(\\omega^{-1}Fu+v) -\\omega v' \\in T\\boldsymbol{C}.\n\t\\end{aligned}\n\t\\end{equation*}\n\t\n\\end{proof}\n\n\\begin{remark}\n\tThere is an ambiguity when writing $Fu$ as a vector of $TT^\\vee$, in fact $Fu\\in T^*C$ whose lift to $T^*T$ is only unique up to a vector in $\\omega TC_{iso}$. In the above calculation, a lift of $Fu$ to $T^*T$ is chosen. Another way is to write $T\\boldsymbol{C}=\\{u+f|=F(u,\\,v)\\ \\forall v\\in TC\\}$.\n\\end{remark}\n\n\\subsubsection{Construction for symplectic torus with B-field}\nNow we deal with the more general situation when the symplectic torus starts with a B-field. Let $(T=V\/\\Lambda,\\, \\omega,\\, B)$ be a linear symplectic torus equipped with a B-field $B\\in H^2(T;\\,\\mathbb{R})$.\n\\begin{assumption}\\label{assump}\n\t$Id+(\\omega^{-1}B)^2$ is invertible.\n\\end{assumption}\nThe dual torus is $(T^\\vee=V^\\vee \/\\Lambda^\\vee,\\ -(\\omega+B\\omega^{-1}B)^{-1},\\ (\\omega+B\\omega^{-1}B)^{-1}B\\omega^{-1})$.\n\n\\begin{remark}\n\tThese formular for the symplectic form and B-field on the dual torus are rather confusing at first sight. In fact, they are the imaginary part and real part of $(B+iw)^{-1}$. Indeed,\n\\begin{equation*}\\begin{aligned}\n(B+i\\omega)^{-1}&=-i(Id-i\\omega^{-1}B)^{-1}\\omega^{-1}=-i\\sum_{k\\geq 0} (i\\omega^{-1} B)^k \\omega^{-1}\\\\\n&=-i(\\omega+B\\omega^{-1}B)^{-1}+(\\omega+B\\omega^{-1}B)^{-1}B\\omega^{-1}. \\end{aligned}\\end{equation*}\n\\end{remark}\n\n\\begin{definition}\n\tThe twisted doubling torus of $(T,\\,\\omega,\\, B)$ is a symplectic torus with a B-field\n\t\\begin{equation*}\n\t(T\\times T^\\vee,\\quad \\frac{1}{2}\n\t\\left(\\begin{array}{cc}\n\t \\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n\t\t -\\omega^{-1}B & -\\omega^{-1}\n\t\t\\end{array}\\right),\n\t\t\\quad\n\t\t \\sigma_0=\\sum_j \\frac{1}{2} dx_j\\wedge d\\hat{x}_j)\t\n\t\\end{equation*}\n\twhere $x_j$ are coordinates on $T$ and $\\hat{x}_j$ are the dual coordinates on $T^\\vee$. $\\sigma_0=\\sum \\frac{1}{2} dx_j\\wedge d\\hat{x}_j$ is the B-field which does not depend on the choice of coordinates.\n\\end{definition}\n\n\\begin{remark}\n\tIf we start with the dual torus $(T^\\vee=V^\\vee \/\\Lambda^\\vee,\\,-(\\omega+B\\omega^{-1}B)^{-1},\\, (\\omega+B\\omega^{-1}B)^{-1}B\\omega^{-1})$, then the twisted double torus of the dual torus is\n\t\t\\begin{equation*}\n\t\t(T^\\vee\\times T,\\quad \\frac{1}{2}\n\t\t\\left(\\begin{array}{cc}\n\t\t-\\omega^{-1} & -\\omega^{-1}B\\\\\n\t\t B\\omega^{-1} & \\omega+B\\omega^{-1}B\n\t\t\\end{array}\\right),\n\t\t\\quad\n\t\t-\\sum_j \\frac{1}{2} dx_j\\wedge d\\hat{x}_j).\t\n\t\t\\end{equation*}\n\tThis is different from the twisted double torus of $(T,\\omega, B)$ by a B-field twist. See Chapter 6 for more discussion.\n\\end{remark}\n\n\\begin{remark}\\label{AS2}\n\tThe twisted doubling torus still comes with a complex structure $J$ which is twisted by the B-field. In matrix notation,\n\t\n\t\\begin{equation*}\t\n\t J=\\left( \\begin{array}{cc}\n\t \\omega^{-1}B & \\omega^{-1} \\\\\n\t -\\omega-B\\omega^{-1}B & -B\\omega^{-1}\n\t\\end{array}\\right)\n\t=\\left( \\begin{array}{cc}\n\t 1 & 0 \\\\\n\t -B & 1\n\t\\end{array}\\right)\n\t\\left( \\begin{array}{cc}\n\t 0 & \\omega^{-1} \\\\\n\t -\\omega & 0\n\t\\end{array}\\right)\n\t\\left( \\begin{array}{cc}\n\t 1 & 0 \\\\\n\t B & 1\n\t\\end{array}\\right).\n\t\\end{equation*}\n\t\n\\end{remark}\n\nThe lift procedures are the same as in the case without B-field, the definition is copied:\n\\begin{definition}\n\tThe lift of a coisotropic (possibly Lagrangian) brane $(C,\\,\\nabla)$ is defined to be\n\t\\begin{equation*}\n\\{(x,\\,\\hat{x})\\in T\\times T^{\\vee}|x\\in C \\text{ and }\\ \\langle\\hat{x},\\,\\gamma_x\\rangle =(-1)^{\\xi(\\gamma_x)}hol_\\nabla(\\gamma_x),\\ \\forall\\ \\gamma_x \\in \\pi_1(C,\\,x) \\},\\quad \\pi_T^*\\nabla\n \\end{equation*}\n\twhere $\\gamma_x$ is any linear circle passing through $x$ within $C$. And $\\xi:\\,H_1(C)\\rightarrow \\mathbb{Z}\/2$ such that\n\t\\begin{equation*} \\xi(\\gamma+\\gamma')-\\xi(\\gamma)-\\xi(\\gamma')=c_1(\\nabla)(\\gamma\\wedge \\gamma') \\ mod \\ 2\n\t\\end{equation*}\n\\end{definition}\nSimilarly to the case without B-field, the lifts are Lagrangian and complex submanifolds of $T\\times T^\\vee $:\n\n\\begin{proposition}\\label{LiftisLag2}\n\tThe lift of a coisotropic (possibly Lagrangian) brane is a Lagrangian brane in the doubling torus. And it is also a $J$-complex (see remark \\ref{AS2}) submanifold of $T\\times T^\\vee$.\n\\end{proposition}\n\\begin{proof}\nNote that\n$$\n\\left(\\begin{array}{cc}\n\t \\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n\t\t -\\omega^{-1}B & -\\omega^{-1}\n\t\t\\end{array}\\right)\n=\\left(\\begin{array}{cc}\n\t1 & -B \\\\\n\t0 & 1\n\t\\end{array}\\right)\n\\left(\\begin{array}{cc}\n\t\\omega & 0 \\\\\n\t0 & -\\omega^{-1}\n\t\\end{array}\\right)\n\\left(\\begin{array}{cc}\n\t1 & 0 \\\\\n\tB & 1\n\t\\end{array}\\right).\n$$\nThe proof goes over the same as for Proposition \\ref{LiftisLag1} if we replace $F$ by $B+F$.\n\\end{proof}\n\n\n\\begin{remark}\nRecall that a space filling coisotropic brane $(C,\\,\\nabla)$ comes with a complex structure $\\omega^{-1}(F+B|_C)$. This complex structure coincides the complex structure \\ref{AS2} on the lift $\\boldsymbol{C}$ under the isomorphism $\\Pi_{T}:\\,\\boldsymbol{C} \\rightarrow C$, where $\\Pi_{T}:\\,T\\times T^\\vee \\rightarrow T$ is the projection to the $T$ factor.\n\\end{remark}\n\n\\subsection{Mirror of the twisted double torus}\n\nWe recall that the mirror of a symplectic torus with B-field $(T,\\,\\omega,\\,B)$ equipped with a SYZ fibration $F\\rightarrow T \\rightarrow Q$ is as follows:\n\\begin{equation}\nH^1(F;\\,\\mathbb{C})\/H^1(F;\\,\\mathbb{Z})+(B+i\\omega) H_1(Q;\\,\\mathbb{Z}).\n\\end{equation}\n\n\\begin{example}\\label{Tndouble}\n\tLet $(T,\\,B+i\\omega)=((\\mathbb{R}\/\\mathbb{Z})^{n}_r\\times (\\mathbb{R}\/\\mathbb{Z})^{n}_\\theta ,\\,\\tau dr \\wedge d\\theta)$ with SYZ fibers $F=T_\\theta \\times \\{r\\}$. The dual torus is $(T^\\vee,\\,(B+i\\omega)^{-1})=((\\mathbb{R}\/\\mathbb{Z})^{n}_{\\hat{\\theta}} \\times (\\mathbb{R}\/\\mathbb{Z})_{\\hat{r}}^{n} ,\\,\\tau^{-1} d\\hat{\\theta}\\wedge d\\hat{r})$ with dual SYZ fibers $F^\\vee =T_{\\hat{r}}^n\\times\\{\\hat{\\theta}\\}$.\n\tThe mirror complex torus for the dual torus is\n\t$$E_{-\\tau^{-1}}=\\mathbb{C}^n\/(\\mathbb{Z}^n-(\\tau^{T})^{-1} (\\mathbb{Z}^n)).$$\n\tThe twisted double torus is\n\t\\begin{equation*}\\begin{aligned}\n\t\\Big(T\\times T^{\\vee}=(\\mathbb{R}\/\\mathbb{Z})^{2n}\\times (\\mathbb{R}\/\\mathbb{Z})^{2n},\n\\quad \\boldsymbol{B}+i\\boldsymbol{\\omega}&=\n\t\\frac{i}{2}(Im\\tau +Re\\tau (Im\\tau)^{-1}Re\\tau)dr\\wedge d\\theta \\\\\n\t&-\\frac{i}{2}\\tau(Im\\tau)^{-1}dr\\wedge d\\hat{r}\\\\\n\t&+\\frac{i}{2}(Im\\tau)^{-1}\\tau d\\hat{\\theta}\\wedge d\\theta \\\\\n\t&-\\frac{i}{2} (Im\\tau)^{-1} d\\hat{\\theta}\\wedge d\\hat{r}\\Big).\n\t\\end{aligned}\\end{equation*}\n\twith SYZ fibers\n\t$\\boldsymbol{F}=\\{r\\}\\times T_\\theta^n\\times\\{\\hat{\\theta}\\}\\times T_{\\hat{r}}^n$.\n\tThe mirror is\n\t\n\t\\begin{equation*} \\mathbb{E}=\\mathbb{C}^{4n}\/(\\mathbb{Z}^{2n}+\\boldsymbol{\\tau}^{T}(\\mathbb{Z}^{2n})), \\quad\\boldsymbol{\\tau}^{T} =\\frac{i}{2}\n\t\\left(\\begin{array}{cc}\n\tIm\\tau^T+Re\\tau^T (Im\\tau^T)^{-1}Re\\tau^T & \\tau^{T}(Im\\tau^T)^{-1} \\\\\n\t-(Im\\tau^{T})^{-1}\\tau^T & -(Im\\tau^T)^{-1}\n\t\\end{array}\\right).\n\t\\end{equation*}\n\tEquipping SYZ fibers\n\t$\\boldsymbol{F}=\\{r\\}\\times T_\\theta^n\\times\\{\\hat{\\theta}\\}\\times T_{\\hat{r}}^n$\n\twith connection $\\nabla=d+2\\pi i (\\phi d\\theta +\\kappa d\\hat{r})$, local coordinates on the mirror are\n\t\\begin{equation*}\\begin{aligned}\n\t&\\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\theta})\n\t=\\frac{i}{2}(Im\\tau^T+Re\\tau^T (Im\\tau^T)^{-1}Re\\tau^T)r\n\t+\\frac{i}{2}\\tau^{T}(Im\\tau^T)^{-1}\\hat{\\theta}-\\phi \\\\\n\t&\\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\hat{r}})\n\t=-\\frac{i}{2}(Im\\tau^{T})^{-1}\\tau^T r\n\t-\\frac{i}{2}(Im\\tau^T)^{-1}\\hat{\\theta} -\\kappa.\n\t\\end{aligned}\\end{equation*}\n\tUsing alternative coordinates\n\t\\begin{equation*}\\begin{aligned}\n\t\t&u=\\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\theta})\n\t\t+\\tau^T \\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\hat{r}})\n\t\t=\\tau^T r-\\tau^T \\kappa -\\phi \\\\\n\t\t&v=\\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\theta})\n\t\t+\\bar{\\tau}^T \\frac{1}{2\\pi i}log(z_\\frac{\\partial}{\\partial \\hat{r}})\n\t\t=-\\hat{\\theta}-\\phi-\\bar{\\tau}^T\\kappa,\n\t\\end{aligned}\\end{equation*}\n\twe have: $\\boldsymbol{E}\\simeq E_\\tau \\times E_{-\\bar{\\tau}}.$\n\t\n\\end{example}\n\\begin{comment}\n\\begin{example}\n\tThe dual torus $T^{\\vee}=(\\mathbb{R}\/\\mathbb{Z})^{2n}, -\\omega^{-1}=-a^{-1}d\\hat{\\theta}\\wedge d\\hat{r} $, with Lagrangian fibers $F^\\vee =T_{\\hat{r}}^n\\times\\{\\hat{\\theta}\\}$, the mirror abelian variety is $E=\\mathbb{C}^n\/(\\mathbb{Z}^n-\\tau^{-1} (\\mathbb{Z}^n))$.\n\\end{example}\n\\begin{example}\n\tThe twisted double torus $\\mathbb{T}=(\\mathbb{R}\/\\mathbb{Z})^{4n},\\mathbf{B}+i\\mathbb{\\omega}=\\frac{1}{2}(\\sigma_0+iadr\\wedge d\\theta+ia^{-1}d\\hat{r}\\wedge d\\hat{\\theta}),$\n\twith Lagrangian torus fibers\n\t$\\mathbf{F}=\\{r\\}\\times T_\\theta^n\\times T_{\\hat{r}}^n\\times\\{\\hat{\\theta}\\}$.\n\tThe mirror is\n\t\n\t\\begin{equation*}\n\t\\mathbb{E}=\\mathbb{C}^{4n}\/(\\mathbb{Z}^{2n}+\\boldsymbol{\\tau}(\\mathbb{Z}^{2n})), \\boldsymbol{\\tau} =\\frac{1}{2}\n\t\\left(\\begin{array}{cc}\n\t&\\tau \\ \\ \\ \\mathbbm{1} \\\\\n\t&-\\mathbbm{1} \\ -\\tau^{-1}\n\t\\end{array}\\right)\n\t\\end{equation*}\n\tUsing coordinates $u=z+\\tau \\hat{z}, v=z-\\tau \\hat{z}$, we have: $\\mathbf{E}\\simeq E\\times E.$\n\\end{example}\n\\end{comment}\n\n\\begin{remark}\n\tThe mirror of the twisted double torus constructed above turns out to be isomorphic to the product of the original mirror with its complex conjugate $E\\times \\bar{E}$. This twisted double torus has the property that, even if a sheaf $\\mathcal{E}\\in Coh(E)$ corresponds to a coisotropic brane in $T$, a closely related sheaf on $E\\times \\bar{E}$ corresponds to a Lagrangian (which is the lift of the coisotropic brane) in $\\boldsymbol{T}$.\n\\end{remark}\nSYZ fibers $F\\subset T$ lift to fibers $\\boldsymbol{F}\\subset \\boldsymbol{T}$ which correspond to points in $E\\times 0$, i.e. $v=0$.\nIn example \\ref{Tndouble}, $(F=\\{r\\}\\times T^n_{\\theta},\\ \\nabla =d+2\\pi i\\phi d\\theta) $ is lifted to\n$(\\boldsymbol{F}=\\{r\\}\\times T^n_{\\theta}\\times\\{\\hat{\\theta}=-\\phi\\}\\times T^n_{\\hat{r}},\\ \\nabla=d+2\\pi i\\phi d\\theta)$ which corresponds to the point $u=\\tau^Tr-\\phi$, $v=0$.\\\\\nSimilarly, if a Lagrangian brane $L$ is mirror to a coherent sheaf $\\mathcal{E}$, then the lift $\\boldsymbol{L}$ is mirror to $\\mathcal{E}\\boxtimes \\mathcal{E}_0$ on $E\\times \\bar{E}$, where $\\mathcal{E}_0$ is a particular sheaf on $\\bar{E}$.\n\n\n\\section{Floer Theory}\n\n\nThe Floer theory of $(T,B+i\\omega)$ and its twisted double torus\n$$(T\\times T^{\\vee},\\quad \\frac{1}{2}\\sigma_0+\\frac{i}{2}\n\\left(\\begin{array}{cc}\n\\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n-\\omega^{-1}B & -\\omega^{-1}\n\\end{array}\\right))$$\nare deeply related to each other. In fact, the Floer cohomology between two Lagrangian brane $HF^*(L,\\; L')$ in $(T,\\; B+i\\omega)$ can be identified with a subspace of the Floer cohomology of the lifts $HF^*(\\boldsymbol{L},\\; \\boldsymbol{L}')$.\n\n\n\n\\subsection{Floer theory in 2-tori and their twisted doubles}\nLet $(T=\\mathbb{R}^2\/\\mathbb{Z}^2,\\ B+i\\omega=(b+ia)dr\\wedge d\\theta)$ equipped with SYZ fibration projecting to $r$ coordinate. \\\\\nThe coordinate on the mirror manifold is\n$$z_{\\theta}=e^{2\\pi i\\int{B+i\\omega} }hol_\\nabla(S^1_{\\theta})\n=e^{2\\pi i\\tau r}e^{-2\\pi i\\phi}=e^{2\\pi i(\\tau r-\\phi)}.$$\nThus the mirror is the elliptic curve $E=\\mathbb{C}\/\\mathbb{Z}+\\tau \\mathbb{Z}$, where $\\tau=b+ia$. \\\\\n\n\\subsubsection{Theta functions on Elliptic curves}\nTheta functions are holomorphic sections of holomorphic line bundles on elliptic curves. They can be constructed with the help of a holomorphic connection by periodizing a holomorphic section on the universal cover of the elliptic curve. If we begin with two gauge equivalent holomorphic connections on a degree $1$ line bundle, we get a priori different holomorhpic sections of the line bundle. The dimension of the space of holomorphic sections, which equals $1$, forces the two sections to be the same up to a constant. We can establish some magic formula for theta functions using this approach.\n\\begin{example}\t\n\tConsider the degree $1$ holomorphic line bundle on the elliptic curve $E_\\tau=\\mathbb{C}\/(\\mathbb{Z}+\\tau\\mathbb{Z})$ where $\\tau =b+ia$ with holomorphic connection $d+\\frac{2\\pi i}{a}ydx$. The transition function of the line bundle is given by\n\t $$ s(z+1)=s(z),\\quad s(z+n\\tau)=e^{-\\pi in^2b}e^{-2\\pi inx}s(z).$$\n\tBy starting with the holomorphic section $e^{-\\frac{\\pi}{a}y^2}$ on the universal cover of the torus and by periodizing it, we get a section of the line bundle given by\n \t $$s=\\sum_n e^{-\\frac{\\pi}{a}(y+na)^2}e^{2\\pi inx}e^{\\pi in^2b}=\\sum_n e^{\\pi in^2\\tau}e^{2\\pi in(x+iy)}e^{-\\frac{\\pi}{a}y^2}.$$\n\\end{example}\n\n\\begin{example}\n\tConsider a gauge equivalent connection $d+\\frac{\\pi i}{a}(ydx-xdy)$. The transition function of the line bundle is given by\n\t$$ s(z+m+n\\tau)=(-1)^{mn}e^{\\frac{\\pi i}{a}my}e^{\\frac{\\pi ib}{a}ny}e^{-\\pi inx}s(z).$$\n\tBy periodizing the holomorphic section $e^{-\\frac{\\pi}{2a}(x^2+y^2)}$ we get a holomorphic section\n\t\\[\n\t\\begin{aligned}\n\t&\\sum_{m,n} (-1)^{mn}e^{\\frac{\\pi i}{a}my}e^{\\frac{\\pi ib}{a}ny}e^{-\\pi inx}e^{-\\frac{\\pi}{2a}((x-m-nb)^2+(y-na)^2)}\\\\\n\t=&\\sum_{m,n}(-1)^{mn}e^{-\\frac{\\pi}{2a}(m+n\\tau)(m+n\\bar{\\tau})}e^{-\\frac{\\pi}{a}(m+n\\bar{\\tau})(x+iy)}e^{-\\frac{\\pi}{2a}(x^2+y^2)}.\n\t\\end{aligned}\n\t\\]\n\\end{example}\n\n\\begin{proposition}\n\tThe holomorphic sections from the above examples are equal to each other up to a factor.\n\t \\begin{equation}\\label{thetaformula1}\n\t\\begin{aligned}\n\t&\\sum_{m,n}(-1)^{mn}e^{-\\frac{\\pi}{2a}(m+n\\tau)(m+n\\bar{\\tau})}e^{-\\frac{\\pi}{a}(m+n\\bar{\\tau})z}e^{-\\frac{\\pi}{2a}z^2}\\\\\n\t=&\\sum_{m,n}e^{-\\frac{\\pi}{2a}(z+m)^2}e^{-\\frac{\\pi}{a}n\\bar{\\tau}(z+m)}e^{-\\frac{\\pi}{2a}n^2\\tau\\bar{\\tau}} \\\\\n\t=&\\sqrt{2a}\\sum_{l}e^{-\\pi i\\bar{\\tau}l^2}\\sum_{k}e^{\\pi i\\tau k^2}e^{2\\pi ikz}.\n\t\\end{aligned}\n\t \\end{equation}\n\t\n\\end{proposition}\n\\begin{remark}\n\tThe right hand side of equation \\eqref{thetaformula1} is (up to a constant factor) the standard formula for the theta function for a degree $1$ line bundle, while the left hand side natrually arises in the Floer products on the twisted double torus. And this formula is the key to relate Floer theory of $(T,\\; B+i\\omega)$ and Floer theory of its twisted doubling torus. We will provide an aternative proof of a generalization of this formula \\eqref{thetaformula2} using Fourier series.\n\\end{remark}\n\n\\subsubsection{Floer products on the 2-torus and on its twisted doubling torus}\n\\begin{example}\n\n\tLet $T=\\mathbb{R}^{2}\/\\mathbb{Z}^{2},\\; B+i\\omega=\\tau dr\\wedge d\\theta$ be a symplectic two torus with coordinates $r,\\theta$. Let $L_0$ be a horizontal Lagrangian $\\{\\theta=0\\}$ (mirror to the structure sheaf), $L_1$ be a slope $-1$ Lagrangian $\\{\\theta=-r\\}$ (mirror to a degree 1 line bundle $\\mathcal{L}^1$), $L_z$ be a vertical Lagrangian with position $r$ and a connection with holonomy $e^{-2\\pi i\\phi}$, $(L_z=\\{r\\}\\times S^1_\\theta,\\; \\nabla=d+2\\pi i\\phi d\\theta)$, where $z=\\tau r-\\phi$.\n\tThe generator $s=(0,\\;0)\\in L_0\\cap L_1$ of\n\t$HF(L_0,\\; L_1)$ correspond to the $\\vartheta$-function in $H^0(E,\\; \\mathcal{L}^1)$\n\t\t\n\t\t$$\\vartheta =\\sum_{n\\in \\mathbb{Z}} e^{\\pi\\tau i n^2}e^{2\\pi in z}.$$\n\n\tThe doubling torus is given by\n\t \\begin{equation*}\n\t (T\\times T^{\\vee},\\; \\frac{1}{2}(\\sigma_0+i\\Omega)=\\frac{i}{2a}(\\tau\\bar{\\tau}dr\\wedge d\\theta+\\tau (d\\hat{r}\\wedge dr+d\\hat{\\theta}\\wedge d\\theta)+d\\hat{r}\\wedge d\\hat{\\theta}))\n\t \\end{equation*}\n\t Complex coordinates on the mirror are given by\n\t \\begin{equation*}\\begin{aligned}\n\t z_{\\frac{\\partial}{\\partial\\theta}}&=e^{2\\pi i \\frac{i}{2a}(\\tau\\bar{\\tau}r+\\tau \\hat{\\theta})}e^{-2\\pi i \\phi};\\\\\n\t z_{\\frac{\\partial}{\\partial\\hat{r}}}&=e^{2\\pi i \\frac{-i}{2a}(\\tau r+\\hat{\\theta})}e^{-2\\pi i \\kappa};\n\t \\end{aligned}\\end{equation*}\n\t Note that\n\t \\begin{equation*}\\begin{aligned}\n\t z_{\\frac{\\partial}{\\partial\\theta}}(z_{\\frac{\\partial}{\\partial\\hat{r}}})^{\\tau}&=e^{2\\pi i(\\tau r-\\tau \\kappa -\\phi)};\\\\\n\t z_{\\frac{\\partial}{\\partial\\theta}}(z_{\\frac{\\partial}{\\partial\\hat{r}}})^{\\bar{\\tau}}&=e^{2\\pi i(-\\hat{\\theta}-\\phi-\\bar{\\tau}\\kappa)}.\n\t \\end{aligned}\\end{equation*}\n\t\n\tLet\n\t$$u=\\tau r-\\tau \\kappa -\\phi,\\quad v=-\\hat{\\theta}-\\phi -\\bar{\\tau}\\kappa$$\n\t be the new coordinates of the mirror manifold. We can see that the mirror manifold is isomorphic to $E_{\\tau}\\times E_{-\\bar{\\tau}}$.\n\t\n\tThe lifts $\\boldsymbol{L}_0=\\{\\theta=0,\\; \\hat{r}=0\\}$ and $\\boldsymbol{L}_1=\\{\\theta=-r,\\; \\hat{r}=\\hat{\\theta}\\}$ of $L_0$ and $L_1$ to the twisted doubling torus intersect in one point, which corresponds to a section of a line bundle on the mirror manifold. Considering the intersection with SYZ fibers $(\\boldsymbol{F}=\\{r\\}\\times S_\\theta^1\\times S_{\\hat{r}}^1\\times\\{\\hat{\\theta}\\},\\; \\nabla=d+2\\pi i \\phi d\\theta+2\\pi i \\kappa d\\hat{r})$, the Flor product $CF(\\boldsymbol{L}_0,\\;\\boldsymbol{L}_1)\\otimes CF(\\boldsymbol{L}_1,\\;\\boldsymbol{F})\\rightarrow CF(\\boldsymbol{L}_0,\\;\\boldsymbol{F})$ is given by the following expression, summing the contribution of holomorphic triangles of edge length $(n+r)$ in $T$ and $(m+\\hat{\\theta})$ in $T^\\vee$:\n\t\\begin{equation*}\\begin{aligned}\ns&=\\sum_{m,n}e^{2\\pi i (\\frac{i}{4a}\\tau\\bar{\\tau}(n+r)^2+\\frac{i}{4a}(m+\\hat{\\theta})^2+\\frac{i}{2a}\\tau(n+r)(m+\\hat{\\theta}))}e^{-2\\pi i(n+r)\\phi}e^{2\\pi i(m+\\hat{\\theta})\\kappa}\\\\\n\t&=\\sum_{m,n}e^{-\\frac{\\pi}{2a}(n^2\\tau\\bar{\\tau}+m^2+2\\tau mn)}e^{-\\frac{\\pi}{a}(m+n\\bar{\\tau})u}e^{\\frac{\\pi}{a}(m+n\\tau)v}e^{-\\frac{\\pi}{2a}(\\tau\\bar{\\tau}r^2+\\hat{\\theta}^2+2\\tau r\\hat{\\theta})}e^{-2\\pi ir\\phi}e^{2\\pi i\\hat{\\theta}\\kappa}\n\\end{aligned}\\end{equation*}\n\tClaim:\n\t\\begin{equation}\\label{thetaformula2}\n\t\\begin{aligned}\n\t\t&\\sum_{m,n}e^{-\\frac{\\pi}{2a}(n^2\\tau\\bar{\\tau}+m^2+2\\tau mn)}e^{-\\frac{\\pi}{a}(m+n\\bar{\\tau})u}e^{\\frac{\\pi}{a}(m+n\\tau)v}e^{-\\frac{\\pi}{2a}u^2}e^{\\frac{\\pi}{a}uv} e^{-\\frac{\\pi}{2a}v^2} \\\\\n\t\t=&\\sqrt{2a}\\sum_k e^{\\pi i\\tau k^2}e^{2\\pi iku}\\sum_l e^{-\\pi i\\bar{\\tau}l^2}e^{2\\pi ilv}\n\t\\end{aligned}\n\t\\end{equation}\n\n\t\\begin{proof}\n\t\t\n\t\tLet $\\displaystyle f(u)=\\sum_m e^{-\\frac{\\pi}{2a}(u+m)^2}e^{-\\frac{\\pi}{a}(n\\bar{\\tau}-v)(u+m)}$. \\\\\n\t\tThen $f(u+1)=f(u)$, hence $f$ is equal to its Fourier series\n\t\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\tf(u)&=\\sum_k\\int_{0}^{1}f(z)e^{-2\\pi ikz}dz\\ e^{2\\pi iku} \\\\\n\t\t&=\\sum_k \\sum_m\\int_{0}^{1}e^{-\\frac{\\pi}{2a}((z+m)^2+(2n\\bar{\\tau}-2v)(z+m))}e^{-2\\pi ikz}dz\\ e^{2\\pi iku} \\\\\n\t\t&=\\sum_k e^{\\frac{\\pi}{2a}(n\\bar{\\tau}-v+2ika)^2}\\sum_m\\int_{0}^{1}e^{-\\frac{\\pi}{2a}((z+m)^2+(2n\\bar{\\tau}-2v+4ika)(z+m)+(n\\bar{\\tau}-v+2ika)^2)}dz\\ e^{2\\pi ik u} \\\\\n\t\t&= \\sum_k e^{\\frac{\\pi}{2a}(n\\bar{\\tau}-v+2ika)^2}\\int_{-\\infty}^{\\infty}e^{-\\frac{\\pi}{2a}((z+n\\bar{\\tau}-v+2ika)^2)}dz\\ e^{2\\pi ik u} \\\\\n\t\t&=\\sum_k \\sqrt{2a}\\ e^{\\frac{\\pi}{2a}(n\\bar{\\tau}-v+2ika)^2} e^{2\\pi ik u}. \\\\\n\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\tThen\n\t\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\tLHS &=\\sum_{m,n}e^{-\\frac{\\pi}{2a}(n^2\\tau\\bar{\\tau}+m^2+2\\tau mn)}e^{-\\frac{\\pi}{a}(m+n\\bar{\\tau})u}e^{\\frac{\\pi}{a}(m+n\\tau)v}e^{-\\frac{\\pi}{2a}u^2}e^{\\frac{\\pi}{a}uv} \\\\\n\t\t&=\\sum_n f(u)e^{-\\frac{\\pi}{2a}\\tau\\bar{\\tau}n^2}e^{\\frac{\\pi}{a}n\\tau v} \\\\\n\t\t&=\\sum_n \\sum_k \\sqrt{2a}\\ e^{\\frac{\\pi}{2a}(n\\bar{\\tau}-v+2ika)^2} e^{2\\pi ik u} e^{-\\frac{\\pi}{2a}\\tau\\bar{\\tau}n^2}e^{\\frac{\\pi}{a}n\\tau v} \\\\\n\t\t&=\\sum_k \\sum_n \\sqrt{2a}\\ e^{-\\pi i\\bar{\\tau}(n-k)^2}e^{\\pi i\\tau k^2}e^{2\\pi i(n-k)v}e^{2\\pi iku} \\\\\n\t\t&=\\sqrt{2a} \\sum_k e^{\\pi i\\tau k^2}e^{2\\pi iku}\\sum_l e^{-\\pi i\\bar{\\tau}l^2}e^{2\\pi ilv}\n\t\t\\end{aligned}\n\t\t\\end{equation*}\n\t\t\n\t\\end{proof}\n\t\n\\end{example}\n\n This calculation shows that, up to a suitable rescaling (due to the discrepancy between the trivialization given by Floer complex and the usual holomorphic trivialization), the generator of $CF(\\boldsymbol{L}_0,\\,\\boldsymbol{L}_1)$ corresponds to the section $\\theta_\\tau(u)\\theta_{-\\bar{\\tau}}(v)$ of $\\mathcal{L}\\boxtimes\\mathcal{L}$ on $E_\\tau\\times E_{-\\bar{\\tau}}$.\n \n In general, let $L_0$ be a horizontal Lagrangian $\\{\\theta=0\\}$ (mirror to the structure sheaf), $L_d$ be a slope $-d$ Lagrangian $\\{\\theta=-dr\\}$ (mirror to the degree d line bundle $\\mathcal{L}^d$), $L_z$ be a vertical Lagrangian with position $r$ and a connection $\\nabla=d+2\\pi i\\phi d\\theta$, where $z=\\tau r-\\phi$.\nThe generators $s_k=(\\frac{k}{d},\\; 0)\\in L_0\\cap L_d$ of\n$HF(L_0,\\; L_d)$ correspond to the $\\vartheta$-basis of $H^0(E,\\; \\mathcal{L}^d)$\n\n$$\\vartheta_{k\/d} =\\sum_{n\\in \\mathbb{Z}} e^{\\pi\\tau id (n+\\frac{k}{d})^2}e^{2\\pi id(n+\\frac{k}{d}) z}.$$\n\nOn the doubling torus\n$(T\\times T^\\vee,\\; \\frac{i}{2a}(\\tau\\bar{\\tau}dr\\wedge d\\theta+\\tau (d\\hat{r}\\wedge dr+d\\hat{\\theta}\\wedge d\\theta)+d\\hat{r}\\wedge d\\hat{\\theta}))$, we get lifts of the above Lagrangian branes:\n\\begin{equation*}\\begin{aligned}\n\\boldsymbol{L}_0&=\\{\\theta=0,\\ \\hat{r}=0\\}\\\\\n\\boldsymbol{L}_d&=\\{\\theta=-dr,\\ \\hat{r}=d\\hat{\\theta}\\}\\\\\n\\boldsymbol{L}_z&=(\\{r\\}\\times S^1_\\theta\\times S^1_{\\hat{r}}\\times\\{ \\hat{\\theta}=-\\phi\\},\\\n\\nabla=d+2\\pi i\\phi d\\theta).\n\\end{aligned}\\end{equation*}\n\nAn argument similar to that given above for the case $d=1$ shows that, in $T\\times T^\\vee$, the generator $s_j\\otimes \\hat{s}_k \\in CF(\\boldsymbol{L}_0,\\; \\boldsymbol{L}_d)$ given by the point of $\\boldsymbol{L}_0\\cap\\boldsymbol{L}_d$ with coordinate $r=j\/d$ and $\\hat{\\theta}=k\/d$ corresponds to\n\\begin{equation*}\n\\sum_{l\\in \\mathbb{Z}\/d}e^{2\\pi ikl\/d}\\vartheta_{(j-l)\/d}(u)\\vartheta_{l\/d}(v)\\in H^0(E_\\tau\\times E_{-\\bar{\\tau}},\\ \\mathcal{L}^{d}\\boxtimes \\mathcal{L}^{d}).\n\\end{equation*}\n\n\\subsection{The general case for $T^{2n}$}\n\nLet $T=(\\mathbb{R}\/\\mathbb{Z})^{2n}$, $ B+i\\omega=\\tau dr\\wedge d\\theta=\\sum \\tau_{jk}dr_j\\wedge d\\theta_k \\ (\\tau \\in M_{n\\times n}(\\mathbb{C}))$, with Lagrangian fibers $F=\\{r\\}\\times T_\\theta ^n$ and base $Q$, the mirror complex torus is\n$$E\\cong H^1(F;\\,\\mathbb{C})\/(H^1(F;\\,\\mathbb{Z})+(B+i\\omega)H_1(Q;\\,\\mathbb{Z}))\\cong \\mathbb{C}^n\/(\\mathbb{Z}^n+\\tau^{T} (\\mathbb{Z}^n)).$$\nWe consider the following three Lagrangian branes in $(T,\\,B+i\\omega)$:\n\\begin{itemize}\n\t\\item $L_0=\\{\\theta=0\\}$ with trivial connection. $L_0$ is mirror to the structure sheaf on $E$.\n\t\n\t\\item $L_z=(\\{r\\}\\times T_\\theta ^n$, $\\nabla=d+2\\pi i(\\phi d\\theta)$. $L_z$ is mirror to the skyscraper sheaf at $z=\\tau^{T}r-\\phi$ on $E$.\n\t\n\t\\item $L_D=(\\{\\theta=-Dr\\}$, $\\nabla=d-\\pi i(r^{T}(Re\\tau D-D^{T}Re\\tau^{T} )dr)$\n where $D\\in GL(n,\\;\\mathbb{Z})$ such that $Im\\tau D=D^{T}Im\\tau^{T}>0$ and $Re\\tau D-D^{T}Re\\tau^{T}\\in M_{n\\times n}(\\mathbb{Z})$. The transition of the line bundle is \n \\begin{equation} s(r+m)=(-1)^{\\xi(m)}e^{\\pi i m^{T}(Re\\tau D-D^{T}Re\\tau^{T})r}s(r), \\end{equation}\n where $\\xi:H_1(T;\\mathbb{Z})\\rightarrow \\mathbb{Z}\/2$ such that \n \\begin{equation}\n \\xi(m_1+m_2)=\\xi(m_1)+\\xi(m_2)+m_1^{T}(Re\\tau D-D^{T}Re\\tau^{T})m_2 \\in \\mathbb{Z}\/2.\n \\end{equation}\n Note that $F=Re\\tau D dr\\wedge dr=-B|_{L_D}$ satisfying the B-field condition. $L_D$ is mirror to a line bundle $\\mathcal{L}_D$ on $E$ with first Chern class \n $ D^{T}dr \\wedge (d\\phi-Re\\tau^{T}dr)$, or equivalently $\\frac{i}{2}(Im\\tau)^{-1}D^{T}dz\\wedge d\\bar{z}$, where $z=\\tau^{T}r-\\phi$.\n\t\n\t\n\\end{itemize}\n\nThe intersections of $L_0$ and $L_D$ have coordinates $(D^{-1}k,\\;0)$, where $k\\in \\mathbb{Z}^n$. Hence\n$HF(L_0,\\;L_D)=CF(L_0,\\; L_D)=span_{\\mathbb{C}}\\{s_{(D,\\;k)}\\}_{k\\in \\mathbb{Z}^n}$. If $k\\equiv k'\\; mod\\, D\\mathbb{Z}^n$, then $s_{(D,k)}$ and $s_{(D,k')}$ correspond to the same intersection point, and the corresponding generators of $CF(L_0,L_D)$ coincide up to a multiplicative factor (which arise from the holonomy of the connection on $L_D$, see \\eqref{eq4.8}).\n\nDenote $e_D=(r,\\;-Dr)$ the generator of $HF(L_D,\\;L_z)$ and $e_0=(r,\\;0)$ the generator of $HF(L_0,\\;L_z)$. We will calculate the Floer product $\\mu_2:HF(L_D,\\;L_z)\\otimes HF(L_0,\\;L_D)\\rightarrow HF(L_0,\\;L_D)$.\\\\\nNote that \n$$hol(L_D)=e^{-\\pi i \\langle m-D^{-1}k,\\;(Re\\tau D-D^{T}Re\\tau^{T})r\\rangle }e^{\\pi i\\langle (D^{-1}k),\\;(Re\\tau D-D^{T}Re\\tau^{T})m\\rangle}(-1)^{\\xi(m)}$$ on the line segment from $s_{(D,\\;k)}$ to $e_D$ along the vector $(m+r-D^{-1}k,\\, -D(m+r-D^{-1}k))$, $m\\in\\mathbb{Z}^n$. (The first term is obtained by integrating the connection form, the rest come from the transition functions of the line bundle over $L_D$.)\n\\begin{equation*}\\begin{aligned}\ne_D\\circ s_{(D,\\,k)}\n&=\\sum_{ \\triangle\\in M(e_D,\\, S_{(D,k)},\\, e_0)} e^{2\\pi i\\int_{\\triangle}B+i\\omega}hol(\\partial \\triangle) e_0 \\\\\n&=\\quad\\sum_{m\\in \\mathbb{Z}^n}\\quad e^{\\pi i(B+i\\omega)(m+r-D^{-1}k,\\,D(m+r-D^{-1}k))}e^{-2\\pi i\\langle D(m+r-D^{-1}k),\\,\\phi \\rangle}hol(L_D)e_0 \\\\\n&=\\quad\\sum_{m\\in \\mathbb{Z}^n}\\quad (-1)^{\\xi(m)}e^{\\pi i\\langle D^{-1}k,\\,(Re\\tau D-D^{T}Re\\tau^{T})m\\rangle}e^{\\pi i\\langle \\tau D(m-D^{-1}k),\\,m-D^{-1}k\\rangle} e^{2\\pi i \\langle Dm-k,\\,\\tau^{T}r-\\phi\\rangle}\\\\\n&\\qquad\\qquad\\qquad e^{\\pi i\\langle \\tau Dr,\\,r\\rangle}e^{-2\\pi i\\langle Dr,\\, \\phi \\rangle}e_0\n\\end{aligned}\\end{equation*}\n\nUp to a rescaling, whole expression coincide with the theta functions\n\\begin{equation*}\n\\vartheta_{D, \\,k}(z)=\\sum_{m\\in \\mathbb{Z}^n} (-1)^{\\xi(m)}e^{\\pi i\\langle D^{-1}k, \\,(Re\\tau D-D^{T}Re\\tau^{T})m\\rangle}e^{\\pi i\\langle \\tau D(m-D^{-1}k), \\,m-D^{-1}k\\rangle} e^{2\\pi i \\langle Dm-k, \\,z\\rangle}\n\\end{equation*}\ncorresponding to the sections of the line bundle $\\mathcal{L}_D$, which satisfy\n\\begin{equation*}\n\\vartheta_{D,\\,k}(z+\\tau^{T}h)=(-1)^{\\xi(h)}e^{-\\pi i\\langle \\tau Dh,\\,h\\rangle}e^{-2\\pi i \\langle Dh,\\,z\\rangle} \\vartheta_{D,\\,k}(z),\\; \\vartheta_{D,\\,k}(z+h)=\\vartheta_{D,\\,k}(z).\n\\end{equation*}\n\\begin{equation}\\label{eq4.8}\n\\vartheta_{D,\\,k+Ds}(z)=(-1)^{\\xi(s)}e^{\\pi i\\langle (Re\\tau D-D^{T}Re\\tau^{T})s,\\,D^{-1}k\\rangle} \\vartheta_{D,\\,k}(z)\n\\end{equation}\n\nThe twisted double torus is given by\n\\begin{equation*}\\begin{aligned}\n\\Big(T\\times T^{\\vee}=(\\mathbb{R}\/\\mathbb{Z})^{2n}\\times (\\mathbb{R}\/\\mathbb{Z})^{2n},\\quad \\frac{1}{2}\\sigma_0+\\frac{i}{2}\\Omega=&\n\\frac{i}{2}(Im\\tau +Re\\tau (Im\\tau)^{-1}Re\\tau)dr\\wedge d\\theta \\\\\n&-\\frac{i}{2}\\tau(Im\\tau)^{-1}dr\\wedge d\\hat{r}\\\\\n&+\\frac{i}{2}(Im\\tau)^{-1}\\tau d\\hat{\\theta}\\wedge d\\theta \\\\\n&-\\frac{i}{2} (Im\\tau)^{-1} d\\hat{\\theta}\\wedge d\\hat{r}\\Big)\n\\end{aligned}\\end{equation*}\n\nRecall that the mirror of the twisted double torus is\n\\begin{equation*}\n\\boldsymbol{E}\\cong \\mathbb{C}^n\/(\\mathbb{Z}^n+\\tau^{T}\\mathbb{Z}^n)\n\\times \\mathbb{C}^n\/(\\mathbb{Z}^n-\\bar{\\tau}^{T}\\mathbb{Z}^n)\n\\end{equation*}\nwith holomorphic coordinates\n\\begin{equation*}\nu=\\tau^{T}(r-\\kappa)-\\phi, \\qquad\nv=-\\bar{\\tau}^T\\kappa-\\hat{\\theta}-\\phi.\n\\end{equation*}\n\nThe lifts of the three Lagrangians above are\n\\begin{enumerate}\n\t\\item $\\boldsymbol{L}_0=\\{\\theta=0,\\;\\hat{r}=0\\}$ with trivial connection. $\\boldsymbol{L}_0$ is mirror to the structure sheaf.\n\t\n\t\\item $\\boldsymbol{L}_z=\\{\\{r\\}\\times T_{\\theta}^n\\times T_{\\hat{r}}\\times \\{\\hat{\\theta}\\} \\}$.\n\t\n\t\\item $\\boldsymbol{L}_D=\\Big(\\{\\theta=-Dr,\\; \\hat{r}-D^{T}\\hat{\\theta}=-(Re\\tau D-D^{T}Re\\tau^{T})r \\},\\quad \\nabla=d-\\pi ir^{T}(Re\\tau D-D^{T}Re\\tau^{T} )dr\\Big)$ .\n\t\n\\end{enumerate}\n\n\n\\begin{equation*}\n\\boldsymbol{L}_0\\cap \\boldsymbol{L}_D\n=\\{(r=D^{-1}k,\\,\\theta=0,\\,\\hat{r}=0,\\ \\hat{\\theta}\n=(D^{T})^{-1}(Re\\tau D-D^{T} Re\\tau^{T})D^{-1}k+(D^{T})^{-1}l)\\}\n\\end{equation*}\n\nDenote $p=D^{-1}k,\\; q=(D^{T})^{-1}(Re\\tau D-D^{T} Re\\tau^{T})D^{-1}k+(D^{T})^{-1}l.$\\\\\nLet $s_{k,\\,l}=(p,\\,0,\\,0,\\,q) \\in CF(\\boldsymbol{L}_0,\\,\\boldsymbol{L}_D),\\;\ne_0=(r,\\,0,\\,0,\\,\\hat{\\theta}) \\in CF(\\boldsymbol{L}_0,\\;\\boldsymbol{L}_z)$ and $$e_{D}=(r,\\;-D(r-p),\\;D^{T}(\\hat{\\theta}-q)-(Re\\tau D-D^{T} Re\\tau^{T})(r-p),\\;\\hat{\\theta}) \\in CF(\\boldsymbol{L}_D,\\;\\boldsymbol{L}_z). $$\nThe coefficient of $e_0$ in $\\mu^2(e_D,\\, s_{k,l})$ is then\n\\begin{equation*}\\begin{aligned}\ns_{k, \\,l}=\n\\sum_{m, \\,n\\in \\mathbb{Z}^n}\n&e^{-\\frac{\\pi }{2}\n\\langle Im\\tau^{-1} D^{T}(\\hat{\\theta}+n-q),\\,\\hat{\\theta}+n-q\\rangle\n-\\frac{\\pi}{2}\n\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}(r+m-p),\\,r+m-p\\rangle\n-\\pi\\langle Im\\tau^{-1}D^{T}\\tau^{T}(r+m-p),\\,\\hat{\\theta}+n-q\\rangle} \\\\\n&e^{-2\\pi i\\langle D(r+m-p),\\,\\phi\\rangle +2\\pi i\\langle D^{T}(\\hat{\\theta}+n-q)-(Re\\tau D-D^{T}Re\\tau^{T})(r+m-p),\\,\\kappa\\rangle}\\\\\n& (-1)^{\\xi(m)}e^{-\\pi i\\langle m-p,\\,(Re\\tau D-D^{T}Re\\tau^{T})r\\rangle}e^{\\pi i\\langle p,\\,(Re\\tau D-D^{T}Re\\tau^{T})m\\rangle} \\\\\n=\\sum_{m,\\,n\\in \\mathbb{Z}^n}\n& (-1)^{\\xi(m)}e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}(n-q),\\,n-q\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}(m-p),\\,m-p\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}(m-p),\\,n-q\\rangle} \\\\\n&e^{\\pi i \\langle p,\\,(Re\\tau D-D^{T}Re\\tau^{T})m\\rangle} e^{-\\pi\\langle Im\\tau^{-1}D^{T}(n-q+\\bar{\\tau}^{T}(m-p)),\\,u\\rangle}e^{\\pi\\langle Im\\tau^{-1}D^{T}(n-q+\\tau^{T}(m-p)),\\,v\\rangle} \\\\\n&e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}\\hat{\\theta},\\,\\hat{\\theta}\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}r,r\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}r,\\,\\hat{\\theta}\\rangle}e^{-2\\pi i\\langle Dr,\\phi \\rangle+2\\pi i\\langle D^{T}\\hat{\\theta}-(Re\\tau D-D^{T}Re\\tau^{T})r,\\,\\kappa\\rangle} \\\\\n=\\sum_{m,\\,n\\in \\mathbb{Z}^n}\n&e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}(u+n-q),\\,u+n-q\\rangle}\ne^{-\\pi \\langle Im\\tau ^{-1}D^{T}(\\bar{\\tau}^{T}(m-p)-v),\\,u+n-q\\rangle}e^{-2\\pi i\\langle D(m-p),\\,n-q\\rangle } \\\\\n& (-1)^{\\xi(m)}e^{\\pi i\\langle p,\\, (Re\\tau D-D^{T}Re\\tau^{T})m\\rangle}\ne^{-\\frac{\\pi}{2}\\langle \\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^T(m-p),\\,m-p\\rangle}\ne^{\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}(m-p),\\,v\\rangle} \\\\\n&e^{\\frac{\\pi}{2}\\langle Im\\tau^{-1}D^{T}u,\\,u\\rangle}\ne^{-\\pi \\langle Im\\tau^{-1}D^{T}u,\\,v\\rangle }\ne^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}\\hat{\\theta},\\,\\hat{\\theta}\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}r,\\,r\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}r,\\,\\hat{\\theta}\\rangle}\\\\\n&e^{-2\\pi i\\langle Dr,\\phi \\rangle+2\\pi i\\langle D^{T}\\hat{\\theta}-(Re\\tau D-D^{T}Re\\tau^{T})r,\\,\\kappa\\rangle}\n\\end{aligned}\\end{equation*}\n\nUsing the same Fourier series manipulation as in the proof of \\eqref{thetaformula2}, this can be rewritten as\n\n\\begin{equation*}\\begin{aligned}\ns_{k, \\,l}=\\sum_{m,\\,s\\in \\mathbb{Z}^n}\n&\\sqrt{\\frac{2}{det(Im\\tau^{-1}D^{T})}}\ne^{\\frac{\\pi}{2}\\langle Im\\tau^{-1}D^{T}(\\bar{\\tau}^{T}(m-p)-v)+2is,\\,\\bar{\\tau}^{T}(m-p)-v+2iIm\\tau^{T}D^{-1}s \\rangle}e^{2\\pi i \\langle s,u-q\\rangle}\\\\\n&(-1)^{\\xi(m)}e^{\\pi i\\langle p,\\, (Re\\tau D-D^{T}Re\\tau^{T})m\\rangle}\ne^{-\\frac{\\pi}{2}\\langle \\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^T(m-p),\\,m-p\\rangle}\ne^{\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}(m-p),\\,v\\rangle} \\\\\n&e^{2\\pi i\\langle D(m-p),\\,q\\rangle}e^{\\frac{\\pi}{2}\\langle Im\\tau^{-1}D^{T}u,\\,u\\rangle}\ne^{-\\pi \\langle Im\\tau^{-1}D^{T}u,\\,v\\rangle }\\\\\n&e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}\\hat{\\theta},\\,\\hat{\\theta}\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}r,\\,r\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}r,\\,\\hat{\\theta}\\rangle}e^{-2\\pi i\\langle Dr,\\,\\phi \\rangle+2\\pi i\\langle D^{T}\\hat{\\theta}-(Re\\tau D-D^{T}Re\\tau^{T})r,\\,\\kappa\\rangle} \\\\\n\\end{aligned}\\end{equation*}\nAfter a change of variables $s\\mapsto Ds-t$ and rearranging, this becomes\n\\begin{equation*}\\begin{aligned}\ns_{k, \\,l}=\\sum_{t\\in \\mathbb{Z}^n\/D\\mathbb{Z}^n}\n&\\sqrt{\\frac{2}{det(Im\\ \\tau^{-1}D^{T})}}\ne^{-2\\pi i\\langle l,\\,D^{-1}(k-t)\\rangle}\ne^{\\pi i\\langle (Re\\tau D-D^{T}Re\\tau^{T})D^{-1}k,\\,D^{-1}t\\rangle}\n\\\\\n&\\sum_{s\\in \\mathbb{Z}^n}\n(-1)^{\\xi(s)}e^{\\pi i\\langle (Re\\tau D-D^{T}Re\\tau^{T})s,\\,D^{-1}t\\rangle}\ne^{\\pi i \\langle \\tau D(s-D^{-1}t),\\,s-D^{-1}t\\rangle}e^{2\\pi i \\langle Ds-t,u\\rangle} \\\\\n&\\sum_{m\\in \\mathbb{Z}^n}\n(-1)^{\\xi(m)}e^{-\\pi i \\langle (Re\\tau D-D^{T}Re\\tau^{T})m,\\,p-D^{-1}t\\rangle}\ne^{-\\pi i \\langle \\bar{\\tau}D(m-p+D^{-1}t),m-p+D^{-1}t\\rangle}\ne^{2\\pi i\\langle D(m-p+D^{-1}t),\\,v\\rangle} \\\\\n&e^{\\frac{\\pi}{2}\\langle Im\\tau^{-1}D^{T}(u-v),\\,(u-v)\\rangle}\\\\\n&e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}\\hat{\\theta},\\,\\hat{\\theta}\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}r,\\,r\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}r,\\,\\hat{\\theta}\\rangle}e^{-2\\pi i\\langle Dr,\\,\\phi \\rangle+2\\pi i\\langle D^{T}\\hat{\\theta}-(Re\\tau D-D^{T}Re\\tau^{T})r,\\,\\kappa\\rangle} \\\\\n=\\sum_{t\\in \\mathbb{Z}^n\/D\\mathbb{Z}^n}\n&\\sqrt{\\frac{2}{det(Im\\ \\tau^{-1}D^{T})}}\ne^{-2\\pi i\\langle l,\\,D^{-1}(k-t)\\rangle}\ne^{\\pi i\\langle (Re\\tau D-D^{T}Re\\tau^{T})D^{-1}k,\\,D^{-1}t\\rangle}\n\\vartheta_{D,t}(u)\\bar{\\vartheta}_{D,\\,k-t}(v)\\\\\n&e^{\\frac{\\pi}{2}\\langle Im\\tau^{-1}D^{T}(u-v),\\,(u-v)\\rangle}\\\\\n&e^{-\\frac{\\pi }{2}\\langle Im\\tau^{-1}D^{T}\\hat{\\theta},\\,\\hat{\\theta}\\rangle-\\frac{\\pi }{2}\\langle\\bar{\\tau}Im\\tau^{-1}D^{T}\\tau^{T}r,\\,r\\rangle-\\pi \\langle Im\\tau^{-1}D^{T}\\tau^{T}r,\\,\\hat{\\theta}\\rangle}e^{-2\\pi i\\langle Dr,\\,\\phi \\rangle+2\\pi i\\langle D^{T}\\hat{\\theta}-(Re\\tau D-D^{T}Re\\tau^{T})r,\\,\\kappa\\rangle} \\\\\n\\end{aligned}\\end{equation*}\nThe factors on the last two lines correspond to the difference between the Floer basis and the usual holomorphic trivialization of $\\mathcal{L}^D\\boxtimes\\mathcal{L}_0^D$ on the mirror, and can be dropped.\nThis leads to the formula:\n\n\\begin{equation}\\label{usub}\n\\sum_{l\\in \\mathbb{Z}^n\/D^{T}\\mathbb{Z}^n}s_{k,l}\n=\\sqrt{2det(Im\\ \\tau D)}\\vartheta_{D,k}(u)\\bar{\\vartheta}_{D,0}(v) \\in H^0(E_\\tau\\times E_{-\\bar{\\tau}}; \\mathcal{L}^{D}\\boxtimes \\mathcal{L}^{D}_0)\n\\end{equation}\nBy evaluating $v$ at $0$, we recover (up to a scaling factor) $\\vartheta_{D,k}(u)$ up to a constant factor which corresponds to the Floer product on the original torus $T$.\n\n\n\n\\section{The ``u-part\" Floer cohomology $HF_u(\\boldsymbol{L},\\,\\boldsymbol{L}')$}\n\nGiven two Lagrangian branes $L$ and $L'$ in $(T,\\,\\omega)$ with their lifts $\\boldsymbol{L}$ and $\\boldsymbol{L}'$, and assuming that $\\mathcal{L},\\,\\mathcal{L'}$ are the mirror sheaves of $L$ and $L'$, we expect \n$$HF^*(\\boldsymbol{L},\\,\\boldsymbol{L}')\\cong Ext_{E}^*(\\mathcal{L},\\,\\mathcal{L}')\\boxtimes Ext_{\\bar{E}}^*(\\mathcal{L}_0,\\,\\mathcal{L}'_0).$$\nThe goal of this chapter is to define a subspace $HF_u(\\boldsymbol{L},\\,\\boldsymbol{L}')$ of $HF^*(\\boldsymbol{L},\\,\\boldsymbol{L}')$ which is isomorphic to $Ext_{E}^*(\\mathcal{L},\\,\\mathcal{L}')$, and thus to $HF(L,L')$.\n\\subsection{The ``u-part\" of $HF^*(\\boldsymbol{L},\\,\\boldsymbol{L})$}\nRecall that the twisted doubling torus $T\\times T^\\vee$ is equipped with a natural complex structure from Remark \\ref{AS2},\n\\begin{equation}\nJ=\\left( \\begin{array}{cc}\n\\omega^{-1}B & \\omega^{-1} \\\\\n-\\omega-B\\omega^{-1}B & -B\\omega^{-1}\n\\end{array}\\right)\n\\end{equation}\nwhich induces a complex structure on the cotangent bundle $T^*(T\\times T^\\vee)$, still denoted by $J$,\n\\begin{equation}\nJ=\\left( \\begin{array}{cc}\nB\\omega^{-1} & \\omega+B\\omega^{-1}B \\\\\n-\\omega^{-1} & -\\omega^{-1}B\n\\end{array}\\right).\n\\end{equation}\n\nGiven a Lagrangian brane $(L,\\,\\nabla)$ in $(T,\\,\\omega)$ with its lift $(\\boldsymbol{L},\\,\\pi_T^*\\nabla)$, we can compare the first order deformation of the objects.\nA first order deformation of $(L,\\,\\nabla)$ is described by $(v;\\,\\alpha)$, where $v$ is a normal vector of $L$, and $\\alpha$ is a real $1$-form. $(v;\\,\\alpha)$ maps to \n$[\\iota_v (\\omega-iB)+i\\alpha]\\in H^1(L;\\,\\mathbb{C})\\cong HF(L,L)$. \nThe corresponding first order deformation of $(\\boldsymbol{L},\\,\\pi_T^*\\nabla)$ is given by \n$(v,\\,-\\tilde{\\alpha};\\,\\alpha,\\,0)$,where $\\tilde{\\alpha}$ is the image of $\\alpha$ under the identification $T^*T\\cong TT^\\vee$. \nThis maps to \n\\begin{equation}\n\\iota_{v-\\tilde{\\alpha}}(\\frac{1}{2}\n\\left(\\begin{array}{cc}\n\\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n-\\omega^{-1}B & -\\omega^{-1}\n\\end{array}\\right)\n-i\\sigma_0)+i\\alpha=\\frac{1}{2}(1+iJ)(\\iota_v (\\omega-iB)+i\\alpha)\n\\in H^1(\\boldsymbol{L};\\,\\mathbb{C}).\n\\end{equation}\n\nNote that the full first order deformation space of $(\\boldsymbol{L},\\,\\pi_T^*\\nabla)$ coincides with $H^1(\\boldsymbol{L};\\, \\mathbb{C})\\cong HF(\\boldsymbol{L},\\boldsymbol{L})$, and the first order deformations coming from lifts are exactly the $(0,1)$ part of $H^1(\\boldsymbol{L},\\,\\mathbb{C})\\cong HF(\\boldsymbol{L},\\,\\boldsymbol{L})$ with respect to $J$ mentioned above. Then \n\\begin{equation}\nHF^*(L,\\,L)\\cong H^*(L;\\,\\mathbb{C})=\\bigwedge H^1(L;\\,\\mathbb{C})\\cong \\bigwedge H^{0,\\,1}_{J}(\\boldsymbol{L})=H^{0,\\,*}_{J}(\\boldsymbol{L})\n\\end{equation}.\n\\begin{definition}\n\t$HF(\\boldsymbol{L},\\,\\boldsymbol{L})_u:=\\, H^{0,\\,*}_{J}(\\boldsymbol{L})\\subset H^*(\\boldsymbol{L};\\,\\mathbb{C})=HF(\\boldsymbol{L},\\,\\boldsymbol{L})$.\n\\end{definition}\n\n\\subsection{The case of transversal intersection $L\\cap L'$}\nAs in Chapter 4, consider a pair of Lagrangian branes $L,\\,L'$, which are mirror to a pair of line bundles $\\mathcal{L},\\,\\mathcal{L'}$. Let $\\boldsymbol{L},\\,\\boldsymbol{L}'$ be the lifts in the double torus. Suppose $\\mathcal{L'}\\otimes \\mathcal{L}^{-1}$ is ample, then the subspace of $HF^*(\\boldsymbol{L},\\,\\boldsymbol{L'})$ spanned by \n$$\\displaystyle \\sum_{l\\in \\mathbb{Z}^n\/(D'-D)^{T}\\mathbb{Z}^n}s_{k,\\,l}=\\sqrt{2det(Im\\ \\tau (D'-D))}\\vartheta_{D'-D,k}(u)\\bar{\\vartheta}_{D'-D,0}(v)$$\nis isomorphic to $HF^*(L,L')$ by evaluation at $v=0$. \nWe define the ``u-part\" cohomology for transversal intersections as follows.\n\n\\begin{definition}\n Suppose $L=\\{\\theta=-Dr+c\\} $ and $ L'=\\{\\theta=-D'r+c'\\}$ intersect transversally, assume $\\boldsymbol{L}\\cap \\boldsymbol{L}'=\\{s_{k,l}\\}$, \n where $k\\in \\mathbb{Z}^n\/(D'-D)\\mathbb{Z}^n,\\,\n l \\in \\mathbb{Z}^n\/(D'-D)^{T}\\mathbb{Z}^n$. Then we define\n\n\\begin{equation}\n HF_u(\\boldsymbol{L},\\boldsymbol{L}'):=\n span\\Big\\{\\sum_{l\\in \\mathbb{Z}^n\/(D'-D)^{T}\\mathbb{Z}^n}s_{k,l}\\Big\\}_{k\\in \\mathbb{Z}^n\/(D'-D)\\mathbb{Z}^n}\n \\subset HF^*(\\boldsymbol{L},\\boldsymbol{L}').\n \\end{equation}\n\\end{definition}\n\\begin{definition}\n Let \n \\begin{equation}\n \\begin{aligned}\n \\Pi_T: HF^*(\\boldsymbol{L},\\boldsymbol{L}') &\\rightarrow \n HF_u^*(\\boldsymbol{L},\\boldsymbol{L}') \\\\\n s_j\\otimes s_h' &\\mapsto s_j\\otimes (\\frac{1}{det(D'-D)}\\sum_l s_l').\n \\end{aligned}\n \\end{equation}\n Define the product structure to be\n \\begin{equation}\n \\begin{aligned}\n \\mu^2_u: HF^*_u(\\boldsymbol{L}',\\,\\boldsymbol{L}\")&\\otimes HF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}') \\rightarrow HF_u(\\boldsymbol{L},\\boldsymbol{L}\") \\\\\n x&\\otimes y \\longmapsto \\Pi_T(\\mu^2(x,y)),\n \\end{aligned}\n \\end{equation}\n where $\\mu^2$ is the usual Floer product in $HF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}')$.\n\\end{definition}\n\nWith these definitions, we state the main theorem:\n\\begin{theorem}\n\tFor a pair of Lagrangian branes $L,\\,L'$ which are mirror to a pair of line bundles $\\mathcal{L},\\,\\mathcal{L'}$, let $\\boldsymbol{L},\\,\\boldsymbol{L}'$ be the lifts in the twisted doubling torus. Suppose $\\mathcal{L'}\\otimes \\mathcal{L}^{-1}$ is ample. Then the ``u-part\" Floer cohomology $HF_u^*(\\boldsymbol{L},\\boldsymbol{L}')$ is isomorphic to $HF^*(L,\\,L')$. And for two such pairs $L,L'$ and $L',L''$, the following diagram commutes\n\t\\begin{equation}\\label{uspscomm}\n\t\\begin{tikzcd}\n\t{HF^*(L',\\,L'')\\otimes HF^*(L,\\,L')} \\arrow[rr] \\arrow[d, \"\\cong\"] & & {HF^*(L,\\,L\")} \\arrow[d, \"\\cong\"] \\\\\n\tHF^*_u(\\boldsymbol{L}',\\,\\boldsymbol{L}'')\\otimes HF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}') \\arrow[rr] & & HF_u(\\boldsymbol{L},\\boldsymbol{L}'').\n\t\\end{tikzcd}\n\t\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nThe isomorphism $HF_u^*(\\boldsymbol{L},\\boldsymbol{L}')\\cong HF^*(L,\\,L')$ is given by\n\\begin{equation}\ns_k \\mapsto \n\\sum_{l\\in \\mathbb{Z}^n\/(D'-D)^{T}\\mathbb{Z}^n}s_{k,l}\/(\\sqrt{2det(Im\\ \\tau (D'-D))}\\bar{\\vartheta}_{D'-D,0}(0))\n\\end{equation}\nThen the commutative diagram follows from \\eqref{usub} and formulas for $\\vartheta$-functions.\n\\end{proof}\n\n\\subsection{Proposal for general case}\n\nIn the general case when $L$ and $L'$ intersect non transversally, we expect there is still a ``u-part\" subspace $HF_u(\\boldsymbol{L},\\boldsymbol{L}')$ isomorphic to $HF(L,L)$, and a similar commutative diagram holds.\nIn a simple case, assume $T,\\,L,\\,L'$ admit a simultaneous decomposition $T=T_1\\times T_2,\\, L=L_1\\times L_2,\\, L'=L_1'\\times L_2'$, such that $L_1,\\,L_2$ (resp. $L_1',\\,L_2'$) are Lagrangian submanifolds of $T_1$ (resp. $T_2$), and assume $L_1=L_1'$, while $L_2$ intersects $L_2'$ transversally.\nThen the twisted doubling torus and lifts of $L,\\,L'$ also admit product structures, and \n\\begin{equation}\n\\begin{aligned}\nHF^*(L,L')& \\cong HF^*(L_1,L_1')\\otimes HF(L_2,L_2') \\\\\n&\\cong HF_u^*(\\boldsymbol{L}_1,\\boldsymbol{L}_1)\\otimes HF_u^*(\\boldsymbol{L}_2,\\boldsymbol{L}_2') \\\\\n&\\cong H^{0,*}(\\boldsymbol{L}_1)\\otimes HF_u(\\boldsymbol{L}_2,\\boldsymbol{L}_2')\n\\end{aligned}\n\\end{equation}\nInspired by this, and combining the definitions of ``u-part\" Floer cohomology in the previous sections, we make the following tentative definition.\n\n\\begin{definition}\n For a pair of Lagrangian branes $L$ and $L'$, and their lift $\\boldsymbol{L}$ and $\\boldsymbol{L}'$, let\n\\begin{equation*} \nHF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}') :=\nG- \\textbf{invariant part of } H^{0,*}(\\boldsymbol{L}\\cap \\boldsymbol{L}')\n\\end{equation*} \nWhere $G$ is the discrete group of translations in the direction of $T^\\vee$ acting on $\\boldsymbol{L}\\cap \\boldsymbol{L}'$.\n\\end{definition}\n\n\\begin{conjecture}\nFor a pair of Lagrangian branes $L,\\,L'$ with their lifts $\\boldsymbol{L},\\,\\boldsymbol{L}'$ in the twisted doubling torus, the \"u-part\" Floer cohomology $HF_u^*(\\boldsymbol{L},\\boldsymbol{L}')$ is isomorphic to $HF^*(L,\\,L')$. And for two such pair $L,L'$ and $L',L\"$, the following diagram commutes:\n\t\\begin{equation}\\label{uspscomm}\n\t\\begin{tikzcd}\n\t{HF^*(L',\\,L\")\\otimes HF^*(L,\\,L')} \\arrow[rr] \\arrow[d, \"\\cong\"] & & {HF^*(L,\\,L\")} \\arrow[d, \"\\cong\"] \\\\\n\tHF^*_u(\\boldsymbol{L}',\\,\\boldsymbol{L}\")\\otimes HF^*_u(\\boldsymbol{L},\\,\\boldsymbol{L}') \\arrow[rr] & & HF_u(\\boldsymbol{L},\\boldsymbol{L}\").\n\t\\end{tikzcd}\n\t\\end{equation}\n\\end{conjecture}\n\n\n\n\\section{Equivalence of $T$ and $T^\\vee$ with B-field Twist}\n\nRecall that the dual torus of $(T=V\/\\Lambda,B+i\\omega)$ is $(T^\\vee=V^\\vee \/\\Lambda^\\vee,\\ (B+i\\omega)^{-1} )$ under assumption \\ref{assump}. Explicitly, $$(B+i\\omega)^{-1}=(\\omega+B\\omega^{-1}B)^{-1}B\\omega^{-1}-i(\\omega+B\\omega^{-1}B)^{-1}.$$\n\n\\begin{example}\nLet $T=\\mathbb{R}^2\/\\mathbb{Z}^2$ with $B+i\\omega=\\tau dr\\wedge d\\theta$, $\\tau=b+ia,\\ a>0$, then its dual torus is $T^\\vee=\\mathbb{R}^2\/\\mathbb{Z}^2$ with $(B+i\\omega)^{-1}=-\\tau^{-1}d\\hat{r}\\wedge d\\hat{\\theta}$. They are non symplectomorphic tori for generic $\\tau$. However, their mirror manifolds are isomorphic as complex manifold.\n\\begin{equation*} \n\\begin{tikzcd}\n\\mathbb{C}\/\\mathbb{Z}+\\tau \\mathbb{Z} & & \\mathbb{C}\/\\mathbb{Z}-\\tau^{-1}\\mathbb{Z}. \\arrow[ll, \"\\times \\tau\"']\n\\end{tikzcd}\n\\end{equation*}\nThis implies that their Fukaya categories are equivalent.\n\\end{example}\nThe phenomenon that dual tori have equivalent Fukaya categories is not obvious without referring to Homological Mirror Symmetry. \nHowever, the twisted doubling tori of $(T,B+i\\omega)$ and $(T^\\vee,\\ (B+i\\omega)^{-1} )$ are the same up to a B-field twist.\nExplicitly, their twisted doubling tori are\n\\begin{equation*}\n\t(T\\times T^\\vee,\\quad \\Omega=\\frac{1}{2}\n\t\\left(\\begin{array}{cc}\n\t \\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n\t\t -\\omega^{-1}B & -\\omega^{-1}\n\t\t\\end{array}\\right),\n\t\t\\quad\n\t\t \\sigma_0=\\sum_j \\frac{1}{2} dx_j\\wedge d\\hat{x}_j)\t\n\t\\end{equation*}\n\tand\n\t\\begin{equation*}\n\t(T\\times T^\\vee,\\quad \\Omega=\\frac{1}{2}\n\t\\left(\\begin{array}{cc}\n\t \\omega+B\\omega^{-1}B & B\\omega^{-1}\\\\\n\t\t -\\omega^{-1}B & -\\omega^{-1}\n\t\t\\end{array}\\right),\n\t\t\\quad\n\t\t -\\sigma_0=-\\sum_j \\frac{1}{2} dx_j\\wedge d\\hat{x}_j)\t\n\t\\end{equation*}\n\tThe difference of B-field is an integral class in $H^2(T\\times T^\\vee; \\mathbb{R})$. Let\n\t\\begin{equation}\n\t\\nabla_0=d-2\\pi i (rd\\hat{r}+\\theta d\\hat{\\theta})\n\t\\end{equation}\n\tbe a $U(1)$ connection on $T\\times T^\\vee$ with curvature $2\\sigma_0$. Then we have an equivalence of Fukaya categories of the two doubling tori by a B-twist:\n\t\\begin{equation}\n\t\\begin{aligned} \n\tFuk(T\\times T^\\vee,\\Omega,\\sigma_0) &\\rightarrow Fuk(T\\times T^\\vee,\\Omega,-\\sigma_0) \\\\\n\t(L,\\nabla) &\\mapsto (L,\\nabla\\otimes \\nabla_0|_L)\n\t\\end{aligned} \n\t\\end{equation}\n\\begin{conjecture}\n\tThe Fukaya category of a torus $(T,B+i\\omega)$ is equivalent to the Fukaya category of its dual $(T^\\vee,(B+i\\omega)^{-1})$.\n\\end{conjecture}\n\n\n\n\\begin{remark}\n\t The isomorphism between morphism spaces in $(T,B+i\\omega)$ and $(T^\\vee,(B+i\\omega)^{-1})$ is not directly given by the above equivalence of doubling tori. It also involves the projection map associated with $HF(\\boldsymbol{C},\\boldsymbol{C}')_u$. \\\\\n\\end{remark}\n\n\n\n\n\\bibliographystyle{amsplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\n\nThis paper will address the question of existence of \\textit{conserved charges on null hypersurfaces} (and their associated \\textit{conservation laws}) for the wave equation\n\\begin{equation}\n\\Box_{g}\\psi=0\n\\label{wave}\n\\end{equation}\non a general four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. \n\nThe simplest example of such a conserved charge arises in Minkowski space and is given by \\[char\\left[\\psi;S_{v}\\right]=\\int_{S_{v}}\\frac{1}{r^{2}}\\partial_{u}(r\\psi) \\]\n which for all solutions of \\eqref{wave} satisfies the conservation law \n\\[\\partial_{v}\\Big(char\\left[\\psi;S_{v}\\right]\\Big)=0.\\]\n Here $u,v$ are standard null coordinates and $S_{v}$ are the spherical sections of the standard null cones $\\left\\{u=c\\right\\}$. Another example is the recently discovered conservation laws on the degenerate event horizons of extremal black hole spacetimes (see \\cite{aretakis4, hj2012, murata2012}). A third (limiting) example are the celebrated \\textit{Newman--Penrose constants} which are conserved along null infinity in any asymptotically flat spacetime (see \\cite{np1,np2}).\n\nIn the present paper, we define a general notion of conserved charges (see Section \\ref{sec:ConservationLawsForTheWaveEquations}) encompassing all the above examples and give a characterization of null hypersurfaces admitting such charges in terms of the kernel of an elliptic operator (defined for the first time here and in our companion paper \\cite{aretakiselliptic}).\\footnote{It will follow in particular from this characterization that generic Lorentzian manifolds do not admit such charges.} In fact,\nwe show that the only information that can be propagated by all solutions to the wave equation along null hypersurfaces is given precisely by these conserved charges. For this, we initiate the general study of \\textit{gluing constructions for the characteristic initial value problem} (see Section \\ref{sec:TheCharacteristicInitialValueProblem}) and we show that the only obstruction to gluing along a null hypersurface $\\mathcal{H}$ is the existence of conserved charges (in our sense) on $\\mathcal{H}$. \n\nPart of the importance of the conservation laws on degenerate horizons referred to above lies in their role in the instability properties of the wave equation on extremal black holes (see Section \\ref{sec:Remarks}). This result led to the so-called ``horizon instability of extremal black holes''. The present general study may thus shed light on new aspects of the global evolution of the wave equation on more general backgrounds. \n\n\n\n\nThe statement of the main result can be found in Section \\ref{sec:TheMainResultxx}. Our proof introduces a new method which we hope will be relevant for applications to other linear and non-linear equations such as the Einstein equations. \n\n\n\\subsection{Conservation laws for the wave equation}\n\\label{sec:ConservationLawsForTheWaveEquations}\n\nWe first present some basic geometric definitions that will be useful for defining the notion of conservation laws on null hypersurfaces. For more details about the geometric setting see Section \\ref{sec:TheDoubleNullFoliation}; our notation follows \\cite{DC09,christab}. \n\n\\paragraph{Null foliations\\medskip \\\\}\n\\label{sec:nullfoliations}\n\n\nLet $\\mathcal{H}$ be a regular null hypersurface of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. A foliation $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$ of $\\mathcal{H}$, that is a collection of sections $S_{v}$ which vary smoothly in $v$ such that $\\cup_{v}S_{v}=\\mathcal{H}$, can be uniquely determined by the choice of one section $S_{0}$, the choice of a smooth function $\\Omega$ on $\\mathcal{H}$ and the choice of a null geodesic vector field $L_{geod}$ tangential to the null generators of $\\mathcal{H}$ and such that \n\\[\\nabla_{L_{geod}}L_{geod}=0.\\]\nIndeed, if we define the vector field \n\\[L=\\Omega^{2}\\cdot L_{geod}\\]\non $\\mathcal{H}$ and consider the affine parameter $v$ of $L$ such that \n\\[Lv=1, \\text{ with }v=0 \\text{ on } S_{0},\\]\nthen the level sets $S_{v}$ of $v$ on $\\mathcal{H}$ are precisely the leaves of the foliation $\\mathcal{S}$. We use the notation\n\\begin{equation}\n\\mathcal{S}=\\Big\\langle S_{0},L_{geod}, \\Omega\\Big\\rangle.\n\\label{foliation}\n\\end{equation}\n \\begin{figure}[H]\n \\centering\n\t\t\\includegraphics[scale=0.06]{glue1.png}\n\t\\label{fig:glue1}\n\\end{figure}\nWe will henceforth restrict to the case that all sections $S_{v}$ above are diffeomorphic (via a diffeomorphism $\\Phi$) to the 2-sphere.\\footnote{ Topologies with higher genus can be treated analogously. Our argument heavily relies on the compactness of the sections and hence the non-compact case remains an open problem.} The flow of $L$ on $\\mathcal{H}$ provides a diffeomorphism $\\Phi_{v}$ between the sections $S_{v}$ and $S_{0}$. In addition to the induced metric on $S_{v}$, which we will denote by $\\mbox{$g \\mkern-8.8mu \/$\\,}$, we can also equip all sections with the standard metric on the unit sphere $\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}$ (via $\\Phi$) such that it is invariant under the flow of $L$. The volume form on $S_{v}$ with respect to $\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}$ will be denoted by $d\\mu_{_{\\mathbb{S}^{2}}}$.\n\nGiven any section $S_{v}$, there is a unique metric $\\hat{g}$ which is conformal to the induced metric $\\mbox{$g \\mkern-8.8mu \/$\\,}$ such that the volume form $d\\mu_{_{\\hat{g}}}$ with respect to $\\hat{g}$ and the volume form $d\\mu_{_{\\mathbb{S}^{2}}}$ with respect to $\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}$ are equal:\n\\[d\\mu_{_{\\hat{g}}}=d\\mu_{_{\\mathbb{S}^{2}}}. \\]We denote by $\\phi$ the conformal factor:\n\\begin{equation}\n\\mbox{$g \\mkern-8.8mu \/$\\,}=\\phi^{2}\\cdot \\hat{g}. \n\\label{eq:theconformalfactorintroduction}\n\\end{equation}\n\nFurthermore, given a foliation $\\mathcal{S}$ we denote by $Y^{\\mathcal{S}}$ the unique null vector field which is normal to the sections $S_{v}$, conjugate to $\\mathcal{H}$ and normalized such that \n\\begin{equation}\ng\\big(L_{geod},Y^{\\mathcal{S}}\\big)=-1.\n\\label{Y}\n\\end{equation}\n \\begin{figure}[H]\n \\centering\n\t\t\\includegraphics[scale=0.06]{glue2.png}\n\t\\label{fig:glue2}\n\\end{figure}\nThe vector field $Y^{\\mathcal{S}}$ is transversal to $\\mathcal{H}$ and can be seen as the generator of an appropriately normalized ``retarded time'' $u$ such that $u=0$ on $\\mathcal{H}$. Specifically, one can construct an optical function $u$ such that $Y^{\\mathcal{S}}u=1$ on $\\mathcal{H}$ and the level sets of $u$ are ``outgoing'' null hypersurfaces $\\mathcal{H}_{u}$ (hence we assume here that $\\mathcal{H}$ is an ``outgoing'' null hypersurface). Note that in a similar fashion as above we can define the conformal factor $\\phi$ of the section $S_{u,v}$ which are the intersections of $\\mathcal{H}_{u}$ and the ``incoming'' null hypersurfaces $\\underline{\\mathcal{H}}_{v}$ generated by the null geodesics normal to $S_{v}$ and conjugate to $\\mathcal{H}$.\n \\begin{figure}[H]\n \\centering\n\t\t\\includegraphics[scale=0.07]{glue3.png}\n\t\\label{fig:glue3}\n\\end{figure}\n\n\\newpage \n\n\\paragraph{Conservation laws on $\\mathcal{H}$\\medskip \\\\}\n\\label{sec:conservation laws}\n\n\n\n\nConsider the linear space $\\mathcal{V}_{\\mathcal{H}}$ consisting of all smooth functions on $\\mathcal{H}$ which are constant along the null generators of $\\mathcal{H}$, i.e.\n\\begin{equation}\n\\mathcal{V}_{\\mathcal{H}}=\\Big\\{f\\in C^{\\infty}(\\mathcal{H})\\, :\\, Lf=0\\Big\\}.\n\\label{linearspace}\n\\end{equation}\nLet $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$ be a foliation of $\\mathcal{H}$ and let $Y^{\\mathcal{S}}$ be the vector field and $\\phi$ the conformal factor defined above. \nWe define the linear space $\\mathcal{W}^{\\mathcal{S}}$ to be the subspace of $\\mathcal{V}_{\\mathcal{H}}$ such that for all $\\Theta^{\\mathcal{S}}\\in \\mathcal{W}^{\\mathcal{S}}$ and for all solutions $\\psi$ to the wave equation \\eqref{wave} the integrals\n\\begin{equation}\n\\int_{S_{v}}Y^{\\mathcal{S}}\\big(\\phi\\cdot\\psi\\big)\\cdot\\Theta^{\\mathcal{S}} \\, d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{eq:integrals}\n\\end{equation}\nare conserved, i.e.~independent of $v$. That is,\n\\begin{equation}\n\\mathcal{W}^{\\mathcal{S}}=\\left\\{\\Theta^{\\mathcal{S}}\\in C^{\\infty}(\\mathcal{H})\\, :\\, L\\Theta^{\\mathcal{S}}=0, \\ \\partial_{v}\\left(\\int_{S_{v}}Y^{\\mathcal{S}}\\big(\\phi\\cdot\\psi\\big)\\cdot\\Theta^{\\mathcal{S}} \\, d\\mu_{_{\\mathbb{S}^{2}}}\\right)=0\\right\\}\\subset \\mathcal{V}_{\\mathcal{H}}. \n\\label{eq:}\n\\end{equation}\n\n\n We make the following definition:\n\\begin{definition} \\textbf{(Conservation laws on $\\mathcal{H}$)}:\nWe say that a null hypersurface $\\mathcal{H}$ admits (first order) conservation laws with respect to a foliation $\\mathcal{S}$ of $\\mathcal{H}$ if\n\\begin{equation}\n\\dim\\mathcal{W}^{\\mathcal{S}}\\geq 1.\n\\label{dimensionofw}\n\\end{equation} \nIf \\eqref{dimensionofw} holds then we will refer to the space $\\mathcal{W}^{\\mathcal{S}}$ and the number $\\dim \\mathcal{W}^{\\mathcal{S}}$ as the kernel and the dimension of the conservation laws, respectively. The integrals of the form \\eqref{eq:integrals} will be called conserved charges and will be denoted by $char\\big(S_{v}\\big)[\\psi; \\Theta^{\\mathcal{S}}]$.\n\\label{definitionconservationlaw}\n\\end{definition}\nA priori Definition \\ref{definitionconservationlaw} appears to be a very restrictive notion of conservation laws. However, as we shall show (see Theorem \\ref{theoremmainintro}), the conservation laws in the sense of Definition \\ref{definitionconservationlaw} are in fact the only type of ``first order'' conservation laws that a null hypersurface $\\mathcal{H}$ might admit.\n\n One could also define higher order conservation laws by considering higher derivatives of $\\phi\\cdot\\psi$ in \\eqref{eq:integrals}. In order to make our method clear, in the bulk of this paper we only consider first order conservation laws and for this reason we will simply refer to them as conservation laws (see however Section \\ref{sec:Genericity} where we consider higher order conservation laws for spherically symmetric geometries). \n\n \n\n\n\\paragraph{Examples \\\\}\n\\label{sec:examples}\n\nWe next consider three main examples of spacetimes admitting conserved charges for the wave equation.\n\n\\paragraph{\\small 1. Minkowski spacetime\\medskip \\\\}\n\\label{sec:MinkowskiSpacetimea}\n\n\\normalsize\n\n\nThe wave equation in double null coordinates $(u,v)$ on the Minkowski spacetime reads\n\\[\\partial_{v}\\partial_{u}(r\\psi)=\\frac{1}{2r^{2}}\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}(r\\psi), \\]\nwhere $\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}$ is the standard Laplacian on the unit sphere. Hence, \n\\[\\int_{S_{v}}\\partial_{v}\\partial_{u}(r\\psi)\\, d\\mu_{_{\\mathbb{S}^{2}}}=\\frac{1}{2r^{2}}\\int_{S_{v}}\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}(r\\psi)\\, d\\mu_{_{\\mathbb{S}^{2}}}=0\\]\nand thus the integral \n\\[\\int_{S_{v}}\\partial_{u}(r\\psi)\\, d\\mu_{_{\\mathbb{S}^{2}}}\\, \\]\nis conserved along the null hypersurfaces $\\left\\{u=c\\right\\}$. This conserved charge can be written in terms of Definition \\ref{definitionconservationlaw}; indeed, if we consider the foliation $\\mathcal{S}=\\left\\langle S_{0}, L_{geod}=\\partial_{v},\\Omega=1\\right\\rangle$ of $\\mathcal{H}$ then $\\phi=r$, $Y^{\\mathcal{S}}=\\partial_{u}$ and $\\left\\langle 1\\right\\rangle\\subset \\mathcal{W}^{\\mathcal{S}}$ and hence $\\dim \\mathcal{W}^{\\mathcal{S}}\\geq 1$. \n\n\n\\paragraph{\\small 2. Extremal black holes\\medskip \\\\}\n\\label{sec:MinkowskiSpacetime4}\n\n\\normalsize\n\nConsider the coordinate vector fields\n\\[T=\\partial_{v}, \\ \\ \\ \\ \\ R=\\partial_{r}, \\ \\ \\ \\ \\ \\Phi=\\partial_{\\phi^{*}} \\]\nwith respect to the ingoing Eddington--Finkelstein coordinates $(v,r,\\theta,\\phi^{*})$ on an extremal Kerr black hole with mass parameter equal to $M$. Then, the quantity\n\\begin{equation}\n\\int_{S_{v}} \\left[R\\psi+\\frac{\\sin^{2}\\theta}{4}\\cdot T\\psi+ \\frac{1}{2M}\\cdot\\psi\\right]\\,d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{eextremalcharge}\n\\end{equation}\nis conserved along the horizon $\\mathcal{H}=\\left\\{r=M\\right\\}$, i.e.~it is independent of $v$. This conservation law was first found in \\cite{aretakis4} and was then generalized to all extremal black holes by Lucietti and Reall \\cite{hj2012} and Murata \\cite{murata2012}. \n\nThe above conserved charge can be written in terms of Definition \\ref{definitionconservationlaw} as follows: Consider the foliation $\\mathcal{S}=\\left\\langle S_{0}, L_{geod}=\\partial_{v},\\Omega=1\\right\\rangle$ of $\\mathcal{H}$. Then \\[Y^{\\mathcal{S}}=\\frac{1}{\\frac{\\sin^{2}\\theta}{2}-1}\\cdot\\left[R+\\frac{\\sin^{2}\\theta}{4}\\cdot T+\\frac{3+\\cos^{2}\\theta}{8M}\\cdot\\Phi\\right]\\] and \n\\[\\left.\\phi\\right|_{\\mathcal{H}}=\\sqrt{2}\\cdot M,\\ \\ \\ \\ \\ \\left.Y^{\\mathcal{S}}\\phi\\right|_{\\mathcal{H}}=\\frac{\\sqrt{2}}{2}\\cdot\\frac{1}{\\frac{\\sin^{2}\\theta}{2}-1}, \\ \\ \\ \\ \\ \\left\\langle \n\\frac{\\sin^{2}\\theta}{2}-1\\right\\rangle\\subset \\mathcal{W}^{\\mathcal{S}} \\]\nand hence $\\dim \\mathcal{W}^{\\mathcal{S}}\\geq 1$. Note that since the integral curves of $\\Phi$ are closed, the $\\Phi$-derivative drops out from \\eqref{eextremalcharge}. \n\nFor more results based on this conservation law see \\ref{sec:Remarks}. \n\n\n\\paragraph{\\small 3. Null infinity of asymptotically flat spacetimes \\medskip \\\\}\n\\label{sec:MinkowskiSpacetime}\n\n\\normalsize\n\n Given sufficient smoothness for $\\psi$ at the null infinity $\\mathcal{I}^{+}$ of an asymptotically flat spacetime we can write\n\\[\\psi\\big(u,r,\\theta^{1},\\theta^{2}\\big)=\\frac{\\alpha_{1}\\big(u,\\theta^{1},\\theta^{2}\\big)}{r}+\\frac{\\alpha_{2}\\big(u,\\theta^{1},\\theta^{2}\\big)}{r^{2}}+O\\left(\\frac{1}{r^{3}}\\right)\\]\nwith respect to outgoing Eddington--Finkelstein coordinates $\\big(u,r,\\theta^{1},\\theta^{2}\\big)$. Here we identity $\\mathcal{I}^{+}=\\left\\{r=\\infty\\right\\}$. If $\\psi$ is a solution to the wave equation, then the quantity\n\\begin{equation}\n\\lim_{r\\rightarrow+\\infty}\\int_{S_{u}}r^{2}\\cdot\\partial_{r}(r\\psi)\\, d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{eq:npc}\n\\end{equation}\ndoes not depend on $u$. This charge as well as other charges involving higher order derivatives were found by Newman and Penrose \\cite{np1,np2} (see also \\cite{npexton}) and are known as Newman--Penrose constants. \n\n\n\nThe origin of these peculiar constants has been the object of intense study. In particular, we mention the work of Goldberg \\cite{goldberg1,goldberg2} who showed that these constants do not arise from non-trivial transformation laws. The same author was able to rederive these constants in the flat case by using Green's theorem in appropriate regions in conjunction with the fundamental solution to the wave equation. See also the related work by Robinson \\cite{robinson}. Further work on the Newman--Penrose constants can be found in \\cite{chrugrav,goldberg3,pressnp,valientenp1, valientenp2,valientenp3} and references there-in. A nice geometric relation of the the Newman--Penrose constants and the charges at the event horizon of extremal Kerr was given by Bizon and Friedrich \\cite{bizon2012} and independently by Lucietti et al \\cite{hm2012}.\n\n\nThe Newman--Penrose constants can be seen as a limiting example of the conserved charges given by Definition \\ref{definitionconservationlaw} as follows: Let $\\mathcal{I}_{S_{0}}$ be an incoming null hypersurface and $\\mathcal{S}=\\left\\langle S_{0}, L_{geod}, \\Omega=1\\right\\rangle$ be a foliation of it such that $\\frac{1}{2}\\big(L_{geod}+Y^{\\mathcal{S}}\\big)\\left.\\!\\right|_{S_{0}}$ is the (unit timelike) binormal of $S_{0}$ (see Section \\ref{sec:TheNullInfinityMathcalI}). Let also $r=\\sqrt{A\/4\\pi}$ be the area-radius function of the sections of $\\mathcal{S}$ on $\\mathcal{I}_{S_{0}}$, where $A$ is the area of the sections. Then $\\mathcal{I}_{S_{0}}\\rightarrow \\mathcal{I}$ as $r\\big(S_{0}\\big)\\rightarrow+\\infty$ and the Newman--Penrose constants can be retrieved in the limit as $r\\rightarrow +\\infty$ by the conservation law of Definition \\ref{definitionconservationlaw} if we take $Y^{\\mathcal{S}}$ to be the limit of $Y^{\\mathcal{S}}\\left.\\right|_{\\mathcal{I}_{S_{0}}}$ and $\\Theta^{S}$ to be the limit of $r^{2}\\left.\\right|_{\\mathcal{I}_{S_{0}}}$ as $\\mathcal{I}_{S_{0}}\\rightarrow\\mathcal{I}$ .\n\nOur general theory (see Theorem \\ref{theo3}) will in particular show that the conserved charge \\eqref{eq:npc} is the \\textbf{only} non-trivial conserved charge along the null infinity $\\mathcal{I}$ which involves the 1-jet of $\\psi$.\n\n\n\n\n\n\n\n\n\n\\subsection{The characteristic gluing problem}\n\\label{sec:TheCharacteristicInitialValueProblem}\n\n\nThe characteristic gluing problem for the wave equation provides a means to formally show that Definition \\ref{definitionconservationlaw} is the right notion of conservation laws on null hypersurfaces. We introduce this problem below. \n\n \nLet $(\\mathcal{M},g)$ be a four-dimensional Lorentzian manifold and $\\mathcal{H}, \\underline{\\mathcal{H}}$ be two regular null hypersurfaces intersecting at a two dimensional sphere $S_{0}$. Characteristic initial data for the wave equation \\eqref{wave} correspond to prescribing the restriction of $\\psi$ on the union $\\mathcal{H}\\cup \\underline{\\mathcal{H}}$. \nIn fact, given smooth data at $\\mathcal{A},\\underline{\\mathcal{A}}$, as depicted below, there is a unique smooth solution to the wave equation in the domain of dependence $\\mathcal{R}$, depicted schematically below:\n\n \\begin{figure}[H]\n \\centering\n\t\t\\includegraphics[scale=0.09]{picture01.png}\n\t\\label{fig:pfsdkfjapwoeijwe45}\n\\end{figure}\n\n\nThe problem of gluing constructions which we wish to formulate is the following: Consider a null hypersurface $\\mathcal{H}$ and two conjugate null hypersurfaces $\\underline{\\mathcal{H}}_{0}$ and $\\underline{\\mathcal{H}}_{1}$ intersecting $\\mathcal{H}$ at the two-dimensional spheres $S_{0}$ and $S_{1}$. We prescribe initial data for the wave equation \\eqref{wave} on the hypersurfaces $\\mathcal{A}_{0},\\underline{\\mathcal{A}}_{0}$ and $\\mathcal{A}_{1},\\underline{\\mathcal{A}}_{1}$ depicted in the figure below\n\n \\begin{figure}[H]\n \\centering\n\t\t\\includegraphics[scale=0.129]{picture02.png}\n\t\\label{figfdsp45picutre02}\n\\end{figure}\n\n\\noindent and we want to extend the data on the truncated hypersurface $\\mathcal{G}$ such that there is a smooth solution $\\psi$ to the wave equation in the region $\\mathcal{R}_{0}\\cup\\mathcal{R}_{\\mathcal{G}}\\cup\\mathcal{R}_{1}$ such that $\\left.\\psi\\right|_{\\underline{A}_{0}\\cup\\underline{A}_{1}}$ and $\\left.\\psi\\right|_{\\mathcal{A}_{0}\\cup\\mathcal{A}_{1}}$ coincide with the prescribed data. \n\nIn this paper we address in fact a weaker version of the above gluing problem, which is however sufficient for the complete classification of all null hypersurfaces admitting conservation laws. We make the following definition\n\\begin{definition} We shall say that ``we can perform first order gluing along $\\mathcal{H}$'' of the characteristic data $(\\underline{\\mathcal{A}}_{0},\\mathcal{A}_{0})$, $(\\underline{\\mathcal{A}}_{1},\\mathcal{A}_{1})$ as defined above if we can smoothly extend the data in $\\mathcal{G}$ such that the arising solutions\n\n\\begin{itemize}\n\t\\item $\\psi_{0}$ with data given on $\\underline{\\mathcal{A}}_{0},\\mathcal{A}_{0}\\cup\\mathcal{G}$, and \n\t\\item $\\psi_{1}$ with data given on $\\underline{\\mathcal{A}}_{1},\\mathcal{A}_{1}$\n\\end{itemize} agree at $S_{1}$ to all orders tangential to $\\mathcal{H}$ and up to first order in directions transversal to $\\mathcal{H}$; that is, $\\psi_{0}=\\psi_{1}$ at $S_{1}$ to all orders tangential to $\\mathcal{H}$ and $Y\\psi_{0}=Y\\psi_{1}$ at $S_{1}$, where $Y$ is a smooth vector field transversal to $\\mathcal{H}$. \n\\label{firstordergluingdefinition}\n\\end{definition}\n\n\n \\begin{figure}[H]\n \\centering\n\t\t\\includegraphics[scale=0.1]{picture03.png}\n\t\\label{figfdsp45picutre03}\n\\end{figure}\n\nIf $\\psi$ solves the wave equation then the transversal derivative $Y\\psi$ on $\\mathcal{H}$ is completely determined by the data $\\left.\\psi\\right|_{\\mathcal{H}}$ on $\\mathcal{H}$ and the transversal derivative $Y\\psi$ at a section $S$ of $\\mathcal{H}$. For this reason, it is convenient to ``forget'' about the incoming null hypersurfaces $\\underline{\\mathcal{H}}_{1},\\underline{\\mathcal{H}}_{2}$ and hence just ``keep'' the following data\n\\[\\psi\\left.\\right|_{\\mathcal{A}_{0}}, \\ \\ \\ Y\\psi\\left.\\right|_{S_{0}}\\]\nand \n\\[\\psi\\left.\\right|_{\\mathcal{A}_{1}}, \\ \\ \\ Y\\psi\\left.\\right|_{S_{1}}.\\]\nIn fact,\\textit{ we can simply think of the data as given at the two spheres $S_{0},S_{1}$ as follows\n\\[ \\text{Data}(S_{0})=\\left\\{ Y\\psi\\left.\\right|_{S_{0}}, \\ L^{n}\\psi\\left.\\right|_{S_{0}},\\, n\\geq 0\\right\\}\\]\nand \n\\[ \\text{Data}(S_{1})=\\left\\{ Y\\psi\\left.\\right|_{S_{1}}, \\ L^{n}\\psi\\left.\\right|_{S_{1}},\\, n\\geq 0\\right\\},\\]\nwhere $L$ is tangential to the null generator of $\\mathcal{H}$. \nThen, our problem is to smoothly extend $\\psi$ on $\\mathcal{H}$ between $S_{0}$ and $S_{1}$ such that the transversal derivative $Y\\psi$ is continuous on $\\mathcal{H}\\cap\\left\\{0\\leq v\\leq 1\\right\\}$ (and hence such that $\\psi$ is $C^{1}$ on $\\mathcal{H}\\cap\\left\\{0\\leq v\\leq 1\\right\\}$). \n}\n\n\n\\medskip\n\n\nNote that the $C^{k}$ case with $k>1$ (where one needs to ``glue'' transversal derivatives up to the $k$'th order) is addressed here, for simplicity, only in the spherically symmetric case; however the general case can be addressed by an amalgamation of the methods presented in Sections \\ref{sec:TheGeneralCase} and \\ref{sec:Genericity}. \n\n\n\nLet $S_{0}$ and $S_{1}$ be two leaves of a foliation $\\mathcal{S}$ of $\\mathcal{H}$. Clearly, if $\\mathcal{H}$ admits non-trivial conservation laws with respect to the foliation $\\mathcal{S}$ in the sense of Definition \\ref{definitionconservationlaw}, then gluing constructions of data at $S_{0}$ and $S_{1}$ are not always possible since the prescribed charges at $S_{0}$ and $S_{1}$ may not coincide. However, it is not a priori obvious if such charges are the only obstruction to gluing of characteristic data. On the other hand, if we can show that we can always glue to first order characteristic data on $\\mathcal{H}$ then it immediately follows that $\\mathcal{H}$ does not admit any non-trivial charges. \n\n\n\\subsection{The main theorems}\n\\label{sec:TheMainResultxx}\n\nThe next theorem 1) characterizes the conservation laws on null hypersurfaces, 2) derives their role in the characteristic gluing problem, 3) provides necessary and sufficient conditions for their existence in terms of the kernel of the elliptic operator $\\mathcal{O}^{\\mathcal{S}}$ defined by \\eqref{adjoint}, 4) uncovers their behavior under change of foliations and 5) establishes their non-genericity. \n\n\\begin{mytheo}\nLet $\\mathcal{H}$ be a regular, free from conjugate or focal points null hypersurface of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. Let also $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$ be a foliation of $\\mathcal{H}$, such that $S_{v}$ are diffeomorphic to $\\mathbb{S}^{2}$, and $\\mathcal{O}^{\\mathcal{S}}$ be the associated elliptic operator given by \\eqref{adjoint}. Then we have the following\n\\begin{itemize}\n\n\t\\item \\textbf{Characteristic gluing constructions and conservations laws:}\t\n\tOne can perform first order gluing constructions on $\\mathcal{H}$ in the sense of Definition \\ref{firstordergluingdefinition} for general characteristic data if and only if there are no first order conservation laws on $\\mathcal{H}$ in the sense of Definition \\ref{definitionconservationlaw}, i.e. $\\mathcal{W}^{\\mathcal{S}}=\\left\\{0\\right\\}$. If $\\mathcal{H}$ admits conservation laws, then we can glue characteristic data if and only if their associated charges are equal. \n\t\t\n\t\\item \\textbf{Classification of null hypersurfaces admitting conservation laws:} Consider the following linear space\n\t\\begin{equation}\n\t\\mathcal{U}^{\\mathcal{S}}=\\left\\{\\Theta^{\\mathcal{S}}\\in C^{\\infty}(\\mathcal{H})\\, :\\ L\\Theta^{\\mathcal{S}}=0, \\ \\mathcal{O}^{\\mathcal{S}}\\left(\\frac{1}{\\phi}\\cdot\\Theta^{\\mathcal{S}}\\right)=0 \\text{ on }\\mathcal{H} \\right\\}\\subset \\mathcal{V}_{\\mathcal{H}},\n\t\\label{newdef}\n\t\\end{equation}\n\twhere $\\phi$ denotes the conformal factor of the sections of $\\mathcal{S}$. Then, the null hypersurface $\\mathcal{H}$ admits first order conservation laws with respect to $\\mathcal{S}$ for the wave equation \\eqref{wave} in the sense of Definition \\ref{definitionconservationlaw} if and only if $\\mathcal{U}^{\\mathcal{S}}\\neq \\left\\{ 0\\right\\}$. In fact, the kernel of the conservation laws satisfies $\\mathcal{W}^{\\mathcal{S}}=\\mathcal{U}^{\\mathcal{S}}$.\n\n\t\n\t\n\t\\item \\textbf{Classification of conservation laws on null hypersurfaces:} No conserved (linear or non-linear) quantities (or, more generally, monotonic in $v$ quantities) involving the 1-jet of solutions to the wave equation exist on $\\mathcal{H}$ apart from the conservation laws given precisely by Definition \\ref{definitionconservationlaw}. Moreover, $\\mathcal{H}$ can only admit finitely many linearly independent conservation laws, i.e.~$\\dim \\mathcal{W}^{\\mathcal{S}}<\\infty$. \n\t\n\t\t\\item \\textbf{Conservation laws and change of foliation:} If $\\mathcal{H}$ admits a conservation law with respect to the foliation $\\mathcal{S}=\\left\\langle S_{0},L_{geod}, \\Omega\\right\\rangle$, then it also admits a conservation law with respect to any other foliation $\\mathcal{S}'=\\big\\langle S'_{0},L_{geod}',\\Omega'\\big\\rangle$. Specifically, the kernels $\\mathcal{W}^{\\mathcal{S}},\\mathcal{W}^{\\mathcal{S}'}$ satisfy $\\mathcal{W}^{\\mathcal{S}'}=f^{2}\\cdot\\mathcal{W}^{\\mathcal{S}}=\\left\\{f^{2}\\cdot \\Theta^{\\mathcal{S}},\\,\\Theta^{\\mathcal{S}}\\in \\mathcal{W}^{\\mathcal{S}}\\right\\}$, where $f\\in\\mathcal{V}_{\\hh}$ such that $L_{geod}'=f^{2}\\cdot L_{geod}$, and so $\\dim \\mathcal{W}^{\\mathcal{S}'}=\\dim \\mathcal{W}^{\\mathcal{S}}$. Moreover, the value of the conserved charges is independent of the choice of foliation. \n\n\n\\item \\textbf{Non-genericity of conservation laws:} A null hypersurface does not admit conservation laws for generic ambient metrics. The same result holds even if we restrict to spacetimes $(\\mathcal{M},g)$ satisfying the Einstein-vacuum equations. \n\\end{itemize}\n\\label{theoremmainintro}\n\\end{mytheo}\n\n\nTheorem \\ref{theoremmainintro} can be used to show that the event horizon of a subextremal black hole does not admit conservation laws and that the event horizon of an extemal black hole admits a unique conservation law: \n\n\n\\begin{mytheo}\n \\textbf{Conservation laws on extremal black holes:}\tThe event horizon ${\\mathcal{H}}^{+}$ of \nan extremal black hole satisfies $\\dim \\mathcal{U}^{\\mathcal{S}}_{{\\mathcal{H}}^{+}} =1$ and the unique corresponding conservation law coincides with the conservation law on extremal black holes found in \\cite{aretakis4, hj2012, murata2012}. On the other hand, any Killing horizon with positive surface gravity and negative transversal null mean curvature (such as the event horizon of any subextremal Kerr black hole) does not admit conservation laws.\n\n\\label{the2}\n\\end{mytheo}\n\t\n\tTheorem \\ref{theoremmainintro} applied in a limiting sense on null infinity of asymptotically flat spacetime recovers the Newman--Penrose constant and in fact shows that it is the \\textbf{only} conserved charge along null infinity:\n\t\n\t\\begin{mytheo}\n\t\\textbf{Conservation laws and the Newman--Penrose constant:}\n\tThe null infinity $\\mathcal{I}$ of an asympotically flat spacetime satisfies $\\dim\\mathcal{U}^{\\mathcal{S}}_{\\mathcal{I}}=1$ with respect to an appropriately rescaled operator $\\mathcal{O}_{\\mathcal{I}}^{\\ \\mathcal{S}}$, and the \\textbf{unique} corresponding conservation law coincides with the (first-order) Newman--Penrose constant on $\\mathcal{I}$.\n\t\\label{theo3}\n\t\\end{mytheo}\n\t\n\tFinally, the following theorem derives necessary and sufficient conditions for the existence of higher order conservation laws in the context of spherical symmetry.\n\n\t \\begin{mytheo}\n\t\\textbf{Higher order gluing constructions and conservation laws in spherical symmetry:} Let $(\\mathcal{M},g)$ be a spherically symmetric spacetime and $\\mathcal{H}$ be a spherically symmetric null hypersurface. Then, for all $k,l\\in\\mathbb{N}$ there is a unique expression $R_{k,l}$ which depends only on the geometry of $\\mathcal{H}$ such that if $R_{i,l}\\neq 0$ for $i=1,...,k-1$ and $R_{k,l}=0$ on $\\mathcal{H}$ then there is a conservation law involving the $k$-jet of solutions to the wave equation. The kernel of this conservation law consists of all the eigenfunctions of the standard spherical Laplacian $\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}$ which correspond to the eigenvalue $-l(l+1)$. If, on the other hand, we have that for all $l\\in \\mathbb{N}$ $R_{i,l}\\neq 0,\\, i=1,...,k$ almost everywhere on $\\mathcal{H}$ then we can glue characteristic data up to the $k$'th order. Moreover, the higher-order Newman--Penrose constants are limiting examples of the above charges. \n\n\\label{theo4}\n\\end{mytheo}\n\n\n\\subsection{PDE aspects of the conservation laws}\n\\label{sec:Remarks}\n\n\nThe existence of the charge \\eqref{eextremalcharge} implies that the solutions to the wave equation on extremal black holes do not disperse along the event horizon. This is in stark contrast to the result of Dafermos and Rodnianski \\cite{tria} who derived decay results for $\\psi$ and all its derivatives up to and including the event horizon for the general subextremal Kerr family $|a|\\epsilon_{1}$ on $S_{v}$ and then take $\\epsilon$ sufficiently small so the coefficient of $\\psi^{2}$ is strictly positive. Hence, for these choices, the operator $\\mathcal{Q}^{\\mathcal{S}}_{temp}$ is negative definite and hence it has trivial kernel. By the Atiyah--Singer theorem we have $ind\\big(\\mathcal{Q}^{\\mathcal{S}}_{temp}\\big)=0$ and hence by the Fredholm alternative the operator $\\mathcal{Q}^{\\mathcal{S}}_{temp}$ is invertible. By the Poincar\\'{e} inequality and Rellich's theorem the inverse is a compact operator $\\big(\\mathcal{Q}^{\\mathcal{S}}_{temp}\\big)^{-1}:L^{2}(S_{v})\\rightarrow L^{2}(S_{v})$. The spectrum of this operator contains zero and only discrete eigenvalues whose limit point is zero. Denote this spectrum by $\\sigma_{temp}$. It follows that the spectrum of $\\mathcal{Q}^{\\mathcal{S}}_{temp}$ is the set $\\frac{1}{\\sigma_{temp}}$ which consists of discrete eigenvalues whose only limit point is infinity. Then, the spectrum of $\\mathcal{Q}^{\\mathcal{S}}$, restricted at the section $S_{v}$, is precisely the set $\\sigma_{v}=\\frac{1}{\\sigma_{temp}}+\\frac{1}{\\epsilon}$.\n\n\nNote also that if $f\\in C^{\\infty}(S_{v})$ then the equation \n\\begin{equation}\n\\mathcal{Q}^{\\mathcal{S}}\\psi=f_{v},\n\\label{tosolveelliptic}\n\\end{equation} \nhas a solution $\\psi$ on $S_{v}$ if and only if $f_{v}$ lies in the orthogonal complement of the kernel of the adjoint of $\\mathcal{Q}^{\\mathcal{S}}$ with respect to the space $\\big(S_{v},\\mbox{$g \\mkern-8.8mu \/$\\,}\\big)$. We have the following definition\n\\begin{definition}\nLet $\\mathcal{S}$ be a foliation of regular null hypersurface $\\mathcal{H}$ of a Lorentzian manifold $(\\mathcal{M},g)$, as defined in Section \\ref{sec:TheDoubleNullFoliation}. We define the operator $\\mathcal{O}^{\\mathcal{S}}:C^{\\infty}(\\mathcal{H})\\rightarrow\\mathbb{R}$ given by \n\\begin{equation}\n\\begin{split}\n\\mathcal{O}^{\\mathcal{S}}\\psi=& \\Omega^{2}\\cdot\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+\\left[\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\Omega^{2}+2\\Omega^{2}\\cdot\\zeta^{\\sharp}\\right]\\cdot\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi\\\\&+\\left[2\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\,\\Big(\\Omega^{2}\\cdot\\zeta^{\\sharp}\\Big)+\\partial_{v}(\\Omega tr\\underline{\\chi})+\\frac{1}{2}(\\Omega tr\\underline{\\chi})(\\Omega tr\\chi)\\right]\\cdot\\psi.\n\\label{adjoint}\n\\end{split}\n\\end{equation}\nWe also denote by \n\\begin{equation}\n\\mathcal{O}^{\\mathcal{S}}_{v}:=\\mathcal{O}^{\\mathcal{S}}\\left.\\right|_{S_{v}}:C^{\\infty}(S_{v})\\rightarrow \\mathbb{R}\n\\label{restofo}\n\\end{equation}\nthe restriction of $\\mathcal{O}^{\\mathcal{S}}$ on a section $S_{v}$. \n\\label{definitiono}\n\\end{definition}\nThe operator $\\mathcal{O}^{\\mathcal{S}}_{v}$ is the adjoint of $\\mathcal{Q}^{\\mathcal{S}}_{v}$ with respect to the space $\\big(S_{v},\\mbox{$g \\mkern-8.8mu \/$\\,}\\big)$. Hence we have \n\\[ Im\\big(\\mathcal{Q}^{\\mathcal{S}}_{v}\\big)=\\left(Ker(\\mathcal{O}^{\\mathcal{S}}_{v})\\right)^{\\perp}. \\]\nTherefore, the equation \\eqref{tosolveelliptic} as a solution if and only if \n\\begin{equation}\nf_{v}\\in \\big(Ker(\\mathcal{O}^{\\mathcal{S}}_{v}) \\big)^{\\perp}.\n\\label{eq:integrability}\n\\end{equation}\nwhere the operator $\\mathcal{O}^{\\mathcal{S}}$ is the adjoint of $\\mathcal{Q}^{\\mathcal{S}}$ with respect to the space $\\big(S_{v},\\mbox{$g \\mkern-8.8mu \/$\\,}\\big)$ and is given by \n\nIf $f_{v}$ depend smoothly on $v$ for all $v\\in I=[v_{1},v_{2}]$ and \\eqref{eq:integrability} is satisfied in $I$, then $\\psi$ can be chosen to depend smoothly on $v$ too. \n\n\n\nFinally, if $S_{v}$ are endowed with the standard unit metric $\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}$ (see Section \\ref{sec:TheDoubleNullFoliation}) and $f(v,\\theta^{1},\\theta^{2}):\\mathcal{H}\\rightarrow \\mathbb{R}$ is a smooth function then \n\\[f(v,\\theta^{1},\\theta^{2})=\\sum_{ml}f_{ml}(v)\\cdot Y_{ml}(\\theta^{1},\\theta^{2}),\\]\nwhere $Y_{ml}$ are the standard spherical harmonics. \n\n\n\n\\subsection{The gluing construction}\n\\label{sec:GeneralizedConservationLaws}\n\nLet $\\mathcal{D}$ be a regular double null foliation such that $\\mathcal{H}=\\left\\{u=0\\right\\}$. Clearly $\\mathcal{D}$ defines a foliation $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$ of $\\mathcal{H}$. Let $S_{0}$ and $S_{1}$ be two sections of $\\mathcal{S}$. We will first show that we can always glue data on $S_{0}$ to data on $S_{1}$ in the sense of Definition \\ref{firstordergluingdefinition} if the operator $\\mathcal{Q}^{\\mathcal{S}}_{v}$ is surjective for some $v\\in[0,1]$. \n\n\n\\begin{proposition}\nLet $\\mathcal{S}$ be a foliation of a regular null hypersurface $\\mathcal{H}$ of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions}. Let also $\\mathcal{O}_{v}^{\\mathcal{S}}$ be the elliptic operator given by \\eqref{adjoint}. If there is $v_{0}\\in[0,1]$ such that \n\\[Ker(\\mathcal{O}_{v_{0}}^{\\mathcal{S}})= \\left\\{0\\right\\},\\]\ni.e.~if $0$ is \\textbf{not} an eigenvalue of $\\mathcal{O}^{\\mathcal{S}}_{v_{0}}$, then we can glue arbitrary data at $S_{0}$ to arbitrary data at $S_{1}$ in the sense of Definition \\ref{firstordergluingdefinition}. \n\\label{pert1prop}\n\\end{proposition}\n\\begin{proof}\n\nSuppose that the spectrum $\\sigma(\\mathcal{O}_{v_{0}}^{\\mathcal{S}})$ does not contain zero for some $v_{0}\\in[0,1]$. Then zero is in the resolvent set $\\rho(\\mathcal{O}_{v_{0}}^{\\mathcal{S}})$. By Kato's upper semicontinuity of spectrum (see \\cite{kato}), the spectrum $\\sigma(\\mathcal{O}_{v}^{\\mathcal{S}})$ of the operator $\\sigma(\\mathcal{O}_{v}^{\\mathcal{S}})$ also does not contain zero for all $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$ with $\\epsilon>0$ sufficiently small. In other words, since the operators $\\mathcal{O}^{\\mathcal{S}}_{v}$ depend smoothly on $v$, there is a sufficiently small $\\epsilon$ such that the resolvent set of the operators $\\mathcal{O}_{v}^{\\mathcal{S}}$ contains zero for all $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$. In view of previous comments, these operators are also surjective.\n\nIf we integrate the wave equation \\eqref{wenull} along the null generators of $\\mathcal{H}$ we obtain\n\\begin{equation}\n2\\partial_{u}(\\phi\\cdot\\psi)_{\\big|_{S_{1}}}-2\\partial_{u}(\\phi\\cdot\\psi)_{\\big|_{S_{0}}} =\\int_{0}^{1}\\phi\\cdot\\mathcal{Q}_{v}^{\\mathcal{S}}\\psi \\ dv.\n\\label{malista}\n\\end{equation}\nIn the context of our gluing problem the first two terms are given. We smoothly extend $\\psi$ in the cylinders\n\\[(v,\\theta^{1},\\theta^{2})\\in[0,v_{0}-\\epsilon]\\times\\mathbb{S}^{2}, \\ \\ \\ (v,\\theta^{1},\\theta^{2})\\in[v_{0}+\\epsilon,1]\\times\\mathbb{S}^{2}.\\]\nsuch that $\\psi$ vanishes at all orders at the spheres $S_{v_{0}-\\epsilon}$ and $S_{v_{0}+\\epsilon}$. \nThen equation \\eqref{malista} is satisfied if \n\\[\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}\\phi(v,\\theta^{1},\\theta^{2})\\cdot\\mathcal{Q}_{v}^{\\mathcal{S}}\\psi(v,\\theta^{1},\\theta^{2}) \\ dv =\\rho(\\theta^{1},\\theta^{2}), \\]\nwhere $\\rho$ is a given (prescribed) function of the sphere (which depends only on the initial data at $S_{0}$ and $S_{1}$ and the extension of $\\psi$ in the complement of the cylinder for which $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$. )\n\n\nWe consider a smooth function $f(v,\\theta^{1},\\theta^{2})$, such that \n\\[\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}\\phi\\big(v,\\theta^{1},\\theta^{2}\\big)\\cdot f\\big(v,\\theta^{1},\\theta^{2}\\big)\\ dv=\\rho\\big(\\theta^{1},\\theta^{2}\\big) \\]\nand $f$ vanishes to all orders at $v=v_{0}-\\epsilon$ and $v=v_{0}+\\epsilon$ (such a function clearly exists). Then, we simply have to solve the equations\n\\[\\mathcal{Q}_{v}^{\\mathcal{S}}\\psi=f\\big(v,\\cdot,\\cdot\\big)\\]\non $S_{v}$ for all $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$. This is clearly possible in view of the fact that the operators $\\mathcal{O}_{v}^{\\mathcal{S}}$ (and hence $\\mathcal{Q}_{v}^{\\mathcal{S}}$) are all invertible and the comments in the Section \\ref{sec:EllipticTheoryOnHh}. Moreover, $\\psi$ must necessarily vanish to all orders at $v=v_{0}-\\epsilon$ and $v=v_{0}+\\epsilon$ and hence extends to a smooth function in the cylinder where $\\big(v,\\theta^{1},\\theta^{2}\\big)\\in [0,1]\\times\\mathbb{S}^{2}$. \n\n\\end{proof}\n\nHence, in order to have a conservation law along $\\mathcal{H}$ we must have that $Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)\\neq \\left\\{0\\right\\}$ for all $v$ (otherwise we can perform gluing). The above result, however, does not exclude the possibility of gluing \\textit{general} characteristic data even if $Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)\\neq \\left\\{0\\right\\}$ for all $v$. For we have the following general result\n\n\\begin{theorem}\nLet $\\mathcal{S}$ be a foliation of a regular null hypersurface $\\mathcal{H}$ of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$, as defined in Section \\ref{sec:TheDoubleNullFoliation}. Let also $S_{0}$ and $S_{1}$ be two sections of $\\mathcal{S}$. Then\n\n\\begin{enumerate}\n\t\\item We can glue (to first order) general data on $S_{0}$ to general data on $S_{1}$ in the sense of Definition \\ref{firstordergluingdefinition} if and only if $\\mathcal{H}$ does not admit conservation laws with respect to $\\mathcal{S}$ in the sense of Definition \\ref{definitionconservationlaw}. If $\\mathcal{H}$ admits conservation laws with respect to $\\mathcal{S}$, then we can glue characteristic data if and only if their associated charges are equal, i.e. if and only if the data at $S_{0}$ and $S_{1}$ are such that\n\t\\[\\int_{S_{0}}Y^{\\mathcal{S}}\\big(\\phi\\cdot\\psi\\big)\\cdot \\Theta^{\\mathcal{S}}\\,d\\mu_{_{\\mathbb{S}^{2}}}=\\int_{S_{1}}Y^{\\mathcal{S}}\\big(\\phi\\cdot\\psi\\big)\\cdot \\Theta^{\\mathcal{S}}\\,d\\mu_{_{\\mathbb{S}^{2}}}, \\]\n\tfor all $\\Theta^{\\mathcal{S}}\\in \\mathcal{W}^{\\mathcal{S}}$, where $\\mathcal{W}^{\\mathcal{S}}$ is the kernel of the conservation laws as defined in Section \\ref{sec:ConservationLawsForTheWaveEquations}. Here $\\phi$ denotes the conformal factor of the sections of an associated double null foliation $\\mathcal{D}$.\n\t\n\t\\item The null hypersurface $\\mathcal{H}$ admits (first-order) conservation laws with respect to $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$ in the sense of Definition \\ref{definitionconservationlaw} if and only if there is a non-trivial linear space $\\mathcal{U}^{\\mathcal{S}}\\subset \\mathcal{V}_{\\mathcal{H}}$, where $\\mathcal{V}_{\\mathcal{H}}$ is the linear space defined in \\eqref{linearspace}, such that \n\t\t\\[\\mathcal{O}^{\\mathcal{S}}\\left(\\frac{1}{\\phi}\\cdot\\Theta^{\\mathcal{S}}\\right)=0 \\text{ on }\\mathcal{H},\\text{ for all }\\Theta^{\\mathcal{S}}\\in \\mathcal{U}^{\\mathcal{S}}.\\] Furthermore, the kernel of the conservation laws satisfies $\\mathcal{W}^{\\mathcal{S}}=\\mathcal{U}^{\\mathcal{S}}$, and moreover, $dim\\, \\mathcal{W}^{\\mathcal{S}}=dim\\, \\mathcal{U}^{\\mathcal{S}}<\\infty$.\n\n\t\\end{enumerate}\n\\label{theorem}\n\\end{theorem}\n\n\\begin{remark}\nThe conserved charges are independent of the choice of the diffeomorphism $\\Phi$ defined by \\eqref{phidiffeo}. \n\\label{rem1giatheorema}\n\\end{remark}\n\n\\begin{remark}\nGluing is \\textbf{not} always possible even if we allow to freely choose the initial data in an angular neighborhood $\\mathcal{X}$ on $S_{0}$ or $S_{1}$. Indeed, if the function $\\Theta^{\\mathcal{S}}$ vanishes in $\\mathcal{X}$, then the charges cannot change value even if we change the data in that region $\\mathcal{X}$. Hence if the charges at $S_{0}$ and $S_{1}$ do not initially coincide then they will not coincide even after changing the data at $\\mathcal{X}$. \n\\label{re2}\n\\end{remark}\n\nBefore we give the proof of the Theorem \\ref{theorem} we present some lemmata. The first one concerns the kernels of the elliptic operators $\\mathcal{O}_{v}^{\\mathcal{S}}$.\n\\begin{lemma}\n\\textbf{(Variation analysis of the kernel of elliptic operators)} There is an upper bound for the dimension of the kernel $K(v)\\subset L^{2}\\big(\\mathbb{S}^{2}\\big)$ of the operator $\\mathcal{O}_{v}^{\\mathcal{S}}$, given by \\eqref{adjoint}, for $v\\in[0,1]$. Moreover, there is a dense set $\\mathcal{Z}\\subseteq [0,1]$ of point $x$ for which there is an open neighborhood $V_{x}$ containing $x$ such that $K(v)$ varies smoothly for $v\\in V_{x}$ (and hence in particular the dimension $\\text{dim}\\big(K(v)\\big)$ is constant for all $v\\in V_{x}$). \n\\label{lemmagiaelliptickernel}\n\\end{lemma}\n\\begin{proof}\nFollowing the idea of Section \\ref{sec:EllipticTheoryOnHh} we have that for sufficiently large $\\lambda >0$ the operator $\\mathcal{O}_{v}^{\\mathcal{S}}-\\lambda\\cdot I$ has a compact inverse $\\mathcal{C}_{v}:L^{2}(\\mathbb{S}^{2})\\rightarrow H^{1}(\\mathbb{S}^{2})\\subset L^{2}(\\mathbb{S}^{2})$. By the continuity of the resolvent theorem (see \\cite{extremumproblemsbook}, Chapter 2) we obtain that the operators $\\mathcal{C}_{v}$ vary continuously in $v$ with respect to the topology of the space $\\mathcal{L}\\big(L^{2},L^{2} \\big)$. \nThe kernel $K(v)$ of $\\mathcal{O}_{v}^{\\mathcal{S}}$ coincides with the kernel of the operator \n\\begin{equation}\n\\mathcal{P}_{v}=\\mathcal{C}_{v}-\\frac{1}{\\lambda}\\cdot I:L^{2}(\\mathbb{S}^{2})\\rightarrow L^{2}(\\mathbb{S}^{2}),\n\\label{operatorp}\n\\end{equation}\nwhich also varies continuously in $v$. \nSince $\\mathcal{C}_{v}$ is compact it is easily seen that the operator \\[\\Big. \\mathcal{P}_{v}\\Big|_{\\big(K(v)\\big)^{\\perp}} :\\big(K(v)\\big)^{\\perp}\\rightarrow L^{2}(\\mathbb{S}^{2})\\]\nis bounded from below. This implies that the mapping\n\\[K:[0,1]\\rightarrow K(v):=ker\\big(\\mathcal{O}_{v}\\big)\\subset L^{2}\\big(\\mathbb{S}^{2}\\big) \\]\nis upper semicontinuous, i.e.~if $B(1)$ is the unit ball in $L^{2}\\big(\\mathbb{S}^{2}\\big)$ then for all $\\epsilon>0$ there is $\\delta>0$ such that if $|v-v_{0}|<\\delta$ then \n\\[ K(v)\\cap B(1)\\subset B_{\\epsilon}\\Big( K(v_{0})\\cap B(1) \\Big), \\] \nwhere $B_{\\epsilon}(S)$ denotes the set of points who distance from $S$ is at most $\\epsilon$. It thus follows that \n\\begin{equation}\n\\limsup_{n}\\text{dim}\\big(K(v_{n})\\big)\\leq \\text{dim}\\big(K(v_{0})\\big).\n\\label{simantikodiastasi}\n\\end{equation}\nDefine now the sets\n\\begin{equation}\nA_{n}=\\Big\\{v\\in[0,1]\\, :\\, \\text{dim}\\big(K(v)\\big)\\geq n\\Big\\}.\n\\label{tasinolaa}\n\\end{equation}\nIn view of \\eqref{simantikodiastasi} the sets $A_{v}$ are closed in [0,1]. Moreover, $A_{n+1}\\subset A_{n}$. By the compactness of $[0,1]$ it follows that there is a $n_{0}\\in \\mathbb{N}$ such that $A_{n}=\\emptyset$ for all $n\\geq n_{0}$. Hence, there is an upper bound on the dimension of the kernel $K(v)$ for all $v\\in [0,1]$. We consider next the following sets\n\\begin{equation}\nB_{n}=A_{n}\/A_{n-1}=\\Big\\{v\\in[0,1]\\, :\\, \\text{dim}\\big(K(v)\\big)=n\\Big\\}.\n\\label{tasinolab}\n\\end{equation}\nClearly, \n\\begin{equation}\n\\bigcup_{n=0}^{n_{0}}B_{n}=[0,1].\n\\label{tasinolab1}\n\\end{equation}\nThe set $B_{0}=[0,1]\/A_{1}$ is open in $[0,1]$ and hence $B_{0}=\\text{int}\\big(B_{0}\\big)$. We will show that for all $1\\leq n\\leq n_{0}$ we have \n\\begin{equation}\n\\text{int}\\big(B_{n}\\big)=B_{n}\/\\text{clos}\\big(B_{0}\\cup...\\cup B_{n-1}\\big).\n\\label{interiorofb}\n\\end{equation}\nThe inclusion $\\text{int}\\big(B_{n}\\big)\\subseteq B_{n}\/\\text{clos}\\big(B_{0}\\cup...\\cup B_{n-1}\\big)$ is trivial. If now there is $x\\in B_{n}\/\\text{clos}\\big(B_{0}\\cup...\\cup B_{n-1}\\big)$ such that $x\\notin \\text{int}\\big(B_{n}\\big)$ then there is a sequence $y_{k}\\rightarrow x$ with $y_{k}\\notin B_{n}$. Clearly, $y_{k}$ cannot have an infinite subsequence in either $B_{0}\\cup ...\\cup B_{n-1}$ since otherwise $x\\in \\text{clos}\\big(B_{0}\\cup...\\cup B_{n-1}\\big)$. Hence, $y_{k}\\in A_{n+1}$ and since $A_{n+1}$ is closed we have $x\\in A_{n+1}$ and hence $x\\notin B_{n}$, contradiction. \n\nDefine the set\n\\begin{equation}\n\\mathcal{Z}=\\bigcup_{n=1}^{n_{0}}\\text{int}\\big(B_{n}\\big).\n\\label{eq:thesetz}\n\\end{equation}\nThe set $\\mathcal{Z}$ is open in $[0,1]$ and dense. Indeed, in view of \\eqref{tasinolab1} and \\eqref{interiorofb} we have\n\\[\\text{clos}(\\mathcal{Z})= \\text{clos}\\big(B_{0}\\cup...\\cup B_{n_{0}}\\big)=[0,1] . \\]\nIn other words, there is a dense set of points $x\\in [0,1]$ for which there is an open neighborhood $V_{x}$ such that $\\text{dim}\\big(K(v)\\big)$ is constant for all $v\\in V_{x}$. It remains to show that $K(v)$ varies smoothly in $v$ for $v\\in V_{x}$, i.e.~the curve $V_{x}\\ni x\\mapsto K(v)$ is smooth in the Grassmannian $Gr(L^{2},n)$ where $n=\\text{dim}\\big(K(v)\\big)$. Using an adaptation of the aforementioned result of \\cite{extremumproblemsbook} and the fact that the coefficients of $\\mathcal{O}_{v}^{\\mathcal{S}}$ depend smoothly in $v$ one can show that $\\mathcal{C}_{v}$, and hence $\\mathcal{P}_{v}$, varies smoothly in $v$ in the space $\\mathcal{L}\\big(L^{2},L^{2}\\big)$. In view of the fact that $K(v)$ is upper semicontinuous and has constant dimension we obtain that if $v_{0}\\in V_{x}$ then \n\\[\\Big. \\text{proj}\\Big|_{K(v_{0})\\cap B(1)}\\Big(K(v)\\cap B(1)\\Big)= K(v_{0})\\cap B(1) \\]\nfor all $v$ sufficiently close to $v_{0}$. Given $x_{v_{0}}\\in K(v_{0})\\cap B(1)$ there is a unique $x_{v}\\in B_{\\epsilon}\\Big(K(v)\\cap B(1) \\Big)$ such that $x_{v}=x_{v_{0}}+a_{v}^{\\perp}$. Since $\\mathcal{P}_{v}=\\mathcal{P}_{v_{0}}+(v-v_{0})\\cdot Z+O\\big((v-v_{0})^{2}\\big)$ and $\\mathcal{P}_{v}(x_{v})=0$ we obtain $\\mathcal{P}_{v_{0}}(a_{v}^{\\perp})=(v_{0}-v)\\cdot Zx_{v}+O(v^{2}) $ and since $\\Big. \\mathcal{P}_{v_{0}}\\Big|_{\\big(K(v)\\big)^{\\perp}}:\\big(K(v)\\big)^{\\perp}\\rightarrow Im(\\mathcal{P}_{0})$ has a bounded inverse we obtain that $a_{v}^{\\perp}\\rightarrow 0$ in $L^{2}$ as $v\\rightarrow v_{0}$ in a differentiable manner. This shows that $K(v)$ varies differentiably in $v$. Similary we can show that $K(v)$ is smooth in $v$. \n\n\n\n\n\n\\end{proof}\n\n\n\n\n\nIf there is $v_{0}\\in[0,1]$ such that $Ker\\big(\\mathcal{O}^{\\mathcal{S}}_{v}\\big)= 0$ then the Theorem \\ref{theorem} follows from Proposition \\ref{pert1prop}. In particular, in this case we have $dim\\,\\mathcal{W}^{\\mathcal{S}}=dim\\,\\mathcal{U}^{\\mathcal{S}}=0$. We assume that $Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)\\neq 0$ for all $v\\in[0,1]$, that is $B_{0}=\\emptyset$, where $B_{0}$ is defined by \\eqref{tasinolab}. Then according to the above lemma there is a dense set of points $x$ which have an open neighborhood $V_{x}$ in which the kernels $K(v)$ vary smoothly. In each of these intervals, we can find a smoothly varying in $v$ basis \n\\[\\mathcal{B}_{v}^{\\mathcal{S}}=\\left\\{ \\big(E_{1}^{\\mathcal{S}}\\big)_{v}, \\big(E_{2}^{\\mathcal{S}}\\big)_{v},\\cdots, \\big(E_{i}^{\\mathcal{S}}\\big)_{v}, \\ i=dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big) \\right\\}\\]\nof $K(v)=Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$. We next localize in each of these intervals. \n\n\nLet $v_{0}\\in V_{x}$ for some $x$ as above. We can smoothly extend $\\psi\\left.\\right|_{S_{0}}$ and $\\psi\\left.\\right|_{S_{1}}$ in the cylinders $[0,v_{0}-\\epsilon]\\times\\mathbb{S}^{2}$ and $[v_{0}+\\epsilon,1]\\times \\mathbb{S}^{2}$, where $\\epsilon>0$ is sufficiently small such that $v_{0}-\\epsilon,v_{0}+\\epsilon\\in V_{x}$, such that $\\psi$ vanishes to infinite order on $\\mathcal{H}$ at $\\left\\{v_{0}-\\epsilon\\right\\}\\times\\mathbb{S}^{2}$ and $\\left\\{v_{0}+\\epsilon\\right\\}\\times\\mathbb{S}^{2}$. Then, $\\mathcal{O}^{\\mathcal{S}}_{v}\\psi$ is also known in the union of these two cylinders and hence, by \\eqref{wenull1}, $\\mathcal{F}^{\\mathcal{D}}\\psi$, given by \\eqref{defF}, is known there. In fact $\\mathcal{F}^{\\mathcal{D}}\\psi$ vanishes to infinite order on $\\mathcal{H}$ at $\\left\\{v_{0}-\\epsilon\\right\\}\\times\\mathbb{S}^{2}$ and $\\left\\{v_{0}+\\epsilon\\right\\}\\times\\mathbb{S}^{2}$. We wish to extend $\\psi$ smoothly in $[0,1]\\times\\mathbb{S}^{2}$ such that $\\psi$ solves \\eqref{wenull1} and the transversal derivatives $\\partial_{u}\\psi\\left.\\right|_{S_{0}}$ and $\\partial_{u}\\psi\\left.\\right|_{S_{1}}$ agree with the given characteristic data. \n\n\n We will do so by first smoothly extending $\\mathcal{F}^{\\mathcal{D}}\\psi$ everywhere in $[0,1]\\times\\mathbb{S}^{2}$ such that the above are satisfied and, using \\eqref{wenull1}, we can solve with respect to $\\psi$. In other words, gluing is possible if we can extend $\\mathcal{F}^{\\mathcal{D}}\\psi$, or equivalently $\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi$, in $[v_{0}-\\epsilon,v_{0}+\\epsilon]\\times\\mathbb{S}^{2}$ such that the following conditions hold:\n\n\\medskip\n\n\\noindent{\\underline{\\textbf{The conditions 1--3:}}}\n\n\\medskip\n\n\\begin{enumerate}\n\t\\item \\textbf{\\underline{Smoothness on $\\mathcal{H}$}:} The function $\\mathcal{F}^{\\mathcal{D}}\\psi$, or equivalently $\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi$, vanishes to all orders on $\\mathcal{H}$ at $\\left\\{v_{0}-\\epsilon\\right\\}\\times\\mathbb{S}^{2}$ and $\\left\\{v_{0}+\\epsilon\\right\\}\\times\\mathbb{S}^{2}$. \n\t\\item \\textbf{\\underline{Gluing for the transerval derivative $\\partial_{u}\\psi$}:} In view of \\eqref{defF} and \\eqref{wenull1}, $\\mathcal{F}^{\\mathcal{D}}\\psi$ satisfies on $\\mathcal{H}$: \\[\\displaystyle\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}\\!\\!\\!\\big(\\mathcal{F}^{\\mathcal{D}}\\psi\\big)(v,\\theta^{1},\\theta^{2})\\cdot \\phi(v,\\theta^{1},\\theta^{2})\\, dv=\\rho(\\theta^{1},\\theta^{2}),\\] where $\\rho$ is a given (prescribed) function of the sphere (which depends only on the initial data at $S_{0}$ and $S_{1}$ and also the extension of $\\psi$ in the complement of the cylinder for which $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$).\n\t\n\t\\item \\textbf{\\underline{Integrability (orthogonality) condition}:} Given $\\mathcal{F}^{\\mathcal{D}}\\psi$ we can solve with respect to $\\psi$ (i.e. ``invert'' the operator $\\mathcal{F}^{\\mathcal{D}}$) if, using \\eqref{wenull1} and the comments of Section \\ref{sec:EllipticTheoryOnHh}, \n\t\\[\\mathcal{F}^{\\mathcal{D}}\\psi \\in Im\\big(\\mathcal{Q}^{\\mathcal{S}}_{v}\\big)=\\left(Ker(\\mathcal{O}^{\\mathcal{S}}_{v})\\right)^{\\perp}, \\]\n\tor equivalently,\n\t\\begin{equation}\n\t\\displaystyle\\int_{S_{v}}\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot \\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\ d\\mu_{_{\\mbox{$g \\mkern-8.8mu \/$\\,}}}=0\n\t\\label{1inte}\n\t\\end{equation} for all $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$ and $n=1,..., dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$.\n\t\\end{enumerate}\n\tNote that since\n\t\\begin{equation*}\n\\begin{split}\n\\int_{S_{v}}\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot \\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\ d\\mu_{_{\\mbox{$g \\mkern-8.8mu \/$\\,}}}=&\\int_{S_{v}}2\\frac{\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}}}{\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}}\\cdot\\partial_{v}\\partial_{u}(\\phi\\cdot\\psi)\\cdot \\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}}\\, d\\theta^{1}\\, d\\theta^{2}\\\\=&\\int_{S_{v}}2\\frac{\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}}}{\\sqrt[4]{\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}}}\\cdot\\partial_{v}\\partial_{u}(\\phi\\cdot\\psi)\\cdot \\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\sqrt{\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\mathbb{S}^{2}}}\\, d\\theta^{1}\\, d\\theta^{2}\\\\=&\\int_{S_{v}}\\Big(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi\\Big)\\cdot\\Big(\\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\cdot \\phi\\Big)\\, d\\mu_{_{\\mathbb{S}^{2}}},\n\\end{split}\n\\end{equation*}\nthe integrability condition \\eqref{1inte} is equivalent to the following:\n\\begin{equation}\n\\int_{S_{v}}\\Big(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi\\Big)\\cdot\\Big(\\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\cdot \\phi\\Big)\\, d\\mu_{_{\\mathbb{S}^{2}}}=0,\n\\label{integrability}\n\\end{equation}\nfor all $n=1,..., dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$. As we shall see, splitting $\\phi$ in both terms is very important as it also is the fact that the above integral is with respect to the standard unit metric on $S_{v}$. We define the functions\n\\begin{equation}\nG_{n}=\\big(E_{n}^{\\mathcal{S}}\\big)_{v}\\cdot\\phi, \n\\label{g0}\n\\end{equation}\nfor $n=1,\\cdots, dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$ and thus \\eqref{1inte} is equivalent to \n\\begin{equation}\n\\int_{S_{v}}\\Big(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi\\Big)\\cdot\\big(G_{n}\\big)\\, d\\mu_{_{\\mathbb{S}^{2}}}=0,\n\\label{integrability}\n\\end{equation}\nfor $n=1,\\cdots, dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$. \n\\bigskip\n\n\\noindent\\underline{ \\textbf{A special case:}} $dim Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=1$ for all $v\\in[0,1]$\n\n\\medskip\n\nBefore we consider the general case let us first consider the special case for which $dim Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=1$ for all $v\\in[0,1]$ (and hence $V_{x}=[0,1]$). This will make our argument clear. We simplify the notation by denoting $G=G_{1}$. \n\n\n\nWe decompose $\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi,\\,G$ in (standard) angular frequencies. Let\n\\begin{equation}\n\\big(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi\\big)(v,\\theta^{1},\\theta^{2})=\\sum_{ml}F_{ml}(v)\\cdot Y^{ml}(\\theta^{1},\\theta^{2})\n\\label{f1}\n\\end{equation}\nand \n\\begin{equation}\nG(v,\\theta^{1},\\theta^{2})=\\sum_{ml}G_{ml}(v)\\cdot Y^{ml}(\\theta^{1},\\theta^{2}),\n\\label{g1}\n\\end{equation}\nwhere $Y^{ml}$ denote the standard spherical harmonics on $\\mathbb{S}^{2}$. \n\nWe can glue to first order characteristic data on $S_{0}$ and $S_{1}$ if there exist smooth functions $F_{ml}(v)\\!:\\![v_{0}-\\epsilon,v_{0}+\\epsilon]\\rightarrow\\mathbb{R}$ such that the following conditions are satisfied\n\\begin{enumerate}\n\\item \\underline{Smoothness on $\\mathcal{H}$}:\nThe functions $F_{ml}(v)$ vanish to all orders at $v=v_{0}-\\epsilon$ and $v=v_{0}+\\epsilon$ for all $m,l$.\n\\item \n\\underline{Gluing for the transversal derivative $\\partial_{u}\\psi$}:\nThe integrals\n\\[\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{ml}(v)\\, dv\\]\n are all given. \n \\item \n\\underline{Orthogonality condition}: The integrability condition \\eqref{integrability} is satisfied:\n\\begin{equation*}\n\\begin{split}\n\\sum_{ml} F_{ml}(v)\\cdot G_{ml}(v)=0.\n\\end{split}\n\\end{equation*}\n\\end{enumerate}\n\n\n\n\\underline{\\textbf{Case I}}:\\medskip \\\\ \nFor all $m,l$ we have $G_{ml}(v)=\\textbf{G}(v)\\cdot c_{ml}$, for all $v\\in[0,1]$, for some function $\\textbf{G}:[0,1]\\rightarrow\\mathbb{R}$ and some constant non-zero $l^{2}$ sequence $c_{ml}$. That is to say, for all $m,l,m',l'$ we have $\\frac{G_{ml}(v)}{G_{m'l'}(v)}=c_{mlm'l'}$, where $c_{mlm'l'}$ is a constant. Then, the condition 3 becomes\n\\[\\sum_{ml}F_{ml}(v)\\cdot c_{ml}=0. \\]\n The above equation implies that the functions $F_{ml}(v)$ are linearly dependent and hence condition 2 \\textbf{cannot} be satisfied \\textit{in general}. In fact, in this case we have that $G(v,\\theta^{1},\\theta^{2})=\\textbf{G}(v)\\cdot{\\Theta}(\\theta^{1},\\theta^{2}), $ for some function ${{\\Theta}}$ which is constant along the null generators, i.e.~$\\Theta\\in\\mathcal{V}_{\\hh}$. Then, by \\eqref{g0}, we have the splitting \n\\[ \\big(E^{\\mathcal{S}}_{v}\\cdot\\phi\\big)(v,\\theta^{1},\\theta^{2})= \\mathbf{G}(v)\\cdot\\Theta(\\theta^{1},\\theta^{2}) \\]\nwhich shows that \n\\[E_{v}^{\\mathcal{S}}(\\theta^{1},\\theta^{2})= \\mathbf{G}(v)\\cdot \\frac{1}{\\phi}\\cdot \\Theta(\\theta^{1},\\theta^{2}), \\]\nand since the function $\\mathbf{G}$ does not depends on the angular coordinates we have that \n\\[\\frac{1}{\\phi}\\cdot\\Theta\\in Ker(\\mathcal{O}_{v}^{\\mathcal{S}}), \\]\nfor all $v\\in[0,1],$ as required. Furthermore, by \\eqref{integrability} and \\eqref{defF} we have \n\\[0=\\int_{S_{v}}\\big(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi\\big)\\cdot\\Theta\\, d\\mu_{_{\\mathbb{S}^{2}}}=\\int_{S_{v}}\\Big(\\partial_{v}\\partial_{u}(\\phi\\cdot\\psi)\\Big)\\cdot\\Theta\\, d\\mu_{_{\\mathbb{S}^{2}}} \\]\nand, since the function $\\Theta$ and the measure of integration do not depend on $v$, we obtain that the quantity\n\\begin{equation}\n\\int_{S_{v}}\\big(\\partial_{u}(\\phi\\cdot\\psi)\\big)\\cdot\\Theta\\ d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{eq:conserved}\n\\end{equation}\nis conserved, i.e.~independent of $v$. Clearly the above conservation law is an obstruction to gluing of general initial data on $S_{0}$ to general initial data on $S_{1}$. \n\nWe will next show that this conservation law is the \\textbf{only} obstruction to gluing. Indeed suppose that the initial data on $S_{0}$ and $S_{1}$ are such that \n\\[\\int_{S_{0}}\\big(\\partial_{u}(\\phi\\cdot\\psi)\\big)\\cdot\\Theta\\ d\\mu_{_{\\mathbb{S}^{2}}}=\\int_{S_{1}}\\big(\\partial_{u}(\\phi\\cdot\\psi)\\big)\\cdot\\Theta\\ d\\mu_{_{\\mathbb{S}^{2}}}.\\]\nIn this case we need to construct functions $F_{ml}:[v_{0}-\\epsilon,v_{0}+\\epsilon]\\rightarrow\\mathbb{R}$ such that \n\\begin{enumerate}\n\\item $F_{ml}(v)$ vanishes to infinite order at $v=v_{0}-\\epsilon$ and $v=v_{0}+\\epsilon$ for all $m,l$.\n\\item \\begin{equation}\n\\sum_{ml}F_{ml}(v)\\cdot c_{ml}=0\n\\label{in1}\n\\end{equation} for $v\\in[v_{0}-\\epsilon,v_{0}+\\epsilon]$, where $c_{ml}$ is a constant non-zero $l^{2}$ sequence. \n\n\\item The integrals\n\\[ \\mathbf{I}_{ml}=\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{ml}(v)\\, dv \\]are all prescribed such that \n\\begin{equation}\n\\sum_{ml}\\mathbf{I}_{ml}\\cdot c_{ml} =0, \n\\label{in2}\n\\end{equation} \nsince\n\\[Char(S_{v_{0}-\\epsilon})=Char(S_{0})=Char(S_{1})=Char(S_{v_{0}+\\epsilon}),\\]\nand using \\eqref{defF} and \\eqref{malista}, where we denote \n\\[char(S_{v})=\\int_{S_{v}}\\big(\\partial_{u}(\\phi\\cdot\\psi)\\big)\\cdot\\Theta\\ d\\mu_{_{\\mathbb{S}^{2}}}.\\] \\end{enumerate}\n\nFor simplicity we rename the sequences $F_{ml},G_{ml,}c_{ml}$ as $F_{i},G_{i},c_{i},i\\geq 0$. We assume without loss of generality that $c_{0}\\neq 0$. We construct the functions $F_{i}, i\\geq 1,$ such that condition 1 is satisfied and such that the integrals $\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{i}(v)\\,dv$ agree with the prescribed values (condition 3). We next construct $F_{0}$ by solving the equation \\eqref{in1} with respect to $F_{0}$. Clearly, condition 1 is satisfied for $F_{0}$. Condition 2 holds by construction. Moreover, the integral $\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{0}(v)\\, dv$ agrees with its prescribed value in view of \\eqref{in2} and the linearity of the integrals. This finishes the construction of $F_{ml}$ which in turn allows us to extend $\\psi$ and hence to obtain the gluing of the characteristic data. \n\nWe remark that as along as gluing is possible then it can be achieved in a highly non-unique way. \n\n\\bigskip\n\n\n\\underline{\\textbf{Case II}}:\\medskip\\\\ Using the above simplified notation, we can assume without loss of generality that \\begin{equation}\nG_{1}(v)=a(v)\\cdot G_{0}(v) \\text{ and } \\partial_{v} a\\neq 0,\\ G_{0}(v)\\neq 0\n\\label{gmalista}\n\\end{equation} in a (sufficiently) small interval $I$ of $v$ (and hence $G_{1}$ is not linearly dependent on $G_{0}$). We choose $v_{0}$ to be in $I$ and we take $\\epsilon>0$ small enough such that $[v_{0}-\\epsilon,v_{0}+\\epsilon]\\subset I$. By relabeling $m,l$, as before, we can rewrite condition 3 as follows \n\\[ F_{0}(v)\\cdot G_{0}(v)+F_{1}(v)\\cdot G_{1}(v)+\\sum_{i\\geq 2}F_{i}(v)\\cdot G_{i}(v)=0 \\]\nand hence \n\\begin{equation}\n F_{0}(v)=-\\frac{G_{1}(v)}{G_{0}(v)}F_{1}(v)-\\sum_{i\\geq 2}\\frac{G_{i}(v)}{G_{0}(v)}F_{i}(v)=-a(v)\\cdot F_{1}(v)-\\sum_{i\\geq 2}\\frac{G_{i}(v)}{G_{0}(v)}F_{i}(v). \n\\label{eq:pro3}\n\\end{equation}\nWe can then prescribe $F_{i}(v)$, $i\\geq 2$, such that condition 1 is satisfied and such that the integrals $\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{i}(v)\\,dv$ agree with the prescribed values (condition 3). \n\nWe can then prescribe $F_{1}$ such that condition 1 is satisfied, the integral $\\int_{v_{0}-\\epsilon}^{v_{0}-\\epsilon}F_{1}(v)\\, dv$ agrees with its prescribed value \\textbf{and} the integral $\\int_{v_{0}-\\epsilon}^{v_{0}-\\epsilon}a(v)\\cdot F_{1}(v)\\, dv$ is such that the integral $\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{0}(v)\\, dv$, computed via \\eqref{eq:pro3}, agrees with its prescribed value. Note that in view of \\eqref{gmalista}, the integrals $\\int_{v_{0}-\\epsilon}^{v_{0}-\\epsilon}a(v)\\cdot F_{1}(v)\\, dv$ and $\\int_{v_{0}-\\epsilon}^{v_{0}+\\epsilon}F_{0}(v)\\, dv$ are independent. In view of the fact that condition 1 is satisfied for all $F_{i},\\, i\\geq 1$, we have that it is automatically satisfied for $F_{0}$, again via \\eqref{eq:pro3}. The remaining conditions hold by construction. This finishes the construction of $F_{ml}$'s for the case where $dimKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=1$ for all $v\\in [0,1]$.\n\n\n\\bigskip\n\n\\noindent\\underline{\\textbf{The general case}}\n\n\\medskip\n\nWe now return to the general case. We first derive the following lemmata.\n\n\\begin{lemma}\nLet $I$ be a closed interval of $\\mathbb{R}$. Given $n\\in\\mathbb{N}$ linearly independent functions $f_{1},f_{2},...,f_{n}\\in C^{\\infty}\\big(I\\big)$ and $\\lambda_{1},\\lambda_{2},\\cdots, \\lambda_{n}\\in \\mathbb{R}$ there is a function $\\alpha\\in C^{\\infty}\\big(I\\big)$ such that \n \\[\\int_{I}\\alpha(v)\\cdot f_{i}(v)\\, dv=\\lambda_{i}, \\text{ for all } i=1,2,...,n.\\]\n\\label{lemma1stoproof}\n\\end{lemma}\n\\begin{proof}\n\n\nLet $V_{n}=\\Big\\langle f_{1},f_{2},\\cdots, f_{n}\\Big\\rangle\\subset L^{2}(I)$\ndenote the $n$-dimensional span of the functions $f_{1},f_{2},\\cdots, f_{n}$. Using the Gram--Schmidt process we produce an orthonormal basis $\\left\\{e_{1},e_{2},\\cdots, e_{n}\\right\\}$ of $V_{n}$. We extend this basis to obtain an orthonormal basis $\\left\\{e_{1},\\cdots, e_{n},e_{n+1},\\cdots\\right\\}$ of $L^{2}(I)$. Clearly for all $i=1,2,...,n$ we have that $e_{i}\\in C^{\\infty}\\big(I\\big)$ and \n\\[f_{i}=\\sum_{k=1}^{i}\\left\\langle e_{k},f_{i}\\right\\rangle\\cdot e_{k} \\]\nwith $\\left\\langle e_{i},f_{i}\\right\\rangle>0$, where $\\left\\langle\\, \\cdot\\, ,\\, \\cdot\\, \\right\\rangle$ denotes the inner product of $L^{2}(I)$.\nWe want to construct a function $\\alpha\\in C^{\\infty}\\big(I\\big)$ such that $\\left\\langle a, f_{i}\\right\\rangle=\\lambda_{i}$ for $i=1,2,...,n$. The system\n\\[\\lambda_{i}= \\sum_{k=1}^{i}\\left\\langle e_{k},f_{i}\\right\\rangle\\cdot x_{i} \\]\nhas a unique solution with respect to $x_{1},x_{2},...,x_{n}$. We then define the function $\\alpha$ such that \n$\\left\\langle \\alpha, e_{i}\\right\\rangle=x_{i}$ for $i=1,2,...,n$ and $\\left\\langle \\alpha, e_{i}\\right\\rangle=0$ for $i\\geq n+1$, which clearly satisfies the required relations. \n\n\n\\end{proof}\n\n\\begin{lemma}\nLet $I$ be a compact interval of $\\mathbb{R}$. Let $G_{1},G_{2},...,G_{n}\\in C^{\\infty}\\big(I\\times \\mathbb{S}^{2}\\big)$ be $n$ functions such that for each $v\\in I$ the functions $G_{1}(v,\\cdot), G_{2}(v,\\cdot),...,G_{n}(v,\\cdot)\\in C^{\\infty}\\big(\\mathbb{S}^{2}\\big)$ are linearly independent and let \\[\\Pi(v)=\\Big\\langle G_{1}(v,\\cdot), G_{2}(v,\\cdot),...,G_{n}(v,\\cdot)\\Big\\rangle\\subset L^{2}\\big(\\mathbb{S}^{2}\\big), \\] denote the ($v$-dependent) $n$-dimensional subspace of $L^{2}\\big(\\mathbb{S}^{2}\\big)$ spanned by them. Then, given $\\rho\\in C^{\\infty}\\big(\\mathbb{S}^{2}\\big)$ there is a function $F_{\\rho}\\in C^{\\infty}\\big(I\\times\\mathbb{S}^{2}\\big)$ which vanishes to infinite order at $\\partial I\\times \\mathbb{S}^{2}$ and is such that \n\\begin{equation}\n\\int_{I}F_{\\rho}(v,\\cdot)\\, dv= \\rho(\\cdot)\n\\label{eq:rel1}\n\\end{equation}\nand \n\\begin{equation}\n\\int_{\\mathbb{S}^{2}}F_{\\rho}(v,\\cdot)\\cdot G_{i}(v,\\cdot)\\, d\\mu_{_{\\mathbb{S}^{2}}}=0 \\text{ for all } i=1,2,...,n,\n\\label{eq:rel2}\n\\end{equation}\nif and only if \n\\[\\rho\\in \\Bigg(\\bigcap_{v\\in I}\\Pi(v)\\Bigg)^{\\perp}\\subset L^{2}\\big(\\mathbb{S}^{2}\\big). \\]\n\\label{deuterolemma}\n\\end{lemma}\n\\begin{proof}\nWe denote \n\\begin{equation}\nV=\\bigcap_{v\\in I}\\Pi(v)\\subset L^{2}\\big(\\mathbb{S}^{2}\\big).\n\\label{sxesi1}\n\\end{equation}\n Clearly, $V$ is finite dimensional. \n\n If given a function $\\rho$ the function $F_{\\rho}$ exists then $F_{\\rho}\\in \\Big(\\Pi(v)\\Big)^{\\perp}$ for all $v$ and hence $F_{\\rho}\\in V^{\\perp}$. Therefore, an immediate application of Fubini's theorem yields that $\\rho\\in V^{\\perp}$.\n\n\nLet us assume now that $\\rho \\in V^{\\perp}$. We will show that a function $F_{\\rho}\\in C^{\\infty}\\big(I\\times \\mathbb{S}^{2}\\big)$ satisfying the above properties exists. \n\nSince $V$ is finite dimensional, we have the decomposition\n\\[L^{2}\\big(\\mathbb{S}^{2} \\big)=V\\oplus V^{\\perp}. \\]\nIf we define the spaces\n\\[\\Big.\\Pi(v)\\Big|_{V^{\\perp}} :=\\Big.\\text{proj}\\Big|_{V^{\\perp}}\\big(\\Pi(v)\\big). \\]\nSince $V\\subset \\Pi(v)$ for all $v\\in I$ the space $\\Big.\\Pi(v)\\Big|_{V^{\\perp}} $ varies smoothly in $v$. Indeed, since $\\Pi(v)$ varies smoothly in $v\\in I$ we can write $\\Pi(v)=V\\oplus T(v)$ where $T(v)$ varies smoothly in $v\\in I$ and $\\text{dim}\\big(T(v)\\big)=n-\\text{dim}(V)$, where $\\text{dim}\\big(\\Pi(v)\\big)=n$ for all $v\\in I$. Then $\\Big.\\Pi(v)\\Big|_{V^{\\perp}}=\\Big.T(v)\\Big|_{V^{\\perp}}$. The projection\n\\[\\Big.\\text{proj}\\Big|_{V^{\\perp}}: T(v)\\rightarrow V^{\\perp} \\]\nhas full rank since otherwise we would have $T(v)\\cap V\\neq \\left\\{0\\right\\}$, contradiction. Therefore, since $T(v)$ varies smoothly in $v$ the space $\\Big.T(v)\\Big|_{V^{\\perp}}$ varies smoothly in $v$ and hence so does $\\Big.\\Pi(v)\\Big|_{V^{\\perp}}$. It also follows that \n\\begin{equation}\n\\text{dim}\\left(\\Big.\\Pi(v)\\Big|_{V^{\\perp}}\\right)=n-\\text{dim}(V).\n\\label{dimerelationw}\n\\end{equation} \nFurthermore, we obtain\n\\begin{equation}\n\\bigcap_{v\\in I}\\left( \\Big.\\Pi(v)\\Big|_{V^{\\perp}} \\right)=\\left\\{0\\right\\}.\n\\label{sxesi2}\n\\end{equation}\nIndeed, if the line $\\left\\langle l\\right\\rangle$ lies in the above intersection then for every $v\\in I$ there is $y_{v}\\in \\Pi(v)$ such that $y_{v}=x_{v}+l,$ where $x_{v}\\in V$ (and $l\\in V^{\\perp}$). However, by \\eqref{sxesi1}, $x_{v}\\in \\Pi(v)$ and hence by linearity $l\\in \\Pi(v)$. Since this holds of all $v$, it immediately contradicts \\eqref{sxesi1}.\n\n\n\nWe next show that there is a finite dimensional subspace $W\\subset V^{\\perp}$ such that if \n\\[\\widetilde{\\Pi}(v):= \\Big.\\text{proj}\\Big|_{W} \\Big( \\Big.\\Pi(v)\\Big|_{V^{\\perp}}\\Big)\\subset W,\\]\nthen \n\\begin{equation}\n\\bigcap_{v\\in[0,1]}\\widetilde{\\Pi}(v)=\\left\\{0\\right\\}.\n\\label{xrisimipi}\n\\end{equation}\nIndeed, in view of \\eqref{dimerelationw},\\eqref{sxesi2}, there are $v_{1},v_{2}\\in I$ such that $K_{1}= \\Big.\\Pi(v_{1})\\Big|_{V^{\\perp}}\\cap \\Big.\\Pi(v_{2})\\Big|_{V^{\\perp}}$ is at most $(n-1)$-dimensional (where the dimension of $\\Pi(v)$ is $n$). In view of \\eqref{sxesi2}, there is $v_{3}\\in I$ such that $K_{2}=K_{1}\\cap \\Big.\\Pi(v_{3})\\Big|_{V^{\\perp}}$ is at most $(n-2)$-dimensional. Continuing inductively we deduce that there are $v_{1},v_{2},...,v_{n}$ such that \n\\begin{equation}\n\\bigcap_{v_{i}}\\Big.\\Pi(v_{i})\\Big|_{V^{\\perp}}=\\left\\{0\\right\\}.\n\\label{finiteintersectionmiden}\n\\end{equation}\n We then define $W$ by\n\\begin{equation}\nW=\\Big\\langle \\Big.\\Pi(v_{1})\\Big|_{V^{\\perp}},\\Big.\\Pi(v_{2})\\Big|_{V^{\\perp}},...,\\Big.\\Pi(v_{n})\\Big|_{V^{\\perp}} \\Big\\rangle,\n\\label{eq:definitionofw}\n\\end{equation}\nwhich is clearly finite dimensional and satisfies \\eqref{xrisimipi}. Indeed, by the construction of $W$, \\eqref{xrisimipi} is satisfied even if we restrict the intersection for the values of $v$ in the set $ \\left\\{v_{1},...,v_{n}\\right\\}$.\n\nWe next consider the projection\n\\[\\Big.\\text{proj}\\Big|_{W}(v): \\Big.\\Pi(v)\\Big|_{V^{\\perp}}\\rightarrow W. \\]\nBy virtue of \\eqref{eq:definitionofw} we have that $\\Big.\\text{proj}\\Big|_{W}(v)$ has full rank for $v=v_{1},...,v_{n}$. Since $\\Big.\\Pi(v)\\Big|_{V^{\\perp}}$ varies smoothly in $v$ we have that $\\Big.\\text{proj}\\Big|_{W}(v)$ has full rank in the union of sufficiently small intervals $J_{i}$ containing $v_{i}$. \n\n\nLet also $\\widetilde{O}(v)$ denote the orthogonal complement of $\\widetilde{\\Pi}(v)$ in $W$. It is important to note that the space $\\widetilde{O}(v)$ varies smoothly for $v\\in \\cup J_{i}$ since $\\widetilde{\\Pi}(v)$ varies smoothly for $v\\in \\cup J_{i}$. In view of \\eqref{finiteintersectionmiden} we have that \n\\begin{equation}\n\\Big\\langle \\widetilde{O}(v_{1}),...,\\widetilde{O}(v_{n}) \\Big\\rangle= W.\n\\label{finitespanofo}\n\\end{equation}\nIndeed, if there is a proper subspace $X\\subset W$ such that $\\widetilde{O}(v)\\subset X$ for all $v\\in \\left\\{v_{1},...v_{n}\\right\\}$ then $X^{\\perp}\\subset \\big(\\widetilde{O}(v)\\big)^{\\perp}\\subset W$ for all $v\\in \\left\\{v_{1},...v_{n}\\right\\}$. This is however contradiction since, by definition, $\\big(\\widetilde{O}(v)\\big)^{\\perp}=\\widetilde{\\Pi}(v)$ and by \\eqref{finiteintersectionmiden} these spaces cannot have non-trivial common intersection. Therefore, there are $u_{1},...,u_{\\text{dim}(W)}\\in\\left\\{v_{1},...,v_{n}\\right\\}\\subset \\cup J_{i}$ and $x_{i}\\in \\widetilde{O}(u_{i})$ with $i=1,...,\\text{dim}(W)$ such that the set $\\left\\{x_{i}, i=1,...,\\text{dim}(W)\\right\\}$ is a basis of $W$. We can assume that $u_{i}$ are pairwise distinct since otherwise we can consider small perturbations $u_{i}^{\\text{perp}}$ of them in the union $\\cup J_{i}$. Then, since $\\widetilde{O}(v)$ varies smoothly in $\\cup J_{i}$, the perturbed vectors $x_{i}^{\\text{perp}}=\\widetilde{O}\\big(u_{i}^{perp}\\big)$ still form a basis of $W$. Moreover, we can assume that $u_{i}^{\\text{perp}}$ lie in the interior of $I$ for all $i=1,...,\\text{dim}(W)$. We can thus define the following closed intervals $I_{i},i=1,...,\\text{dim}(W)$ as follows:\n\\begin{equation}\nI_{i}\\subset J_{i}\\cap \\text{int}I \\text{ such that } u_{i}^{\\text{perp}}\\in I_{i},\n\\label{orismosdiastimatos}\n\\end{equation}\nwhere $\\text{int}I$ denotes the interior of $I$. \n\nLet now $W^{\\perp}$ denote the orthogonal complement of $W$ in $V^{\\perp}$ and hence \n\\[L^{2}\\big(\\mathbb{S}^{2}\\big)=V\\oplus W \\oplus W^{\\perp}. \\] Since $W=\\widetilde{\\Pi}(v)\\oplus \\widetilde{O}(v)$ we have\n\n\\begin{equation}\nL^{2}\\big(\\mathbb{S}^{2}\\big)= V\\oplus \\widetilde{\\Pi}(v)\\oplus \\widetilde{O}(v)\\oplus W^{\\perp}. \n\\label{sxesi3}\n\\end{equation}\nThen, for any $F\\in C^{\\infty}\\big(I\\times \\mathbb{S}^{2}\\big)$ we have\n\\begin{equation}\nF(v,\\cdot)=\\Big.F\\Big|_{V}+\\Big.F\\Big|_{\\widetilde{\\Pi}(v)}+\\Big.F\\Big|_{\\widetilde{O}(v)}+\\Big.F\\Big|_{W^{\\perp}}\n\\label{sxesiF}\n\\end{equation}\nwhere we denote $\\Big.F\\Big|_{Z}=\\Big.\\text{proj}\\Big|_{Z}\\big(F\\big)$.\n\n\nGiven a function $\\rho\\in V^{\\perp}\\cap C^{\\infty}\\big(\\mathbb{S}^{2}\\big)$ we want to construct a function $F_{\\rho}\\in C^{\\infty}\\big(I\\times \\mathbb{S}^{2}\\big)$ such that \n\\begin{equation}\n\\int_{I}F_{\\rho}(v,\\cdot)dv=\\rho\\in V^{\\perp} \n\\label{edw1}\n\\end{equation}and\n\\begin{equation}\nF_{\\rho}(v,\\cdot)\\in \\Big(\\Pi(v)\\Big)^{\\perp}\\subset V^{\\perp}\\ \\text{ for all }\\ v\\in I.\n\\label{edw2}\n\\end{equation}\nThe first condition we impose on $F_{\\rho}$ is:\n\\begin{equation}\n\\Big.F_{\\rho}(v,\\cdot)\\Big|_{V}=0 \\text{ for all } v\\in I.\n\\label{conditionF1}\n\\end{equation}\nThen condition \\eqref{edw1} is equivalent to \n\\begin{equation}\n\\int_{I}\\Big.F_{\\rho}(v,\\cdot)\\Big|_{W^{\\perp}}dv=\\Big.\\rho\\Big|_{W^{\\perp}} \n\\label{zitoumeno1}\n\\end{equation}\nand \n\\begin{equation}\n\\int_{I}\\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{\\Pi}(v)}dv+\\int_{I}\\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{O}(v)}dv=\\Big.\\rho\\Big|_{W}.\n\\label{zitoumeno2}\n\\end{equation}\nMoreover, condition \\eqref{edw2}, using \\eqref{conditionF1}, is equivalent to the following: \n\\begin{equation}\n\\Big\\langle \\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{\\Pi}(v)}, \\big. w\\big|_{\\widetilde{\\Pi}(v)} \\Big\\rangle =- \\Big\\langle \\Big. F_{\\rho}(v,\\cdot)\\Big|_{W^{\\perp}}, \\big. w\\big|_{W^{\\perp}} \\Big\\rangle\\text{ for any }w\\in \\Pi(v). \n\\label{conditionF2}\n\\end{equation}\nIndeed, for any $w\\in \\Pi(v)$ we have\n\\begin{equation*}\n\\begin{split}\n0=&\\Big\\langle F_{\\rho}(v,\\cdot), w\\Big\\rangle=\\Big\\langle \\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}+\\Big.F_{\\rho}\\Big|_{\\widetilde{O}(v)}+\\Big.F_{\\rho}\\Big|_{W^{\\perp}}, \\big.w\\big|_{V^{\\perp}} \\Big\\rangle \\\\\n& =\\Big\\langle \\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}+\\Big.F_{\\rho}\\Big|_{\\widetilde{O}(v)}+\\Big.F_{\\rho}\\Big|_{W^{\\perp}}, \\big. w\\big|_{\\widetilde{\\Pi}(v)} + \\big. w\\big|_{W^{\\perp}} \\Big\\rangle\\\\\n&=\\Big\\langle \\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)},\\big. w\\big|_{\\widetilde{\\Pi}(v)} \\Big\\rangle+\\Big\\langle \\Big.F_{\\rho}\\Big|_{W^{\\perp}}, \\big. w\\big|_{W^{\\perp}} \\Big\\rangle,\n\\end{split}\n\\end{equation*}\nsince $\\Big.F_{\\rho}(v,\\cdot)\\Big|_{V}=\\big.w\\big|_{\\widetilde{O}(v)}=0$. Note thus that condition \\eqref{conditionF2} (and hence \\eqref{edw2}) is independent of $\\Big. F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{O}(v)}$. \n\n\\bigskip\n\nLet $\\mathcal{B}_{1}=\\left\\{ e_{1},...,e_{\\text{dim}(V)} \\right\\}$, $\\mathcal{B}_{2}=\\left\\{ e_{\\text{dim}(V)+1},...,e_{\\text{dim}(V)+\\text{dim}(W)} \\right\\}$, $\\mathcal{B}_{3}=\\left\\{ e_{\\text{dim}(V)+\\text{dim}(W)+1},... \\right\\}$ be orthonormal bases of the spaces $V,W, W^{\\perp}$, respectively. Clearly, $\\mathcal{B}_{1}\\cup\\mathcal{B}_{2}\\cup\\mathcal{B}_{3}$ is an orthonormal basis of $L^{2}\\big(\\mathbb{S}^{2}\\big)$. Since $\\Pi(v)$ are spanned by smooth functions, it is easy to see that we can take $e_{i}$ to be smooth functions on $\\mathbb{S}^{2}$. Any function $F\\in C^{\\infty}\\big(I\\times \\mathbb{S}^{2}\\big)$ can be written as\n\\[F(v,\\cdot)=\\sum_{i\\geq 1} F_{i}(v)\\cdot e_{i}. \\]\nWe impose $\\big(F_{\\rho}\\big)_{i}(v)$ such that for $1\\leq i\\leq \\text{dim}(V)$ as follows:\n\\begin{itemize}\n\t\\item $\\big(F_{\\rho}\\big)_{i}(v)=0$ for all $v\\in I$, in accordance with \\eqref{conditionF1}. \n\\end{itemize}\n Moreover, we prescribe $\\big(F_{\\rho}\\big)_{i}(v)$ for $i\\geq \\text{dim}(V)+\\text{dim}(W)+1$ such that: \n\\begin{itemize}\n\t\\item $\\left\\{\\big(F_{\\rho}\\big)_{i}(v) \\right\\}_{i}\\in \\ell^{2}$ for all $v\\in I,$\n\\item \t$\\big(F_{\\rho}\\big)_{i}(v)=0 $ for all $v\\in I\/ \\big(I_{1}\\cup ....\\cup I_{\\text{dim}(W)}\\big),$ where $I_{k}$ as defined in \\eqref{orismosdiastimatos},\n\\item $\\big(F_{\\rho}\\big)_{i}(v)$ for $v\\in I_{1}\\cup ....\\cup I_{\\text{dim}(W)}$ are such that $\\int_{I}\\big(F_{\\rho}\\big)_{i}(v)dv$ are imposed by condition \\eqref{zitoumeno1}.\n\\end{itemize}\nClearly such functions exist and hence the functions $\\Big.F_{\\rho}\\Big|_{W^{\\perp}}(v)\\in C^{\\infty}\\big(\\mathbb{S}^{2}\\big)$ are determined and, in particular are such that they vanish for $v\\in I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$. \n\nWe next construct $\\Big.F_{\\rho}\\Big|_{W}(v)=\\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}(v)+\\Big.F_{\\rho}\\Big|_{\\widetilde{O}(v)}(v)$ or, equivalently, to construct $\\big(F_{\\rho}\\big)_{i}(v)$ for $\\text{dim}(V)+1\\leq i\\leq \\text{dim}(V)+\\text{dim}(W)+1$. We first impose \n\\begin{itemize}\n\t\\item $\\Big.F_{\\rho}\\Big|_{W}(v)=0$ for all $v\\in I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ and hence $\\big(F_{\\rho}\\big)_{i}(v)=0$ for $\\text{dim}(V)+1\\leq i\\leq \\text{dim}(V)+\\text{dim}(W)+1$ and $v\\in I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$. \n\\end{itemize}\nClearly, condition \\eqref{conditionF2} is then satisfied for all $v\\in I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$.\nIt remains to construct the functions $\\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}(v),\\, \\Big.F_{\\rho}\\Big|_{\\widetilde{O}(v)}(v)$ for $v\\in \\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$.\n\nWe note that condition \\eqref{conditionF2} uniquely determines a well-defined $\\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}(v)$ for all $v\\in\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$. Indeed, if $\\widetilde{\\Pi}(v)=\\left\\{0\\right\\}$ for some $v\\in \\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ then we have $\\widetilde{\\Pi}(v)=\\left\\{0\\right\\}$ for all $v\\in\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ and hence $\\Pi(v)=V$ for all $v\\in I$. In this case we take $W=\\left\\{0\\right\\}$ and \\eqref{conditionF2} holds trivially. If, on the other hand, the spaces $\\widetilde{\\Pi}(v)$ are non-trivial and vary smoothly for all $v\\in \\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ then condition \\eqref{conditionF2} determines $\\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}(v)$ for $v\\in\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$. Since $\\Big.F_{\\rho}\\Big|_{W^{\\perp}}(v)$ a smooth function in $I$ which vanishes in $I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ we obtain that $\\Big.F_{\\rho}\\Big|_{\\widetilde{\\Pi}(v)}(v)$, as defined above, is also a smooth function in $I$ which vanishes in $I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$. \n\nWe finally construct the function $\\Big.F_{\\rho}\\Big|_{\\widetilde{O}(v)}(v)$ for $v\\in \\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$ such that \\eqref{zitoumeno2} holds (recall that this function vanishes in the complement of this union in $I$). We will indeed show that this is possible.\n\nRecall that $\\widetilde{O}(v)$ varies smoothly in $v\\in I_{1}\\cup...\\cup I_{\\text{dim}(W)}$. Moreover, by the definition of the intervals $I_{1},...,I_{\\text{dim}(W)}$, there are vectors $x_{i}\\in \\widetilde{O}(v_{i})$ with $v_{i}\\in I_{i}, i=1,...,\\text{dim}(W)$ such that $\\left\\{x_{i},i=1,...,\\text{dim}(W)\\right\\}$ is a basis of $W$. We can construct a smooth curve \n\\[\\gamma:I\\rightarrow W \\]\nsuch that \n\\begin{equation}\n\\gamma(v_{i})=x_{i} \n\\label{gam1}\n\\end{equation}\nand \n\\begin{equation}\n\\gamma(v)=0 \\text{ for all } v\\in I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big).\n\\label{gam2}\n\\end{equation}\nThe smoothness of the curve $\\gamma$ can be guaranteed by the smoothness of $\\widetilde{O}(v)$ in $I_{1}\\cup...\\cup I_{\\text{dim}(W)}$. If $\\big(f_{1}(v),f_{2}(v),...,f_{\\text{dim}(W)}(v) \\big)$ are the coordinates of $\\gamma(v)$ with respect to the basis $\\mathcal{B}_{2}$ of $W$, then, in view of \\eqref{gam1}, the smooth functions $f_{i}:I\\rightarrow\\mathbb{R},i=1,...,\\text{dim}(W),$ are linearly independent. We will find an appropriate smooth function $\\alpha:I\\rightarrow\\mathbb{R}$ such that \n\\begin{equation}\n\\Big. F_{\\rho}\\Big|_{\\widetilde{O}(v)}=\\alpha(v)\\cdot \\gamma(v)\\in \\widetilde{O}(v). \n\\label{ansatz1}\n\\end{equation}\nLet us assume that $\\Big(\\big(F_{\\rho}\\big)_{1}(v),...,\\big(F_{\\rho}\\big)_{\\text{dim}(W)}(v) \\Big)$ are the coordinates of $\\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{O}(v)}$ with respect to the basis $\\mathcal{B}_{2}$ of $W$. Condition \\eqref{zitoumeno2} is satisfied if we choose these functions such that \n\\[ \\int_{I}\\big(F_{\\rho}\\big)_{i}(v)\\, dv=\\lambda_{i},\\]for all $i=1,2,...,\\text{dim}(W)$, \nwhere $\\lambda_{i}$ is completely determined by \\eqref{zitoumeno2} and our previous constructions. Equivalently, in view of our ansatz \\eqref{ansatz1}, it suffices to show the existence of a smooth function $\\alpha:I\\rightarrow\\mathbb{R}$ such that\\begin{equation}\n\\int_{I}\\alpha(v)\\cdot f_{i}(v)\\, dv=\\lambda_{i},\n\\label{eq:nai}\n\\end{equation}\nfor all $i=1,2,...,\\text{dim}(W)$. Since the functions $f_{i}$ are linearly independent, the existence of $\\alpha$ follows from Lemma \\ref{lemma1stoproof}. This completes the construction of the function $\\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{O}(v)}$. Note that in view of \\eqref{gam2} the function $\\Big.F_{\\rho}(v,\\cdot)\\Big|_{\\widetilde{O}(v)}$ vanishes in $I\/\\big(I_{1}\\cup...\\cup I_{\\text{dim}(W)}\\big)$. \n\nThis completes the construction of the function $F_{\\rho}$ with all the required properties.\n\\end{proof}\nLemma \\ref{lemma1stoproof} holds not only for intervals but also for union of intervals. Moreover, a modification of the above proof yields the following result\n\\begin{lemma}\nLet $I=\\bigcup_{k=1}^{n}I_{k}$, where $I_{k}$, with $k=1,2,...,n$, are compact intervals of $\\mathbb{R}$. For each $k\\in\\left\\{1,2,...,n\\right\\}$ consider $\\Pi_{k}(v), v\\in I_{k}$, to be a smoothly varying $n_{k}$-dimensional subspace of $L^{2}\\big(\\mathbb{S}^{2} \\big)$ spanned by $k$ smooth functions on $\\mathbb{S}^{2}$. Define the subspaces $V_{k}\\subset L^{2}\\big(\\mathbb{S}^{2} \\big)$ as follows\n\\[ V_{k}=\\bigcap_{v\\in I_{k}}\\Pi_{k}(v). \\]\nGiven a function $\\rho\\in C^{\\infty}\\big(\\mathbb{S}^{2}\\big)$ there is a function $F_{\\rho}\\in C^{\\infty}\\big(I\\times\\mathbb{S}^{2}\\big)$ which vanishes to infinite order at $\\partial I\\times \\mathbb{S}^{2}$ and is such that \n\\begin{equation}\n\\int_{I}F_{\\rho}(v,\\cdot)\\, dv= \\rho(\\cdot)\n\\label{eq:rel1}\n\\end{equation}\nand \n\\begin{equation}\nF_{\\rho}(v,\\cdot)\\in \\Big( \\Pi_{k}(v) \\Big)^{\\perp} \\text{ for all }v\\in I_{k} \\text{ and }k=1,2...,n\n\\label{eq:arbitragelemma}\n\\end{equation}\nif and only if \n\\[\\rho\\in \\Big(V_{1}\\cap V_{2}\\cap...\\cap V_{n}\\Big)^{\\perp}\\subset L^{2}\\big(\\mathbb{S}^{2}\\big). \\]\n\\label{argitragelemma}\n\\end{lemma}\n\\begin{proof}\nThe above lemma is proved using the same arguments as in the proof of Lemma \\ref{deuterolemma} if we replace the space $V$ by $V_{1}\\cap V_{2}\\cap...\\cap V_{n}$.\n\\end{proof}\n\n\\bigskip\n\nWe now have all the tools needed for the proof of Theorem \\ref{theorem}.\n\n Let $x\\in\\mathcal{Z}$ (where $\\mathcal{Z}$ is given by Lemma \\ref{lemmagiaelliptickernel}) and $V_{x}$ is a closed interval containing $x$ and such that $dimKer(\\mathcal{O}_{v}^{*})=n\\geq 1$. If we apply Lemma \\ref{deuterolemma} for $I=V_{x}$ and $\\Pi(v)=\\left\\langle G_{1}(v),...,G_{n}(v)\\right\\rangle$, where the functions $G_{n}$ given by \\eqref{g0}, then the only obstruction to satisfying conditions 1--3 (and thus to gluing) is the existence of the following intersection\n\\begin{equation}\n\\mathcal{W}(V_{x})=\\bigcap_{v\\in V_{x}}\\Pi(v). \n\\label{subkernel}\n\\end{equation}\nIn other words, if there is $x\\in \\mathcal{Z}$ such that $\\mathcal{W}(V_{x})=\\left\\{0\\right\\}$ then we can perform gluing. Suppose now that for all $x\\in\\mathcal{Z}$ we have $\\mathcal{W}(V_{x})\\neq \\left\\{0\\right\\}$. Let us fix an $x\\in\\mathcal{Z}$ and let $\\Theta\\in \\mathcal{W}(V_{x})$ such that $\\Theta\\neq 0$. By \\eqref{defF} and \\eqref{integrability}, and since $\\partial_{v}\\Theta=0$ for $v\\in V_{x}$, we have \n\\[0=\\int_{S_{v}}(\\mathcal{F}^{\\mathcal{D}}\\psi\\cdot\\phi)\\cdot\\Theta\\, d\\mu_{_{\\mathbb{S}^{2}}}=\\int_{S_{v}}\\Big(\\partial_{v}\\partial_{u}(\\phi\\cdot\\psi)\\Big)\\cdot\\Theta\\, d\\mu_{_{\\mathbb{S}^{2}}}=\\partial_{v}\\left(\\int_{S_{v}}\\Big(\\partial_{u}(\\phi\\cdot\\psi)\\Big)\\cdot\\Theta\\, d\\mu_{_{\\mathbb{S}^{2}}}\\right), \\]\nwhere we have also used that the measure of integration does not depend on $v$. Hence, the quantity\n\\begin{equation}\n\\int_{S_{v}}\\Big(\\partial_{u}(\\phi\\cdot\\psi)\\Big)\\cdot\\Theta\\ d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{eq:conserved}\n\\end{equation}\nis conserved in $V_{x}$, i.e.~independent of $v$ for all $v\\in V_{x}$. Therefore, we have conservation laws in each interval $V_{x}$ and the kernel of the conservation laws is precisely the space\n $\\mathcal{W}(V_{x})$. \n\nWe now consider two cases.\n\n\\medskip\n\n\\underline{ \\textbf{Case I:}} \n\n\\medskip\n\nSuppose that \n\\begin{equation}\n\\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x})=\\left\\{0\\right\\}.\n\\label{suppose1}\n\\end{equation}\nBy virtue of Lemma \\ref{lemmagiaelliptickernel}, we have that there is $n\\in\\mathbb{N}$ such that $\\text{dim}\\big(\\mathcal{W}(V_{x})\\big)\\leq n$ for all $x\\in\\mathcal{Z}$. In view of \\eqref{suppose1}, there are $x_{1},x_{2}\\in\\mathcal{Z}$ such that $\\mathcal{W}(V_{x_{1}})\\cap\\mathcal{W}(V_{x_{2}})$ is at most $(n-1)$-dimensional. In view again of \\eqref{suppose1}, there is $x_{3}\\in\\mathcal{Z}$ such that $\\mathcal{W}(V_{x_{1}})\\cap\\mathcal{W}(V_{x_{2}})\\cap\\mathcal{W}(V_{x_{3}})$ is at most $(n-2)$-dimensional. Continuing inductively we obtain $x_{1},...,x_{n+1}\\in\\mathcal{Z}$ such that $\\mathcal{W}(V_{x_{1}})\\cap\\mathcal{W}(V_{x_{2}})\\cap...\\cap\\mathcal{W}(V_{x_{n+1}})=\\left\\{0\\right\\}$. Thus, applying Lemma \\ref{argitragelemma} for $V_{x_{1}},...,V_{x_{n+1}}$ we can satisfy conditions 1--3 and thus perform gluing. The smoothness of the extension of $\\psi$ on $\\mathcal{H}$ follows from the results of Lemma \\ref{lemmagiaelliptickernel} and their analogue in higher Sobolev spaces. \n\n\n\\medskip\n\n\\underline{ \\textbf{Case II:}} \n\n\\medskip\n\nSuppose that \n\\begin{equation}\n\\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x})\\neq\\left\\{0\\right\\}.\n\\label{suppose1}\n\\end{equation}\nLet $\\Theta \\in \\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x})$ with $\\Theta\\neq 0$. Then the integrals \\eqref{eq:conserved} are conserved for $v\\in V_{x}$ for all $x\\in \\mathcal{Z}$. For any $x,y\\in\\mathcal{Z}$ with $V_{x}\\cap V_{y}\\neq \\emptyset$ we have that the integrals \\eqref{eq:conserved} are conserved for $v\\in V_{x}\\cup V_{y}$. We will show that the integrals \\eqref{eq:conserved} are in fact conserved for all $v\\in[0,1]$. Let us denote the integral \\eqref{eq:conserved} over $S_{v}$ by $f(v)$. Then $f$ is a smooth function on $[0,1]$ which is constant on the connected components of the open and dense subset $\\mathcal{Z}$ of $[0,1]$. If $f$ is not constant then there is $v_{0}\\in [0,1]$ for which $\\big.\\partial_{v}f\\big|_{v_{0}}\\neq 0$ and hence there is an interval containing $v_{0}$ where $f$ is strictly monotonic. This is however contradiction since, by assumption, for every interval there is a subinterval where $f$ is constant. \n\n\nTherefore, the integrals \\eqref{eq:conserved} are conserved for $v\\in[0,1]$ for all $\\Theta \\in \\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x})$. Clearly, these conservation laws are legitimate obstructions to gluing since we cannot do gluing unless the corresponding integrals of the initial data at $S_{0}$ and $S_{1}$ are equal. Suppose, therefore, that the initial data at $S_{0}$ and $S_{1}$ are such that the corresponding integrals \\eqref{eq:conserved} are equal. Then we will show that we can perform gluing (and hence there are no additional obstructions to gluing apart from the above conservation laws). \n\nIn view of \\eqref{wenull1} and \\eqref{malista} and the second condition for the gluing of transversal derivative we need to construct a function $C^{\\infty}\\big([0,1]\\times\\mathbb{S}^{2}\\big)$ such that \n\\begin{equation}\n\\int_{0}^{1}F(v,\\cdot)\\, dv= 2\\partial_{u}(\\phi\\cdot\\psi)_{\\big|_{S_{1}}}-2\\partial_{u}(\\phi\\cdot\\psi)_{\\big|_{S_{0}}}.\n\\label{apopano}\n\\end{equation}\nIn view of our assumption on the initial data at $S_{0},S_{1}$ we have that the right hand side of \\eqref{apopano} is orthogonal to the space $\\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x})$ with respect to the inner product of $L^{2}\\big(\\mathbb{S}^{2}\\big)$. Using a similar argument as above, we can choose finitely many $x_{1},...,x_{m}\\in \\mathcal{Z}$ such that $V_{x_{i}}\\cap V_{x_{j}}=\\emptyset$ for $i\\neq j$ and \n\\[\\mathcal{W}(V_{x_{1}})\\cap...\\cap\\mathcal{W}(V_{x_{m}})=\\bigcap_{x\\in\\mathcal{Z}}\\mathcal{W}(V_{x}). \\]\nIt now suffices to extend $\\psi$ in the complement of $V_{x_{1}}\\cup...\\cup V_{x_{m}}$ in $[0,1]$ and apply Lemma \\ref{argitragelemma} for the intervals $V_{x_{1}}, ...., V_{x_{m}}$ in order to complete the gluing construction. This finishes the proof of Theorem \\ref{theorem}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Change of foliation and conservation laws}\n\\label{sec:EffectOfGaugeOnConservationLaws}\n\nSo far we have consider characteristic gluing constructions and conservation laws on null hypersurfaces with respect to a \\textbf{fixed} foliation. In particular, it is clear the the conserved charges depend, at least a priori, on the choice of foliation\\footnote{After all, the conserved charges are appropriate integrals over the leaves of the foliation.}. In this subsection we address the issue of change of foliation and its effect on the conservation laws. \n\nLet $\\mathcal{S}=\\Big\\langle S_{0}, L_{geod}, \\Omega\\Big\\rangle$ and $\\mathcal{S}'=\\left\\langle S_{0}', L_{geod}', \\Omega'\\right\\rangle$ be two foliations of a regular null hypersurface $\\mathcal{H}$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions}. Let $\\mathcal{V}_{\\hh}$ be the linear space defined by \\eqref{linearspace}. Consider the kernels $\\mathcal{W}^{\\mathcal{S}},\\mathcal{W}^{\\mathcal{S}'}\\subset \\mathcal{V}_{\\hh}$ of the conservation laws with respect to the foliations $\\mathcal{S},\\mathcal{S}'$, respectively, as defined in Section \\ref{sec:ConservationLawsForTheWaveEquations}. Given also the operators $\\mathcal{O}^{\\mathcal{S}},\\mathcal{O}^{\\mathcal{S}'}$ associated to the foliations $\\mathcal{S},\\mathcal{S}'$, respectively, and defined by \\eqref{adjoint} we consider the linear spaces $\\mathcal{U}^{\\mathcal{S}},\\mathcal{U}^{\\mathcal{S}'}\\subset\\mathcal{V}_{\\hh}$ which are (appropriately rescaled) subspaces of $Ker\\big(\\mathcal{O}^{\\mathcal{S}}\\big),Ker\\big(\\mathcal{O}^{\\mathcal{S}'}\\big)$ defined as in Theorem \\ref{theorem}. Then, according to Theorem \\ref{theorem} we have\n\\begin{equation}\n\\mathcal{W}^{\\mathcal{S}}=\\mathcal{U}^{\\mathcal{S}}\\ \\text{ and }\\ \\mathcal{W}^{\\mathcal{S}'}=\\mathcal{U}^{\\mathcal{S}'}.\n\\label{eq:change1}\n\\end{equation}\nThe following proposition derives the relation between the spaces $\\mathcal{U}^{\\mathcal{S}}$ and $\\mathcal{U}^{\\mathcal{S}'}$. \n\\begin{proposition}\nLet $\\mathcal{S}=\\Big\\langle S_{0}, L_{geod}, \\Omega\\Big\\rangle$ and $\\mathcal{S}'=\\left\\langle S_{0}', L_{geod}', \\Omega'\\right\\rangle$ be two foliations of a (regular) null hypersurface $\\mathcal{H}$ of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions}. Let $f\\in\\mathcal{V}_{\\hh}$ be such that \n\\begin{equation}\nL_{geod}'=f^{2}\\cdot L_{geod}\\ : \\text{ on }\\, \\mathcal{H}.\n\\label{geodesicf}\n\\end{equation}\nThen, we have\n\\begin{equation}\n\\mathcal{U}^{\\mathcal{S}'}=f^{2}\\cdot\\mathcal{U}^{\\mathcal{S}}=\\Big\\{f^{2}\\cdot\\Theta\\ :\\ \\Theta\\in\\mathcal{U}^{\\mathcal{S}} \\Big\\}.\n\\label{changef12}\n\\end{equation}\n\\label{corm1}\n\\end{proposition}\n\\begin{proof}\nAccording to the main result of \\cite{aretakiselliptic}, the operators $\\mathcal{O}^{\\mathcal{S}}$ and $\\mathcal{O}^{\\mathcal{S}'}$ satisfy the following relation\n\\begin{equation}\n\\mathcal{O}^{\\mathcal{S}}\\left(\\frac{1}{\\phi}\\cdot\\Theta\\right)=\\left(\\frac{\\Omega}{\\Omega'}\\right)^{2}\\cdot\\frac{1}{f^{2}}\\cdot \\mathcal{O}^{\\mathcal{S}'}\\left(f^{2}\\cdot\\frac{1}{\\phi}\\cdot\\Theta \\right),\n\\label{fromotherpaper}\n\\end{equation}\non $\\mathcal{H}$, for all functions $\\Theta\\in\\mathcal{V}_{\\hh}$. Let now $\\Theta\\in\\mathcal{U}^{\\mathcal{S}}$ and $S'_{v'}$ be a section of the foliation $\\mathcal{S}'$. Then, $S'_{v'}$ can be sweeped by the sections $S_{v}$ of the foliation $\\mathcal{S}$ as depicted below. \n\\begin{figure}[H]\n \\centering\n\t\t\\includegraphics[scale=0.13]{nullsection2.png}\n\t\\label{fig:nullsectionxax}\n\\end{figure}\nFor each intersection point of $S'_{v'}$ with the sections of $\\mathcal{S}$ we apply the relation \\eqref{fromotherpaper} to deduce that \n\\[\\text{ If }\\ \\mathcal{O}_{v}^{\\mathcal{S}}\\left(\\frac{1}{\\phi}\\cdot \\Theta \\right)=0 \\ \\text{for all }v\\in\\mathbb{R}\\ \\text{ then }\\ \\mathcal{O}^{\\mathcal{S}'}_{v'}\\left(\\frac{1}{\\phi}\\cdot \\big(f^{2}\\cdot\\Theta \\big)\\right)=0. \\]\nHence, $ f^{2}\\cdot\\mathcal{U}^{\\mathcal{S}} \\subseteq \\mathcal{U}^{\\mathcal{S}'}$. The inverse inclusion can be similarly shown by sweeping the section of the foliation $\\mathcal{S}$ with the sections of $\\mathcal{S}'$, yielding the required result. \n\\end{proof}\nWe remark that the equation \\eqref{changef12} holds only if we can sweep the sections of the foliation $\\mathcal{S}$ with those of $\\mathcal{S}'$ and vice versa. This shows that the existence of conservation laws (and hence the obstructions to gluing) are due to global properties of the sections of the null hypersurface (and not just pointwise or local; cf. Remark 1 in Section \\ref{sec:Remarks}). Specifically, they are related with the properties of the kernels of the elliptic operators $\\mathcal{O}^{\\mathcal{S}}_{v}, v\\in\\mathbb{R}$. The calculation of \\cite{aretakiselliptic} showed that these elliptic operators are covariant (in the sense of \\eqref{fromotherpaper}) under change of foliation. This covariance property holds also for the kernels of these operators as long as the sweeping property holds. Hence, although capturing the elliptic structure associated to the sections of $\\mathcal{S}$ is of fundamental importance in the present paper, it turns out this elliptic structure does not depend on the choice of the foliation of $\\mathcal{H}$.\n\nIn view of the results of Theorem \\ref{theorem}, an immediate corollary of Proposition \\ref{corm1} is the following\n\\begin{corollary}\nA null hypersurface $\\mathcal{H}$ of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$ admits conservation laws with respect to a foliation $\\mathcal{S}=\\Big\\langle S_{0}, L_{geod}, \\Omega\\Big\\rangle$ in the sense of Definition \\ref{definitionconservationlaw} if and only it admits conservation laws with respect to any other foliation $\\mathcal{S}'=\\Big\\langle S_{0}', L_{geod}', \\Omega'\\Big\\rangle$. Specifically, the kernels $\\mathcal{W}^{S},\\mathcal{W}^{\\mathcal{S}'}$ of the conservation laws satisfy\n\\begin{equation}\n\\mathcal{W}^{\\mathcal{S}'}=f^{2}\\cdot\\mathcal{W}^{\\mathcal{S}}=\\Big\\{f^{2}\\cdot\\Theta\\ :\\ \\Theta\\in\\mathcal{W}^{\\mathcal{S}} \\Big\\},\n\\label{corequchangeoffoliation}\n\\end{equation}\nwhere $f$ is given by \\eqref{geodesicf}. In other words, the integrals \n\\begin{equation}\nchar\\big(S_{v}\\big)[\\psi]=\\int_{S_{v}}Y^{\\mathcal{S}}\\big(\\phi\\cdot\\psi\\big)\\cdot\\Theta \\, d\\mu_{_{\\mathbb{S}^{2}}}\n\\label{corollaryequation1}\n\\end{equation}\nwith $\\Theta\\in\\mathcal{V}_{\\hh}$ are conserved (i.e.~independent of $v$), for all solutions $\\psi$ to the wave equation, if and only if the integrals \n\\begin{equation}\nchar\\big(S'_{v'}\\big)[\\psi;\\Theta]=\\int_{S'_{v'}}f^{2}\\cdot Y^{\\mathcal{S}'}\\big(\\phi\\cdot\\psi\\big)\\cdot\\Theta \\, d\\mu_{_{\\mathbb{S}^{2}}} \n\\label{corollaryequation2}\n\\end{equation}\nare conserved, i.e.~independent of $v$, for all solutions $\\psi$ to the wave equation. The vector fields $Y^{\\mathcal{S}},\\, f^{2}\\cdot Y^{\\mathcal{S}'}$ are null and normal to the sections of $\\mathcal{S},\\mathcal{S}'$, respectively, conjugate to $\\mathcal{H}$ and normalized such that \n\\[g\\big(Y^{\\mathcal{S}},L_{geod}\\big)=g\\big(f^{2}\\cdot Y^{\\mathcal{S}'},L_{geod}\\big)=-1. \\]\nMoreover, if $\\mathcal{H}$ admits conservation laws then we in fact have\n\\begin{equation}\nchar\\big(S_{v}\\big)[\\psi;\\Theta]=char\\big(S'_{v'}\\big)[\\psi;\\Theta]\n\\label{corollaryequation3}\n\\end{equation}\nand hence the value of the conserved charges is independent of the choice of foliation. \n\\label{corollarychangeofoliation}\n\\end{corollary}\n\n\n\n\n\n\n\\subsection{Perturbation analysis}\n\\label{sec:PerturbativeAnalysis}\n\nWe next investigate the stability\/genericity properties of the conservation laws on a null hypersurface $\\mathcal{H}$ of a Lorentzian manifold $(\\mathcal{M},g)$ under perturbations of the ambient metric $g$ (i.e.~under perturbations of the geometry of $\\mathcal{H}$). We have the following\n\n\n\\begin{proposition}\nLet $\\mathcal{H}$ be a null hypersurface as in Theorem \\ref{theoremmainintro} which admits conservation laws for the wave equation on a Lorentzian manifold $(\\mathcal{M},g)$. There are arbitrarily small perturbations of the ambient metric $g$ for which $\\mathcal{H}$ is a null hypersurface without, however, admitting conservation laws (and hence gluing is possible on these perturbed null hypersurfaces). In fact, the set of all ambient metrics for which $\\mathcal{H}$ admits consevation laws is of positive codimension. \n\nFurthermore, if the underlying Lorentzian manifold $(\\mathcal{M},g)$ satisfies the Einstein--vacuum equations then we can perturb the metric $g$ in the class of spacetimes satisfying the Einstein--vacuum equations such that the above conclusions still hold.\n\n\\label{prop2pert}\n\\end{proposition} \n\\begin{proof}\nIn view of the results of Section \\ref{sec:EffectOfGaugeOnConservationLaws} it suffices to consider a geodesic foliation $\\mathcal{S}=\\left\\langle S_{0}, \\,L_{geod}, \\, \\Omega=1\\right\\rangle$ of $\\mathcal{H}$. By assumption we have that $dim\\mathcal{W}^{\\mathcal{S}}=\\dim\\mathcal{U}^{\\mathcal{S}}\\geq 1$, which, in particular, implies that $Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)\\neq \\left\\{0\\right\\}$ for all $v\\in\\mathbb{R}$. Recall that (for the unperturbed metric):\n\\begin{equation}\n\\mathcal{O}_{v}^{\\mathcal{S}}\\psi=\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+2\\zeta^{\\sharp}\\cdot\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi+\\Big[2\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\zeta+\\partial_{v}( tr\\underline{\\chi})+\\frac{1}{2}( tr\\underline{\\chi})\\cdot( tr\\chi) \\Big]\\cdot\\psi \n\\label{toxreiazomai}\n\\end{equation}\nWe will show that for any fixed $v$ there is an $\\epsilon_{1}>0$ such that for all $\\epsilon\\in(0,\\epsilon_{1})$ the operator\n\\[\\mathcal{O}_{v}^{\\mathcal{S},\\epsilon}\\psi=\\mathcal{O}_{v}^{\\mathcal{S}}\\psi-\\epsilon\\cdot \\psi\\]\nhas \\textit{trivial} kernel. Indeed, if we consider the operator\n\\[\\mathcal{O}_{temp}\\psi=\\mathcal{O}_{v}^{\\mathcal{S}}\\psi-\\frac{1}{\\epsilon_{0}}\\cdot \\psi \\]\nthen we can show as before that if we take $\\epsilon_{0}$ sufficiently small (depending on $v$) then the operator $\\mathcal{O}_{temp}$ is invertible and hence has a discrete spectrum consisting only of eigenvalues. Therefore, the operator $\\mathcal{O}_{v}^{\\mathcal{S}}$ has a discrete set of eigenvalues in the spectrum (whose limit point is infinity). By assumption, one of these eigenvalues is zero. By the discreteness of the spectrum, if we take $\\epsilon_{1}$ sufficiently small, then the operator $\\mathcal{O}_{v}^{\\mathcal{S},\\epsilon}$ has trivial kernel. \n\nTherefore, if we consider a perturbed metric $g_{\\epsilon}$ for which $\\mbox{$g \\mkern-8.8mu \/$\\,}_{\\epsilon}\\!\\left.\\right|_{S_{0}}=\\mbox{$g \\mkern-8.8mu \/$\\,}\\!\\left.\\right|_{S_{0}}$, $\\zeta_{\\epsilon}\\!\\left.\\right|_{\\mathcal{H}}=\\zeta\\!\\left.\\right|_{\\mathcal{H}},$ $\\mbox{tr}\\chi_{\\epsilon}\\!\\left.\\right|_{\\mathcal{H}}=\\mbox{tr}\\chi\\!\\left.\\right|_{\\mathcal{H}},$ $tr\\underline{\\chi}_{\\epsilon}\\!\\left.\\right|_{S_{0}}=tr\\underline{\\chi}\\!\\left.\\right|_{S_{0}}$, $\\partial_{v}tr\\underline{\\chi}_{\\epsilon}\\!\\left.\\right|_{S_{0}}=\\partial_{v}tr\\underline{\\chi}\\!\\left.\\right|_{S_{0}}+\\epsilon$ for some $v$. Then, \n\\[\\widetilde{\\mathcal{O}}_{v}^{\\mathcal{S}}=\\mathcal{O}^{\\mathcal{S},\\epsilon}_{v},\\]\nwhere $\\widetilde{\\mathcal{O}}_{v}^{\\mathcal{S}}$ denotes the elliptic operator $\\mathcal{O}_{v}^{\\mathcal{S}}$ with respect to the metric $g_{\\epsilon}$. By the above discussion and Proposition \\ref{pert1prop} we deduce that $\\mathcal{H}$ does not admit conservation laws as a null hypersurface embedded in the Lorentzian manifold $\\big(\\mathcal{M},g_{\\epsilon}\\big)$. \n\n\n\nWe next consider perturbations in the class of spacetimes satisfying the Einstein--vacuum equations. The freely prescribable initial data of the geometry of $\\mathcal{H}$ in the context of the characteristic problem of the Einstein--vacuum equations is the following:\n\\begin{enumerate}\n\t\\item The conformal geometry of $\\mathcal{H}$,\n\\item The metric of $S_{0}$,\n\t\\item The expansions $tr\\chi$ and $tr\\underline{\\chi}$ at $S_{0}$ and\n\t\t\\item The torsion $\\zeta$ at $S_{0}$.\n\\end{enumerate}\nIn our perturbation argument, we will fix the conformal geometry of $\\mathcal{H}$, the metric of $S_{0}$ and the torsion $\\zeta$ at $S_{0}$ and we will perturb the initial expansions $tr\\chi,tr\\underline{\\chi}$ at $S_{0}$.\n\nThe propagation equation for the transversal null second fundamental form yields (see \\cite{DC09})\n\\[\\partial_{v}tr\\underline{\\chi}=\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\zeta+|\\zeta|^{2}+\\rho-(\\chi,\\underline{\\chi}). \\]\nFurthermore, the Gauss equation gives us\n\\[\\rho-(\\chi,\\underline{\\chi})=-K-tr\\chi tr\\underline{\\chi}. \\]\nTherefore, by eliminating the left hand side of the above equation we obtain a linear propagation equation for $tr\\underline{\\chi}$ along $\\mathcal{H}$:\n\\begin{equation}\n\\partial_{v}tr\\underline{\\chi}=\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\zeta+|\\zeta|^{2}-K-tr\\chi tr\\underline{\\chi}.\n\\label{propaforzeta}\n\\end{equation}\nHence the coefficient of the zeroth order term in \\eqref{toxreiazomai} becomes\n\\begin{equation}\nw=3\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\zeta+|\\zeta|^{2}-K-\\frac{1}{2}( tr\\underline{\\chi})\\cdot( tr\\chi). \n\\label{eq:w}\n\\end{equation}\nBy assumption the first three terms on the right hand side are fixed under our perturbations. We can then freely perturb $tr\\chi$ and $tr\\underline{\\chi}$ such that \n\\[w_{\\epsilon}=w+\\epsilon\\]\nat $S_{0}$. The result then follows from the above discussion that Proposition \\ref{pert1prop}.\n\n\n\\end{proof}\n\n\n\\section{Black hole spacetimes}\n\\label{sec:KillingHorizons}\n\nLet $(\\mathcal{M},g)$ be a stationary spacetime with a black hole region (for the relevant definitions see \\cite{haw}). For such a spacetime the event horizon $\\mathcal{H}$ is a Killing horizon, that is there is a Killing vector field $\\xi$ normal to $\\mathcal{H}$. In this case $\\xi$ satisfies\n\\begin{equation}\n\\nabla_{\\xi}\\xi=\\kappa\\cdot \\xi\\, : \\text{ on }\\mathcal{H},\n\\label{surface}\n\\end{equation}\nwhere $\\kappa$ is constant along the null generators of $\\mathcal{H}$. In consistency with the zeroth law of black hole mechanics, we will assume that $\\kappa$ is globally constant on $\\mathcal{H}$ in which case $\\kappa$ is the so-called \\textit{surface gravity} of $\\mathcal{H}$. If $\\kappa=0$, then $\\mathcal{H}$ is called an \\textit{extremal horizon}. We will next investigate the existence of conservation laws on the event horizon $\\mathcal{H}$. \n\nThe following properties of Killing horizons were shown in \\cite{aretakiselliptic}:\n\\begin{lemma}\nLet $\\mathcal{H}$ be a Killing horizon and $\\mathcal{S}=\\left\\langle S_{0},L_{geod}, \\Omega=1\\right\\rangle$ be a geodesic foliation of $\\mathcal{H}$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions}. Assume that $\\xi$ is a Killing vector field normal to $\\mathcal{H}$ and such that \\eqref{surface} is satisfied. Then, the following relations hold on $\\mathcal{H}$: \n\\begin{enumerate}\n\t\\item $\\chi=0$, \n\\item $\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}\\mbox{$g \\mkern-8.8mu \/$\\,}=0$,\n\t\\item $\\mbox{$d \\mkern-9.2mu \/$\\,}\\kappa=g(\\xi, \\underline{L})\\cdot \\beta$, where the curvature component $\\beta$ is given by \\eqref{curvcompdeflist},\n\t\\item $\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}\\zeta=\\mbox{$\\nabla \\mkern-13mu \/$\\,}_{L}\\zeta=-\\beta$,\n\t\\item If, in addition, we take $\\left.L_{geod}\\right|_{S_{0}}=\\left.\\xi\\right|_{S_{0}}$ and $\\kappa$ is constant on $\\mathcal{H}$, then\n\t\n\t$\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}\\underline{\\chi}=\\mbox{$\\nabla \\mkern-13mu \/$\\,}_{L}\\underline{\\chi}=\\displaystyle\\frac{\\kappa}{f}\\cdot\\underline{\\chi}$, where $f$ is such that $\\xi=f\\cdot {L}_{geod}$ on $\\mathcal{H}$.\n\n\n\\end{enumerate}\n\\label{lemma}\n\\end{lemma}\n\n\n\n\n\\noindent Recall that since $\\Omega=1$ we have $L_{geod}=L=\\partial_{v}$. If we trace the last identity of the above lemma we obtain\n\\[L tr\\underline{\\chi}=\\frac{\\kappa}{f}\\cdot tr\\underline{\\chi}. \\]\nSince $Lf=\\kappa$ we obtain\n\\begin{equation}\ntr\\underline{\\chi}=\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot f\n\\label{trchibarlambda}\n\\end{equation} and so \n\\begin{equation}\n\\partial_{v} tr\\underline{\\chi}=\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot\\kappa.\n\\label{tracechibar}\n\\end{equation}\nTherefore, we obtain\n\\begin{equation}\n\t\\begin{split}\n\\mathcal{O}^{\\mathcal{S}}\\psi=\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+2\\zeta^{\\sharp}\\cdot\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi+\\left[2\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\,\\zeta^{\\sharp}+\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot\\kappa\\right]\\cdot\\psi,\n\\end{split}\n\\label{killingoperator}\n\\end{equation}\nwith respect to the foliation \\begin{equation}\n\\mathcal{S}=\\left\\langle S_{0}, \\left.L_{geod}\\right|_{S_{0}}=\\left.\\xi\\right|_{S_{0}},\\, \\Omega=1\\right\\rangle.\n\\label{foliationkilling}\n\\end{equation} \n\nSince $\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}\\mbox{$g \\mkern-8.8mu \/$\\,}=0$ all the sections of $\\mathcal{H}$ are isometric. Moreover, since $\\kappa$ is constant on $\\mathcal{H}$, and hence $\\mbox{$d \\mkern-9.2mu \/$\\,} k=0$, by Lemma \\ref{lemma}, we obtain that $\\beta=0$ on $\\mathcal{H}$ and hence $\\zeta$ is conserved on $\\mathcal{H}$, i.e.~$\\mbox{${\\cal L} \\mkern-9.5mu \/$}_{L}\\zeta=0$. \tFrom \\eqref{tracechibar}, $\\partial_{v} tr\\underline{\\chi}$ does not depend on $v$. By virtue of the equation $L\\phi=\\frac{1}{2}tr\\chi\\cdot\\phi$, the conformal factor $\\phi$ also does not depend on $v$. Therefore, the operators $\\mathcal{O}_{v}^{\\mathcal{S}}$ do \\textbf{not} depend on $v$ (modulo identifying the sections $S_{v}$ with $S_{0}$ via the diffeomorphisms $\\Phi_{v}$). \n\nIf we now consider a general foliation ${\\mathcal{S}'}=\\left\\langle S_{0},\\left.L_{geod}\\right|_{S_{0}}=\\left.\\xi\\right|_{S_{0}},\\Omega\\right\\rangle$ where $\\Omega$ is a smooth function on $\\mathcal{H}$, then by virtue of \\eqref{fromotherpaper} and using the fact that the geodesic vector field $L_{geod}$ is the same for both foliations $\\mathcal{S},\\mathcal{S}'$, we obtain\n\\[\\frac{1}{\\Omega^{2}}\\cdot \\mathcal{O}^{{\\mathcal{S}'}}\\Psi=\\mathcal{O}^{{\\mathcal{S}}}\\Psi\\ :\\ \\text{ for all }\\, \\Psi\\in\\mathcal{V}_{\\hh}. \\]\nand hence the operators $\\frac{1}{\\Omega^{2}}\\cdot \\mathcal{O}^{{\\mathcal{S}}'}_{v}$ do not depend on $v$ (again modulo identifying the sections $S_{v}'$ of $\\mathcal{S}'$ with $S_{0}$ via the diffeomorphisms $\\Phi_{v}$). \n\n\\begin{remark}\nGiven a foliation $\\mathcal{S}=\\left\\langle S_{0}, \\left.L_{geod}\\right|_{S_{0}}=\\left.\\xi\\right|_{S_{0}},\\, \\Omega=1\\right\\rangle$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions}, of a Killing horizon of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$, we can rewrite the operator $\\mathcal{O}^{\\mathcal{S}}_{v}$ given by \\eqref{killingoperator} as follows\n\\begin{equation}\n\\mathcal{O}_{v}^{\\mathcal{S}}\\psi=\\underbrace{\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+2\\zeta^{\\sharp}\\cdot\\mbox{$\\nabla \\mkern-13mu \/$\\,}\\psi+2\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\,\\zeta^{\\sharp}\\cdot \\psi}_{{\\mathcal{K}_{v}^{\\mathcal{S}}\\psi}}+\\underbrace{\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot\\kappa\\cdot\\psi}_{\\mathcal{T}_{v}^{\\mathcal{S}}\\psi},\n\\label{killingoperator1}\n\\end{equation}\nThe section $S_{0}$ can be freely chosen for the foliation $\\mathcal{S}$. In view of the above discussion the operator $\\mathcal{K}_{v}^{\\mathcal{S}}$ does not depend on the choice of the section $S_{0}$ (again, modulo identifying all sections of $\\mathcal{H}$ via the flow of the null generators). However, clearly the operator $\\mathcal{T}_{v}^{\\mathcal{S}}$ depends on $S_{0}$. Specifically, if we consider another foliation $\\mathcal{S}'=\\left\\langle S_{0}', \\left.L_{geod}\\right|_{S_{0}'}=\\left.\\xi\\right|_{S_{0}'},\\, \\Omega=1\\right\\rangle$ then in view of \\eqref{fromotherpaper}, and recalling that $L\\phi=0$, we have that\n\\begin{equation}\n\\mathcal{O}_{v}^{\\mathcal{S}}(\\Psi)=\\frac{1}{f^{2}}\\cdot\\mathcal{O}_{v}^{\\mathcal{S}'}\\big(f^{2}\\cdot\\Psi\\big)\\ :\\ \\text{ for all }\\, \\Psi\\in\\mathcal{V}_{\\hh},\n\\label{changeofsection}\n\\end{equation}\nwhere $f$ is such that $L_{geod}'=f^{2}\\cdot L_{geod}$, where $L_{geod}',L_{geod}$ denote the geodesic vector fields of $\\mathcal{S}',\\mathcal{S}$, respectively.\n\\label{remarkgiazeta}\n\\end{remark}\nIn view of the above proposition, the main Theorem \\ref{theorem} and the fact that $L\\phi=0$, i.e.~$\\phi\\in\\mathcal{V}_{\\hh}$, the existence of conservation laws along Killing horizons is equivalent to the non-triviality of the kernel $Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)$ for some fixed $v$. Specifically, we have shown that \n\\begin{equation}\ndim\\, \\mathcal{W}^{\\mathcal{S}}= dim\\, \\mathcal{U}^{\\mathcal{S}}= \\phi\\cdot Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big).\n\\label{killingapotelesma}\n\\end{equation}\n\n\nSummarizing we have shown the following\n\\begin{proposition}\nLet $\\mathcal{H}$ be a Killing horizon with constant surface gravity $\\kappa$ of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. Let also $S=\\left\\langle S_{0},\\left.L_{geod}\\right|_{S_{0}}=\\left.\\xi\\right|_{S_{0}},\\Omega\\right\\rangle$ be a foliation of $\\mathcal{H}$, as defined in Section \\ref{sec:NullFoliationsandOpticalFunctions}. Then the operators $\\frac{1}{\\Omega^{2}}\\cdot\\mathcal{O}^{\\mathcal{S}}_{v}$ do not depend on $v$ modulo identifying $S_{v}$ with $S_{0}$ via the diffeomorphism $\\Phi_{v}$, i.e.\n\\[ \\big(\\Phi_{v}\\big)^{*}\\left(\\frac{1}{\\Omega^{2}}\\cdot\\mathcal{O}_{v}^{\\mathcal{S}}\\right)=\\frac{1}{\\Omega^{2}}\\cdot\\mathcal{O}_{0}^{\\mathcal{S}} .\\] \nUnder the same identification we have \n\\begin{equation}\nKer\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=Ker\\big(\\mathcal{O}_{0}^{\\mathcal{S}}\\big),\n\\label{kernelrelation}\n\\end{equation}\nfor all $v\\in\\mathbb{R}$. Moreover, the kernel of the conservation laws along $\\mathcal{H}$ satisfies\n\\[dim\\, \\mathcal{W}^{\\mathcal{S}}= dim\\, \\mathcal{U}^{\\mathcal{S}}= \\phi\\cdot Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=\\left\\{\\phi\\cdot f\\, :\\, f\\in Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)\\right\\}.\\]\n\\label{lastprop}\n\\end{proposition}\n\n\n\n\n\n\n\n\\subsection{Conservation laws on extremal black holes}\n\\label{sec:ConservationLawsOnExtremalHorizons}\n\nBy definition, extremal black holes satisfy $\\kappa=0$. In this case the operator $\\mathcal{O}_{v}^{\\mathcal{S}}$ takes the form \n\\begin{equation}\n\\mathcal{O}_{v}^{\\mathcal{S}}\\psi=\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\big(2\\psi\\cdot\\zeta\\big)\n\\label{operatorextremal}\n\\end{equation}\nwith respect to the foliation $\\mathcal{S}$ given by \\eqref{foliationkilling}. \nFollowing the argument of Lucietti and Reall \\cite{hj2012} one obtains that $dim\\mathcal{U}^{\\mathcal{S}}=dimKer(\\mathcal{O}_{v}^{\\mathcal{S}})=1$ for all $v$. Indeed, since $\\int_{S_{v}}\\mathcal{O}^{\\mathcal{S}}_{v}\\psi=0$ for all $\\psi$, the unique positive principal eigenfunction $\\Psi$ of $\\mathcal{O}_{v}^{\\mathcal{S}}$ (see \\cite{andersson, evans}) must lie in the kernel of $\\mathcal{O}_{v}^{\\mathcal{S}}$. This shows the following\n\\begin{proposition}\nLet $\\mathcal{H}$ be an extremal horizon (i.e.~$\\kappa=0$) of a four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. If $\\mathcal{S}$ is the foliation of $\\mathcal{H}$ given by \\eqref{foliationkilling} then \n\\[dim\\,\\mathcal{W}^{\\mathcal{S}}=dim\\,\\mathcal{U}^{\\mathcal{S}}=dimKer(\\mathcal{O}_{v}^{\\mathcal{S}})=1.\\]\nTherefore, $\\mathcal{H}$ admits a unique conservation law with respect to the foliation $\\mathcal{S}$ (and thus with respect to any foliation). This law coincides with the \nconservation law found in \\cite{aretakis4, hj2012, murata2012}.\n\\label{propositionextremal}\n\\end{proposition}\n\n\nThe above conservation law coupled with dispersive estimates away from the event horizon forces higher order derivatives of generic solutions to the wave equation to blow up asymptotically along the event horizon (see \\cite{aretakis1,aretakis3}). We remark that the previous result is in stark contrast with the subextremal case for which Dafermos and Rodnianski \\cite{lecturesMD,enadio,tria} have derived quantitative decay estimates for all higher order derivatives in the exterior region up to and including the event horizon. \n\n\n\n\\subsection{Gluing constructions for sub-extremal black holes}\n\\label{sec:GluingConstructionsForNonExtremalHorizons}\n\nWe next consider sub-extremal Killing horizons and specifically such that $\\kappa>0$. We also assume that there exists a section $S_{0}$ such that $\\left.tr\\underline{\\chi}\\right|_{S_{0}}<0$ on $S_{0}$.\\footnote{Note that under these assumptions we can use the calculations in \\cite{aretakiselliptic} and the method of the present subsection to deduce that there must exist a section $S$ such that $\\left.tr\\underline{\\chi}\\right|=c,$ where $c<0$ is constant on $S$. } Let $\\mathcal{S}$ be the foliation given by \\eqref{foliationkilling} such that its `initial' section is the above one. In view of \\eqref{killingoperator} the operator $\\mathcal{O}_{v}^{\\mathcal{S}}$ can then be written as\n\\[\\mathcal{O}_{v}^{\\mathcal{S}}\\psi=\\mbox{$\\triangle \\mkern-13mu \/$\\,}\\psi+\\mbox{$\\text{div} \\mkern-16mu \/$\\,\\,}\\big(2\\psi\\cdot\\zeta\\big)+\\left[\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot\\kappa\\right]\\cdot\\psi.\\]\nLet $\\Psi>0$ be the unique (up to rescaling) positive principal eigenfunction of $\\mathcal{O}_{v}^{\\mathcal{S}}$ and let $\\lambda$ be its principal (maximum) eigenvalue. Then, we immediately obtain\n\\[ \\int_{S_{v}}\\left(\\left.tr\\underline{\\chi}\\right|_{S_{0}}\\cdot\\kappa\\right)\\cdot \\Psi \\, d\\mu_{_{\\mbox{$g \\mkern-8.8mu \/$\\,}}}=\\lambda \\cdot\\int_{S_{v}}\\Psi\\, d\\mu_{_{\\mbox{$g \\mkern-8.8mu \/$\\,}}}. \\]\nThe left hand side is manifestly negative which forces the maximum eigenvalue $\\lambda$ to be strictly negative and hence $ Ker\\big(\\mathcal{O}_{v}^{\\mathcal{S}}\\big)=\\left\\{0\\right\\}$. We have thus shown the following\n\\begin{proposition}\nLet $\\mathcal{H}$ be a Killing horizon with positive surface gravity $\\kappa>0$. We additionally assume that there is a spherical section $S_{0}$ of $\\mathcal{H}$ with negative transversal null expansion, i.e.~$\\big.tr\\underline{\\chi}\\big|_{S_{0}}<0$. Then $\\mathcal{H}$ does not admit any conservation laws (i.e.~$dim\\,\\mathcal{W}^{\\mathcal{S}}=0$) and hence gluing in the sense of Definition \\ref{firstordergluingdefinition} of characteristic data is always possible on $\\mathcal{H}$. \n\\label{gluingnonextremal}\n\\end{proposition}\nWe note that the non-existence of conservation laws on the event horizon $\\mathcal{H}$ of a subextremal Kerr black hole $(|a|0$, we obtain\n\\begin{equation*}\n\\begin{split} r^{2}\\partial_{u}\\partial_{v}(\\phi\\cdot\\psi)&=\\partial_{u}\\big(r^{2}\\partial_{v}(\\phi\\cdot\\psi)\\big)-2r\\partial_{u}r\\partial_{v}(\\phi\\cdot\\psi)\\\\\n&=\\left[\\partial_{u}\\big(r^{2}\\partial_{v}(r\\psi)\\big)-2r\\partial_{u}r\\partial_{v}(r\\psi)+\\partial_{u}\\big(r^{2}\\partial_{v}(r^{-a}\\psi)\\big)-2r\\partial_{u}r\\partial_{v}(r^{-a}\\psi)\\right]\\\\\n&\\rightarrow \\partial_{\nu}\\big(r^{2}\\partial_{v}(r\\psi)\\big).\n\\end{split}\n\\end{equation*}\n\\label{remarknkconse}\n\\end{remark}\n\\noindent The restriction $\\mathcal{O}_{u}^{\\underline{\\mathcal{S}}}$ on $S_{u}$ of the operator $\\mathcal{O}^{\\underline{\\mathcal{S}}}$ on $\\mathcal{I}$ is given by\n\\begin{equation}\n\\mathcal{O}_{u}^{\\underline{\\mathcal{S}}}=\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}\n\\label{eq:oni}\n\\end{equation}\nand hence is independent of $u$. This ``$u$-invariance'' of the operator $\\mathcal{O}^{\\underline{\\mathcal{S}}}_{u}$ is due to the BMS symmetry group of $\\mathcal{I}$ (see \\cite{wald}). Recall from Section \\ref{sec:KillingHorizons} that a similar result holds for Killing horizon and in fact the kernel of $\\mathcal{O}_{u}^{\\underline{\\mathcal{S}}}$ is isomorphic to the kernel of the associated operator on extremal horizons. This reveals yet another common property of $\\mathcal{I}$ and extremal horizons. \n\nIf $\\mathcal{W}^{\\underline{\\mathcal{S}}}$ denotes the kernel of the (limiting) conservation laws on $\\mathcal{I}$ and $\\mathcal{U}^{\\underline{\\mathcal{S}}}$ denotes the appropriately rescaled kernel of the operator $\\mathcal{O}^{\\underline{\\mathcal{S}}}$ (see \\eqref{rescaledu}) then, in view of \\eqref{operatorfori}, we have\n\\[dim\\,\\mathcal{W}^{\\underline{\\mathcal{S}}}=dim\\,\\mathcal{U}^{\\underline{\\mathcal{S}}}=1\\]\nand, therefore, there exists \\textbf{only} one (non-trivial limiting) conservation law on $\\mathcal{I}$, namely that given by the Newman--Penrose constants. Moreover, the result of Section \\ref{sec:EffectOfGaugeOnConservationLaws} applies for this conservation law. Specifically, if ${\\underline{\\mathcal{S}}}'$ is a foliation of $\\mathcal{I}$ and $\\widetilde{Y}^{\\underline{\\mathcal{S}}'}$ denotes the unique null conjugate to ${\\underline{\\mathcal{S}}}'$ vector field normalized such that $\\widetilde{Y}^{\\underline{\\mathcal{S}}'}r=1$ then the integrals\n\\[\\lim_{r\\rightarrow+\\infty}\\int_{S'_{u'}}r^{2}\\cdot \\widetilde{Y}^{\\underline{\\mathcal{S}}'}(r\\cdot\\psi) \\, d\\mu_{_{\\mathbb{S}^{2}}} \\]\nare conserved, i.e.~independent of $u$. \n\nSummarizing we have shown the following\n\\begin{proposition}\nLet $\\underline{\\mathcal{S}}=\\big(S_{u}\\big)_{u\\in\\mathbb{R}}$ be a foliation of the null infinity $\\mathcal{I}$ of an asymptotically flat spacetime $(\\mathcal{M},g)$, as defined in Section \\ref{sec:TheAsympoticGauge}. Then, the appropriately rescaled operator $\\mathcal{O}^{\\underline{\\mathcal{S}}}$ given by \\eqref{operatorfori} is $u$-invariant, i.e.~the operators $\\mathcal{O}^{\\underline{\\mathcal{S}}}_{u}$ do not depend on $u$. Moreover, $dim\\,\\mathcal{W}^{\\underline{\\mathcal{S}}}=dim\\,\\mathcal{U}^{\\underline{\\mathcal{S}}}=1$ and the unique associated conservation law on $\\mathcal{I}$ gives rise to the (first-order) Newman--Penrose constant. \n\\label{nullinfiprop}\n\\end{proposition}\n\n\n\n Note that all the higher order Newman--Penrose constants can be obtained by commuting the wave equation with $\\partial_{v}^{k}$ (see also Section \\ref{sec:Genericity}). \n\n\n\n\n\\section{Spherical symmetry}\n\\label{sec:Genericity}\n\nIn this section we investigate the existence of higher order conservation laws. Although our method applies for general spacetimes for the sake of simplicity we focus on spherically symmetric backgrounds. \n\n\n\nLet $\\mathcal{H}$ be a spherically symmetric null hypersurface and $\\mathcal{S}=\\big(S_{v}\\big)_{v\\in\\mathbb{R}}$ be a spherically symmetric foliation on $\\mathcal{H}$ in a spherically symmetric four-dimensional Lorentzian manifold $(\\mathcal{M},g)$. Then the wave equation restricted on $\\mathcal{H}$ can be written as\n\\begin{equation*}\n\\begin{split}\n\\Box_{g}\\psi\n=&-2\\partial_{u}\\partial_{v}(r\\psi)+ \\mathcal{O}^{\\mathcal{S}}(r\\psi)=0, \n\\end{split}\n\\end{equation*}\nwhere \\[\\mathcal{O}^{\\mathcal{S}}\\psi=\\Omega^{2}\\frac{1}{r^{2}}\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}\\psi +2\\Omega^{2}\\cdot \\frac{(\\partial_{u}\\partial_{v}r)}{r}\\cdot \\psi.\\]\nWe next assume that $\\Omega=1$ on $\\mathcal{H}$. Since all the expressions are spherically symmetric, in view of Theorem \\ref{theorem}, we have a (first order) conservation law if and only if \n\\[\\frac{2\\partial_{u}\\partial_{v} r}{r}=\\frac{l(l+1)}{r^{2}}, \\]\nfor some $l\\in\\mathbb{N}$. \n\n\n\\subsection{Higher order conservation laws}\n\\label{sec:HigherOrderConservationLaws}\n\nFor a given $n\\in\\mathbb{N}$, we want to find necessary and sufficient conditions under which we can glue general data \n\\begin{equation}\n\\big.\\psi\\big|_{S_{0}}, \\ \\ \\ \\left.\\partial_{u}^{k}\\psi\\right|_{S_{0}},\\ \\ \\ 1\\leq k\\leq n\n\\label{data1}\n\\end{equation}\non $S_{0}$ to general data \n\\begin{equation}\n\\big.\\psi\\big|_{S_{1}}, \\ \\ \\ \\left.\\partial_{u}^{k}\\psi\\right|_{S_{1}},\\ \\ 1\\leq k\\leq n, \n\\label{data2}\n\\end{equation}\non $S_{1}$ as explained in Section \\ref{sec:TheCharacteristicInitialValueProblem}. \n\n\n\nAs we shall see the only obstruction to such gluings is higher order conservation laws. By decomposing $\\psi$ in angular frequencies we can assume that it is supported on the $l$ angular frequency. Then, the wave equation on $\\mathcal{H}$ reads\n\\begin{equation*}\n\\begin{split}\n\\Box_{g}\\psi\n=&-2\\partial_{u}\\partial_{v}(r\\psi)+ \\mathcal{O}^{\\mathcal{S}}(r\\psi)=0, \n\\end{split}\n\\end{equation*}\nwhere \\[\\mathcal{O}^{\\mathcal{S}}\\psi=\\Omega^{2}\\cdot\\left[2\\frac{\\partial_{u}\\partial_{v} r}{r}-\\frac{l(l+1)}{r^{2}} \\right]\\cdot\\psi.\\]\nSet \n\\begin{equation}\n\\Psi=r\\psi, \\ \\ \\ c_{1}=\\Omega^{2}\\cdot\\left[\\frac{\\partial_{u}\\partial_{v} r}{r}-\\frac{l(l+1)}{2r^{2}} \\right],\\ \\ \\ Y=\\partial_{u}.\n\\label{phi}\n\\end{equation}\nThe data for $\\Psi $ must satisfy the constraint equations\n\\begin{equation}\n\\partial_{v}\\big(Y\\Psi \\big)=c_{1}\\cdot\\Psi , \\ \\ \\ \\partial_{v}\\big(Y^{k+1}_{u}\\Psi \\big)=Y^{k}\\big(c_{1}\\cdot\\Psi \\big), \\ k\\geq 1,\n\\label{constraint}\n\\end{equation}\n By integrating along the null generators we obtain\n\\begin{equation}\n(Y\\Psi )(\\tau)=(Y\\Psi )(0)+\\int_{0}^{1}c_{1}\\cdot\\Psi ,\n\\label{con1}\n\\end{equation}\nand more generally\n\\begin{equation}\n(Y^{k+1}\\Psi )(\\tau)=(Y^{k+1}\\Psi )(0)+ \\int_{0}^{1}Y^{k}(c_{1}\\cdot\\Psi )\n\\label{conk}\n\\end{equation}\nWe consider first the case $k=1$. Clearly, we want to construct $\\Psi $ such that \n\\begin{equation*}\n\\begin{split}\n\\int_{0}^{1}c_{1}\\cdot\\Psi =\\alpha_{1}, \\ \\ \\ \n\\int_{0}^{1}\\Big((Yc_{1})\\cdot\\Psi + c_{1}\\cdot (Y\\Psi )\\Big)d\\tilde{\\tau}=\\alpha_{2}, \n \\end{split}\n\\end{equation*}\nwhere $\\alpha_{1},\\alpha_{2}$ are arbitrary real numbers. Using \\eqref{con1}, with $\\tau\\mapsto\\tilde{\\tau}$, the above is equivalent to \n\\begin{equation*}\n\\begin{split}\n\\int_{0}^{1}\\bigg[(Y c_{1} )\\cdot\\Psi + c_{1}\\cdot (Y\\Psi )(0)+ c_{1}\\cdot \\int_{0}^{\\tilde{\\tau}} c_{1}\\cdot \\Psi \\bigg]d\\tilde{\\tau}=\\alpha_{2}.\n\\end{split}\n\\end{equation*}\nIf $c_{1}=0$ but and $Yc_{1}=0$ then we have two conservation laws. If $c_{1}=0$ but $Yc_{1}\\neq 0$ then we have a first order conservation law and gluing for second order derivatives. If $c_{1}\\neq 0$ for all $v\\in[a,b]$ then we compute\n\\begin{equation*}\n\\begin{split}\n\\alpha_{2}=&\\int_{0}^{a}\\bigg[(Y c_{1} )\\cdot\\Psi + c_{1}\\cdot (Y\\Psi )(0) + c_{1}\\cdot \\int_{0}^{\\tilde{\\tau}} c_{1}\\cdot \\Psi \\bigg]d\\tilde{\\tau}+\\int_{b}^{1}\\bigg[(Y c_{1} )\\cdot\\Psi + c_{1}\\cdot (Y\\Psi )(0) \\bigg]d\\tilde{\\tau}\\\\\n&+\\int_{b}^{1}\\bigg[ c_{1}\\cdot \\int_{0}^{a} c_{1}\\cdot \\Psi + c_{1}\\cdot \\int_{b}^{\\tilde{\\tau}} c_{1}\\cdot \\Psi \\bigg]d\\tilde{\\tau}\\\\\n&+\\int_{a}^{b}\\bigg[ c_{1}\\cdot (Y\\Psi )(0)+ c_{1}\\cdot \\int_{0}^{a} c_{1}\\cdot \\Psi \\bigg]d\\tilde{\\tau} \\\\\n&+\\int_{a}^{b}\\bigg[(Y c_{1} )\\cdot\\Psi + c_{1}\\cdot \\int_{a}^{\\tilde{\\tau}} c_{1}\\cdot \\Psi \\bigg]d\\tilde{\\tau}+\n\\Bigg(\\int_{a}^{b} c_{1}\\cdot \\Psi \\Bigg)\\cdot\\Bigg(\\int_{b}^{1}c_{1}\\Bigg). \n \\end{split}\n\\end{equation*}\nNote that the quantities in the first three lines depend only of the values of $\\Psi $ in the region $\\mathcal{C}=[0,a]\\cup[b,1]$ and on $(Y\\Psi )(0)$. We want to extend $\\Psi$ everywhere in $[0,1]$ so we can do gluing. It suffices to construct $\\Psi $ in $[a,b]$ such that \n\\begin{equation*}\n\\begin{split}\n\\int_{a}^{b} c_{1}\\cdot \\Psi =\\beta_{1},\\\\\n\\int_{a}^{b}\\bigg[(Y c_{1} )\\cdot\\Psi &+ c_{1}\\cdot \\int_{a}^{\\tilde{\\tau}} c_{1}\\cdot \\Psi \\bigg]d\\tilde{\\tau}=\\beta_{2},\n\\end{split}\n\\end{equation*}\nwhere $\\beta_{1},\\beta_{2}$ are given.\nDefine the function\n\\begin{equation}\n\\Phi_{1} :[a,b]\\rightarrow\\mathbb{R}: \\ \\ \\ \\Phi_{1} (t)=\\int_{a}^{t} c_{1}\\cdot \\Psi .\n\\label{phidef}\n\\end{equation}\nNote $\\Phi_{1} '(t)=c_{1}(t)\\cdot\\Psi (t)$ and hence, since $c_{1}\\neq 0$ in $[a,b]$, the function $\\Phi_{1} $ determines $\\Psi $ in $[a,b]$. Therefore, it suffices to construct $\\Phi_{1}$ such that $\\Phi_{1} (b)=\\beta_{1}$ and all derivatives of $\\Phi$ at $b$ are prescribed and such that \n\\begin{equation}\n\\begin{split}\n\\int_{a}^{b}\\bigg[\\frac{Y c_{1} }{c_{1}}\\cdot \\Phi_{1} '+ c_{1}\\cdot \\Phi_{1} \\bigg]d\\tilde{\\tau}=\\beta_{2}\n\\end{split}\n\\label{conditions1}\n\\end{equation}\nNow, by integration by parts we obtain\n\\[\\int_{a}^{b}\\frac{Y c_{1} }{c_{1}}\\cdot \\Phi_{1} '=\\frac{(Y c_{1} )(b)}{c_{1}(b)}\\cdot\\Phi_{1} (b)-\\int_{a}^{b}\\bigg(\\frac{Y c_{1} }{c_{1}}\\Bigg)'\\cdot\\Phi_{1} . \\]\nTherefore, if $f=-\\bigg(\\frac{Y c_{1} }{c_{1}}\\Bigg)'$ then \\eqref{conditions1} can be rewritten\n\\begin{equation}\n\\begin{split}\n\\int_{a}^{b}\\bigg[(f+ c_{1}) \\cdot\\Phi_{1} \\bigg]d\\tilde{\\tau}=\\beta_{3}.\n\\end{split}\n\\label{conditions2}\n\\end{equation}\nClearly, this has a solution if and only if $(f+ c_{1}) \\neq 0$ at a point $\\tau_{0}\\in[a,b]$. Since\n\\begin{equation*}\n\\begin{split}\nf+c_{1}=-\\frac{(Y c_{1} )'\\cdot c_{1}-c_{1}'\\cdot(Y c_{1} )-c_{1}^{3}}{c_{1}^{2}}\n\\end{split}\n\\end{equation*}\nwe obtain that we have glue data to second order if $c_{1}\\neq 0$ and \n\\begin{equation}c_{2}=\nc_{1}^{3}+(\\partial_{v} c_{1}) (\\partial_{u} c_{1}) - c_{1}\\cdot (\\partial_{v}\\partial_{u} c_{1}) \\neq 0. \n\\label{h1condition}\n\\end{equation}\nWe will next show that if $c_{2}=0$ (and $c_{1}\\neq 0$) on $\\mathcal{H}$ then we have a second order conservation law. \n\n Recall the definition \\eqref{phidef} for the function $\\Phi_{1} $. Then, in view of \\eqref{con1} and \\eqref{conk} we obtain\n\\[\\Phi_{1} '= c_{1}\\cdot \\Psi ,\\]\n\\begin{equation*}\n\\begin{split}\n (Y\\Psi )(\\tau)&=(Y\\Psi )(0)+\\Phi_{1} (\\tau)\n \\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split} (YY\\Psi)(\\tau)&=(YY\\Psi)(0)+\\int_{0}^{\\tau}\\bigg[(Yc_{1})\\cdot\\Psi+c_{1}\\cdot (Y\\Psi)(0)+c_{1}\\cdot \\int_{0}^{\\tilde{\\tau}}c_{1}\\cdot\\Psi \\bigg]d\\tilde{\\tau}\\\\\n &=(YY\\Psi)(0)+\\int_{0}^{\\tau}\\bigg[\\frac{Yc_{1}}{c_{1}}\\cdot\\Phi_{1}' +c_{1}\\cdot(Y\\Psi)(0)+c_{1}\\cdot\\Phi_{1} \\bigg]d\\tilde{\\tau}\\\\\n &=(YY\\Psi)(0)+(Y\\Psi)(0)\\cdot\\bigg(\\int_{0}^{\\tau}c_{1}\\bigg) +\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\cdot\\Phi_{1}(\\tau)+\\int_{0}^{\\tau}\\bigg[\\bigg(-\\left(\\frac{Yc_{1}}{c_{1}}\\right)'+c_{1}\\bigg)\\cdot\\Phi_{1} \\bigg]d\\tilde{\\tau}\\\\\n &=(YY\\Psi)(0)+(Y\\Psi)(0)\\cdot\\bigg(\\int_{0}^{\\tau}c_{1}\\bigg) +\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\cdot\\Phi_{1}(\\tau)\\\\\n &=(YY\\Psi)(0)+(Y\\Psi)(0)\\cdot\\bigg(\\int_{0}^{\\tau}c_{1}\\bigg) +\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\cdot\\Big((Y\\Psi)(\\tau)-(Y\\Psi)(0)\\Big)\\\\\n&=(YY\\Psi)(0)+\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\cdot\\big((Y\\Psi)(\\tau)\\big)+(Y\\Psi)(0)\\cdot\\bigg[\\bigg(\\int_{0}^{\\tau}c_{1}\\bigg)-\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\bigg]\\\\\n&=(YY\\Psi)(0)+\\frac{(Yc_{1})(\\tau)}{c_{1}(\\tau)}\\cdot\\big((Y\\Psi)(\\tau)\\big)-(Y\\Psi)(0)\\cdot\\frac{(Yc_{1})(0)}{c_{1}(0)},\\\\\n \\end{split}\n\\end{equation*}\nwhere we repeatedly used that $c_{1}=\\left(\\frac{Y c_{1} }{c_{1}}\\right)'$. Therefore, if $c_{1}\\neq 0$ and $c_{1}=\\left(\\frac{Y c_{1} }{c_{1}}\\right)'$ then the quantity\n\\begin{equation}\nY^{2}\\Psi -\\frac{Y c_{1} }{c_{1}}\\cdot Y\\Psi \n\\label{newcon1}\n\\end{equation}\nis conserved on $\\mathcal{H}$. \n\nThe above provides a scheme in order to find necessary and sufficient conditions for the existence of higher order conservations laws. Clearly, these conservation laws are the only obstruction to gluing.\n\n\nWe next consider third order gluing constructions. We have\n\\begin{equation*}\n\\begin{split}\n(Y^{3}\\Psi )(\\tau)-(Y^{3}\\Psi )(0)=&\\int_{0}^{\\tau}\\bigg[c_{1}\\cdot(Y^{2}\\Psi )(0)+2\\cdot(Y c_{1} )\\cdot(Y\\Psi )(0)+(Y^{2} c_{1}) \\cdot\\Psi \\bigg]d\\tau_{1}\\\\\n&+ \\int_{0}^{\\tau}2(Y c_{1} )\\Bigg[\\int_{0}^{\\tau_{1}} c_{1}\\cdot \\Psi \\Bigg]d\\tau_{1}\\\\\n&+\\int_{0}^{\\tau}\\Bigg[c_{1}\\cdot\\int_{0}^{\\tau_{1}}\\bigg[\\bigg(\n(Y c_{1} )\\cdot\\Psi +c_{1}\\cdot(Y\\Psi )(0)+c_{1}\\cdot\\int_{0}^{\\tau_{2}}c_{1}\\cdot\\Psi \\Bigg)\\bigg]d\\tau_{2} \\Bigg]d\\tau_{1}.\n\\end{split}\n\\end{equation*}\nThen,\n\\begin{equation*}\n\\begin{split}\n&(Y^{3}\\Psi )(\\tau)-(Y^{3}\\Psi )(0)\n\\\\=& \\int_{0}^{\\tau}\\bigg[ c_{1}\\cdot (Y^{2}\\Psi )(0)+2\\cdot(Y c_{1} )\\cdot(Y\\Psi )(0)+\\frac{Y^{2}c_{1}}{c_{1}}\\cdot (\\Phi _{1})'+2(Y c_{1} )\\cdot\\Phi _{1} +(Y\\Psi )(0)\\cdot c_{1}\\cdot\\int_{0}^{\\tau_{1}}c_{1}\\bigg]d\\tau_{1}\n\\\\&+\\int_{0}^{\\tau}\\Bigg[(Y c_{1} )\\cdot\\Phi _{1}+c_{1}\\cdot\\int_{0}^{\\tau_{1}}\\bigg[ \\bigg(-\\left(\\frac{Y c_{1} }{c_{1}}\\right)'+c_{1}\\Bigg)\\cdot\\Phi _{1} \\bigg]d\\tau_{2}\\Bigg]d\\tau_{1}.\\\\\n\\end{split}\n\\end{equation*}\nWe now define the function\n\\[\\Phi _{2}(t)=\\int_{0}^{t}\\bigg(-\\left(\\frac{Y c_{1} }{c_{1}}\\right)'+c_{1}\\Bigg)\\cdot\\Phi _{1}\\]\nand the function\n\\[c_{2}=\\bigg(-\\left(\\frac{Y c_{1} }{c_{1}}\\right)'+c_{1}\\Bigg)\\neq 0.\\]\nWe thus obtain\n\\begin{equation*}\n\\begin{split}\n(Y^{3}\\Psi )(\\tau)-(Y^{3}\\Psi )(0)=&\\int_{0}^{\\tau}\\bigg[ c_{1}\\cdot (Y^{2}\\Psi )(0)+2\\cdot(Y c_{1} )\\cdot(Y\\Psi )(0)+(Y\\Psi )(0)\\cdot c_{1}\\cdot\\int_{0}^{\\tau_{1}}c_{1}\\bigg]d\\tau_{1}\n\\\\&+\\frac{(Y^{2} c_{1}) (\\tau)}{c_{1}(\\tau)\\cdot c_{2}(\\tau)}\\cdot\\Phi '_{2}(\\tau)\n+\\frac{1}{c_{2}(\\tau)}\\Bigg(-\\left(\\frac{Y^{2}c_{1}}{c_{1}}\\right)'+3(Y c_{1} )\\Bigg)(\\tau)\\cdot\\Phi _{2}(\\tau)\\\\&\n+\\int_{0}^{\\tau}\\Bigg[\\Bigg(\\Bigg(\\frac{1}{c_{2}}\\Bigg(\\left(\\frac{Y^{2}c_{1}}{c_{1}}\\right)'-3(Y c_{1} )\\Bigg)\\Bigg)'+c_{1}\\Bigg)\\cdot\\Phi _{2}\\Bigg]d\\tau_{1}.\\\\\n\\end{split}\n\\end{equation*}\nThe first line is completely determined by the initial data and the geometry of $\\mathcal{H}$. The second line is determined (by the geometry of $\\mathcal{H}$) and by $\\Phi _{2}(\\tau), \\Phi '_{2}(\\tau)$. \n\n \n Hence, we can arbitrarily prescribe $\\Psi (\\tau), Y\\Psi (\\tau),Y^{2}\\Psi (\\tau),Y^{3}\\Psi (\\tau)$ if and only if the function\n \\begin{equation}\nc_{3}=\\Bigg(\\Bigg(\\frac{1}{c_{2}}\\Bigg(\\left(\\frac{Y^{2}g}{g}\\right)'-3(Y c_{1} )\\Bigg) \\Bigg) '+g\\Bigg)\n\\label{eq:3}\n\\end{equation}\nis non-zero at at least a point on $\\mathcal{H}$. If, on the other hand, we have $c_{3}=0$ everywhere on $\\mathcal{H}$ then the quantity\n\\begin{equation*}\n Y^{3}\\Psi +\\frac{1}{c_{2}}\\cdot\\left(\\left(\\frac{Y^{2}c_{1}}{c_{1}}\\right)'-3(Y c_{1} )\\right)\\cdot Y^{2}\\Psi +\\Bigg[\\frac{Y c_{1} }{c_{1}}\\cdot\\left[\\frac{1}{c_{2}}\\left(\\left(\\frac{Y^{2}c_{1}}{c_{1}}\\right)'-3(Y c_{1} )\\right)\\right]+\\frac{Y^{2}c_{1}}{c_{1}}\\Bigg] \\cdot Y\\Psi \n\\label{phi3conserved}\n\\end{equation*}\nis conserved, i.e.~independent of $\\tau$. Using the above scheme, Theorem \\ref{theo4} can be proved inductively.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Some examples}\n\\label{sec:AnExampleOfAMetricForWhichC1Neq0AndC20}\n\n\n\\paragraph{1. An example of a metric for which $c_{1}\\neq 0$ and $c_{2}=0$ for $l=0$\\medskip\\\\}\n\\label{sec:AnExampleOfAMetricForWhichC1Neq0AndC20}\n\n\n\nConsider a spherically symmetric metric such that \n\\begin{equation}\n\\Omega(u,v)=1, \\ \\ \\ r(u,v)=\\frac{1}{2}+\\frac{1}{2}(1+uv)^{2}>0. \n\\label{examplec1c2}\n\\end{equation}\nLet now $\\mathcal{H}=\\left\\{u=0\\right\\}$ along which $r=1, \\partial_{u}r=v, \\partial_{u}\\partial_{u}r=v^{2}$. Then along $\\mathcal{H}$ we obtain $c_{1}=1$ and $c_{2}=0$. Therefore, $\\mathcal{H}$ admits a second-order conservation law for all spherically symmetric solutions to the wave equation. This example shows that the order of the conservation law and the angular frequency associated to its kernel are independent.\n\n\\paragraph{2. Conservation law on the Cauchy horizon of Reissner--Nordstr\\\"{o}m\\medskip\\\\}\n\\label{sec:AnExampleOfdsaf}\n\n\nThe Reissner--Norstr\\\"{o}m metric satisfies:\n\\[\\partial_{u}\\partial_{v}r=-\\frac{\\Omega^{2}}{4r}-\\frac{1}{r}\\partial_{v}r\\partial_{u}r+\\frac{1}{4}\\Omega^{2}r^{-3}e^{2}. \\]\nOn the (inner) horizon we have $\\partial_{v}r=0$ and in fact \n\\[r=M-\\sqrt{M^{2}-e^{2}}.\\]\nThen, we have a first-order conservation law if:\n\\begin{equation}\n\\partial_{v}\\partial_{u}r=\\Omega^{2}\\cdot l(l+1)\\cdot\\frac{1}{2r}. \n\\label{malistatwra}\n\\end{equation}\nEquation \\eqref{malistatwra} is satisfied for a discrete set of values for the charge $e$. Indeed, we need $e$ to satisfy:\n\\[\\left(\\frac{e}{r}\\right)^{2}=\\frac{2l(l+1)+1}{2}.\\]\nIn this case the kernel of the conservation law consists of all the eigenfunctions of the standard spherical Laplacian $\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}$ which correspond to the eigenvalue $-l(l+1)$.\n\n\\paragraph{3. A hierarchy of conservation laws for some spacetimes\\medskip\\\\}\n\\label{sec:AnExampleOfdsaf}\n\nThe spherically symmetric null hypersurfaces of Minkowski spacetimes, the null infinity of asymptotically flat spacetimes and the event horizon of extremal Reissner--Norstr\\\"{o}m and extremal Kerr black holes admit the following hierarchy of conservation laws. Specifically, we have $R_{l+1,l}=0$ for all $l\\in\\mathbb{N}$. Hence, for all $l\\in\\mathbb{N}$ there is an $(l+1)$-order conservation law and its kernel consists of all the eigenfunctions of the standard spherical Laplacian $\\mbox{$\\triangle \\mkern-13mu \/$\\,}_{\\mathbb{S}^{2}}$ which correspond to the eigenvalue $-l(l+1)$.\n\n\n\n\n\\section{Acknowledgements}\n\\label{sec:Acknowledgements}\n\tI would like to thank Mihalis Dafermos, Georgios Moschidis, Willie Wong and Shiwu Yang for their help and insights. I would also like to thank Harvey Reall, Sergiu Klainerman, Jan Sbierski and Jeremy Szeftel for several very stimulating discussions and comments. I acknowledge support through NSF grant DMS-1265538.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\n\nDynamic systems have been investigated since the time of Newton, but the interest for numerical experiments grew considerably from the works realised by Lorenz \\citep{Lorenz} in 1960. These works were carried out with the purpose of understanding the behaviour of nonlinear dynamic systems \\citep{Grebogi}. Among the dynamic systems, those without an analytical solution are usually investigated by means of numerical computation, such as nonlinear systems with chaotic behaviour. Thus, the numerical computation plays a fundamental role in the analysis of nonlinear dynamic systems \\citep{Lozi,Nepomuceno}\n \nOne of the most important way to study dynamic systems are the discrete maps.\nThese maps can also be seen as recursive functions, which allow the description and solution to a vast set of problems \\citep{Feigenbaum}. They are the basis for the resolution of most nonlinear dynamic systems \\citep{Lozi}. Using discrete maps it is possible to build bifurcation diagrams, which although much studied, do not have a well-defined set of rules for its application. As it has been already seen in \\citep{Paiva}, the initial condition can be crucial in the constructing bifurcation diagram of the logistic map. In this paper, we are interested in doing a similar investigation of \\citep{Paiva} for quadratic map \\citep{Lorenz,Galias}, which has an important application in encryption of images based on chaos \\citep{Ramadan,Kar}.\n\n\\section{PRELIMINARY CONCEPTS}\n\\label{sec:concepts}\n\n\\subsection{Recursive Functions}\nRecursive functions that take the form $x_{n+1}=f(x_n)$ \ndescribe a wide range of problems. Being $\\mathbb{I} \\subseteq \\mathbb{R}$ a metric space with $f:\\mathbb{I} \\rightarrow \\mathbb{R}$ recursive function can be described as \\citep{Nepomuceno}:\n\n\\begin{equation}\n\\centering\nx_n=f(x_{n-1}). \n\\label{eq01}\n\\end{equation}\n\n\\subsection{Fixed Point}\n\nThe fixed point $x^*$ of a map is the point such that $x_{n+1}=x_n=x^*$. That is, starting from $x^*$, \nremains in $x^*$ in the next iteration. Therefore, $x^*$ satisfies the following relation $x^*=f(x^*)$, \\citep{Monteiro}.\n\n\\subsection{Quadratic Map}\n The quadratic map \\citep{Lorenz} is described by the following function:\n \\begin{equation}\n x_{n+1}=a-x_n^2, \n \\label{eq02}\n\\end{equation}\n\n\\noindent\nwhere, \\textit{n} \nis the number of iterations and \\textit{a} is the bifurcation parameter.\n\\vspace{-3.3cm}\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[angle=0, scale=0.5 ]{pontofixo1.pdf}\n\\vspace{-3.3cm}\n\\caption{Result of the values of $x_n$ for n=50 and $a=1,9$.}\n\\label{fig1}\n\\end{figure}\n\n\\subsection{Bifurcation Diagram}\n\nThe term bifurcation is generally used to refer to the qualitative transition from regular to chaotic behavior by changing the control parameter \\citep{Ramadan}, \ngiven any initial condition.The bifurcation diagram is used to study the system in function of its control parameter, allowing to know regions of the system that converge to bifurcation or even to chaos, depending on the parameter. \\citep{Monteiro}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[angle=0, scale=0.4 ]{diagrama_mapa_quadratico.pdf}\n\\caption{Diagram of bifurcation for the quadratic map.}\n\\label{fig2}\n\\end{figure}\n\n\\subsection{Floating-Point Arithmetic}\n\nNumerical computing is a critical part of dynamic systems analysis. Therefore, it is necessary to emphasize the IEEE 754-2008 standard \\citep{IEEE} for floating-point arithmetic, since a large majority of numerical computation makes use of this norm. It establishes the rounding behavior, error handling and degree of accuracy of the calculations performed, causing the results to approach the expected value. Even with satisfactory results, numerical computation is not yet able to present results that are totally in agreement with reality, due to the fact that it has a finite representation of real numbers. \\citep{Rodrigues,IEEE}.\n\n\\section{METHODOLOGY}\n\\label{sec:methodology}\n\nThe behaviour of the quadratic map is determined by the variation of the bifurcation parameter $a$, \nwhere $-2.5 < a <2.5$. Thus, given an initial condition belonging to the defined interval, the function will have its behaviour as shown in the bifurcation diagram of Fig. (1a).\n\nGiven the equation $x_{n+1}=a-x_n^2$ which describes the Quadratic Map, it is possible to obtain its fixed point $x^*$ doing $x_{n+1}=x_n=x^*$. Thus the map equation takes the following form:\n \\begin{equation}\n \\centering\n x^2 + x - a =0. \n \\label{eq03}\n\\end{equation}\nFrom Eq.(\\ref{eq03}), it is possible to find a value for \\textit{x} depending on the parameter \\textit{a} which is a fixed point, shown by Eq. (\\ref{eq04}).\n\\begin{equation}\n\\centering\n x_1=\\frac{\\sqrt{4a+1}-1}{2};\n x_2=-\\left ( \\frac{1+\\sqrt{4a+1}}{2} \\right )\n \\label{eq04}.\n\\end{equation}\n\nThe use of the inverse of the function \\eqref{eq03}, presented in Eq. (\\ref{eq05}), allows to verify the existence of a set of points that converge to the fixed point $x^*$. \n\n\\begin{equation}\n\\centering\n f^{-1}(x^*)=\\pm \\sqrt{a-x^*}.\n\\label{eq05}\n\\end{equation}\n\n\\thispagestyle{empty} \nFrom Eq. (\\ref{eq05}) the following result was obtained:\n\\begin{equation}\n\\centering\n x = \\pm \\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}.\n \\label{eq06}\n\\end{equation}\nEq. (\\ref{eq06}) is so used as the initial condition $x_0$ to build the bifurcation diagram. Thereby, \nit was possible to show that $\\pm \\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}$ belongs to the set of points that converge to the fixed point $x^*$.\n\n\\begin{equation*}\n x_1 = f(x) = a - \\left [\\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}\\right ]^2 = \\frac{\\sqrt{4a+1}-1}{2};\n\\end{equation*}\n\n\\begin{equation}\n\\centering\n x_2 = f(x_1) = a - \\left [\\frac{\\sqrt{4a+1}-1}{2}\\right ]^2 = a - \\left [\\frac{4a+1 -2\\sqrt{4a+1} +1}{4}\\right] = \\frac{\\sqrt{4a+1}-1}{2};\n \\label{eq07}\n\\end{equation}\n\\vspace{0.1cm}\n\n\\noindent\nand $x_1 = x_2 = ... = x_n$, then $d(f^p(x),x^*) \\to 0$ when $p \\to \\infty$. \n\n\n\\section{RESULTS}\n\nThe bifurcation diagrams are shown in Fig. (\\ref{fig5}) analysis of the behavior of the map given a value for \\textit{a}. When $a=1,9$ the map assumes an unstable fixed-point region, shown in Fig.(\\ref{fig1}). The effect of the application of $ x_0=\\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}$ as initial condition can be observed by Fig. (\\ref{fig3}) where to $a=1,9$, constraint to the initial condition, it was possible to find a convergence for $x^*$, showing a result that is considerably different from that found in Fig. (\\ref{fig1}), \nwhere no restrictions were made for the initial condition and only 50 iterations were performed.\n\n\\newpage\n\\begin{figure}[h!]\n\\centering\n\\vspace{-3.3cm}\n\\includegraphics[angle=0, scale=0.5 ]{pontofixo2.pdf}\n\\vspace{-3.3cm}\n\\caption{Result of the values of \\textit{x} for n=200 and $a=1,9$.}\n\\label{fig3}\n\\end{figure}\n\nFrom these results, it is necessary to obtain a new bifurcation diagram from the initial condition $ x_0=\\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}$. \nThis diagram is shown in Fig.(\\ref{fig4}). The diagrams shown in Figs.(\\ref{fig4}) and (\\ref{fig2}) were obtained in conventional manner in the literature using Matlab and double precision floating point arithmetic.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[angle=0, scale=0.4 ]{novo_diagrama_mq.pdf}\n\\caption{Bifurcation Diagram for Initial Condition $x_0=\\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}$.}\n\\label{fig4}\n\\end{figure}\n\nThe result shown in Fig.(\\ref{fig4}) is largely incorrect, although, due to its similarity with Fig.(\\ref{fig2}), could be easily understood as correct. With initial condition given by Eq.(\\ref{eq06}), the correct answer is shown in Fig. (\\ref{fig5}), which the transient is not discharged, and on the contrary, the bifurcation diagram has been built using only the first result of the iteration process, that is, using only $x_1$. Similar results were found by \\citep{Paiva}, for the logistic map.\n\\newpage\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[angle=0, scale=0.4 ]{diagrama_real.eps}\n\\caption{Bifurcation diagram for initial condition $x_0=\\sqrt{\\left ( a -\\frac{\\sqrt{4a+1}-1}{2} \\right )}$.}\n\n\\label{fig5}\n\\end{figure}\n\n\\section{CONCLUSIONS} \n\\label{sec:conclusions}\nAfter all the analyses, it is concluded that for the construction of the bifurcation diagram of the quadratic map the value of the initial condition has a fundamental role. The diagrams shown in Figs. (\\ref{fig2}), (\\ref{fig4}) and (\\ref{fig5}) show different results. For Fig. (\\ref{fig2}), the diagram was obtained following the methodology proposed in the literature for an equal initial condition $x_0 = 0.2$. While in the simulations of the diagrams shown in Figs. (\\ref{fig4}) and (\\ref{fig5}), a constraint associated with the parameter $ a $ has been applied. In Fig. (\\ref{fig4} the numerical simulation did not present the expected results because using the proposed initial condition, for any value of $ a $, convergence occurs at the first iteration. However, it was observed that after a number of iterations, the result diverges. This divergence can be understood as the result of a numerical error due to the finite-precision of the floating-point arithmetic. Therefore, it is necessary more care on the constructing bifurcation diagram of the quadratic map.\n\n\\section{ACKNOWLEDGEMENTS}\n\nWe thank CAPES, Fapemig and UFSJ for the support.\n\\section{REFERENCES} \n\\bibliographystyle{abcm}\n\\renewcommand{\\refname}{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}