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+{"text":"\\section{Introduction}\nFor a conventional orthogonal class system, it is believed that an arbitrarily weakly uncorrelated diagonal disorder in one and two dimension \\cite{Economou} can result in the Anderson localization \\cite{Anderson1957}.\nIn three dimension, there exists mobility edge $E_c$ which separates the localized states from extended states \\cite{Economou1972}. When the eigenenergies approach the mobility edge $E_c$, the localization length of localized states  would diverge. Interestingly, in the presence of off-diagonal uncorrelated disorders, one-dimensional system can have a singular density of states near the zero energy \\cite{Dyson,Eggarter1978,Balents1997}, which also results in an anomalous localization \\cite{Theodorou1976,Antoaiou1977} that the localization length is proportional to the square root of system  size \\cite{Fleishman1977,Inui1992,Izrailev2012}.\nIf the energy deviates from zero, the eigenstates are usually localized states.\nIf the off-diagonal disorder is correlated, the system can have localized-extended transition \\cite{Cheraghchi2005}.\nIn the presence of both diagonal and correlated off-diagonal disorders, the one-dimensional system can also have extended states \\cite{Zhangwei2004}.\nRecently, a so-called mosaic lattice model with diagonal quasiperiodic disorder has been proposed \\cite{Wangyucheng2020}. It is found that this model  has mobility edges and the mobility edges can be exactly obtained with Avila's theory \\cite{Liu2021,Avila2015}.\n\n\n\n\nSince there exist above anomalous properties of  localizations in the  model with a pure off-diagonal uncorrelated disorder,\na natural question arises, how is it if there the off-diagonal hopping is quasiperiodic?\n One may wonder whether there exist mobility edges for off-diagonal quasiperiodic disorder (hopping). What are the localization properties of eigenstates?\n\n\n\n\n\n\n\n\n\n\n\n\nIn this work, we try to answer the above questions by exploring a quasiperiodic off-diagonal disorder model with mosaic modulation. The model is\n \\begin{align}\\label{2}\nV_{i,i+1}\\psi(i+1)+V_{i,i-1}\\psi(i-1)=E\\psi(i).\n\\end{align}\nwhere\n \\begin{align}\nV_{i,i+1}=V_{i+1,i}\n\\left\\{\\begin{array}{cc}\nt,  & for \\ i\\neq 0 \\ mod \\ \\kappa\\\\\n\\frac{2\\lambda \\cos(2\\pi \\beta i+\\phi)}{\\sqrt{1-\\tau \\cos^2(2\\pi \\beta i+\\phi)}},& for \\ i= 0 \\ mod \\ \\kappa\n  \\end{array}\\right.\n\\end{align}\nwhere $t>0$ is constant hopping strength,  $\\lambda$ describes the quasi-periodic hopping strength, positive integer $\\kappa$ is mosaic period, $\\beta$ is an irrational number, and the parameter $\\tau$ is a real number.\nIn this work,  we only consider the parameter $\\tau$ is not larger than $1$, i.e., $\\tau\\leq 1$.\nIn addition, we should remark that there is no extended state in the above model Eq.(1). This is because by \\cite{Barry1989} (or the mechanism in \\cite{xwzl}), the absolutely continuous spectrum which corresponds extended states is empty since there exists a sequence $\\{n_k\\}$ such that $V_{n_k,n_{k+1}}\\rightarrow0$. Thus the mobility edges (if any ) would separate localized states and critical states.\nIn the whole paper, we take $\\beta=(\\sqrt{5}-1)\/2$ and  use the units of $t=1$ for $\\kappa>1$ (or $\\lambda=1$ for $\\kappa=1$).\n\n\n\n\n\nIt is found that the parity of mosaic period $\\kappa$ has important influences on the localization of eigenstates near zero-energy. To be specific, if mosaic period $\\kappa$ is odd, there is no Anderson localization for arbitrarily strong hopping strength.  While for even mosaic period, the system undergoes Anderson localization as the quasiperiodic hopping increases.\nIn addition, the Lyapunov exponent $\\gamma(E)$ and mobility edges are also exactly obtained with the Avila's theory.\nWith Lyapunov exponent, we find that,  some critical regions in the parameter plane would appear.\n In comparison with the localized states, the spatial extensions of eigenstates and their fluctuations in the critical region are much larger.\nNear localized-critical transition points ($E_c$),  the localization length diverges, i.e.,\n\\begin{align}\\label{10}\n\\xi(E)\\equiv1\/\\gamma(E)\\propto|E-E_c|^{-\\nu}\\rightarrow\\infty, \\ \\ as \\ E\\rightarrow E_c,\n\\end{align}\nwhere the critical index \\cite{Huckestein1990} $\\nu=1$.\nFinally, we show that the systems with different parameter $E$ can be systematically classified  by Lyapunov exponent and Avila acceleration.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThe work is organized as follows. First of all, we discuss the localization properties of zero-energy states for both odd and even number $\\kappa$ in Sec.\\textbf{II}.  In Sec.\\textbf{III}, the Lyapunov exponent is calculated.  Next, with the Lyapunov exponent, we  determine the mobility edges and critical region  in Sec.\\textbf{IV}. In addition, Avila's acceleration is also calculated.\n At the end, a summary is given in Sec.\\textbf{V}.\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{bu0.eps}\n\\end{center}\n\\caption{ The average growth\/decreasing ratio of zero-energy wave function for $\\kappa=2$, and $\\tau=-2$.  The critical hopping strength $\\lambda=\\lambda_c\\simeq\\pm1.366t$ where $f$ is exactly zero (indicated by black arrows in the figure).  }\n\\label{bu0}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{bu1.eps}\n\\end{center}\n\\caption{Several typical zero-energy wave functions of critical states and localized edge states. Panels (a) and (c) are critical zero-energy wave functions where $f=0$. Panel(b) [(d)] is  localized  right (left)-hand edge states where $f>0$ ($f<0$).}\n\\end{figure}\n\n\n\\section{localization of zero-energy state}\nIn this section, we discuss the influences of parity of integer $\\kappa$ on the localization properties of the zero energy states.\nFor the quasi-periodic model Eq.(1), we note that if one applies a transform $\\psi(n)\\rightarrow(-1)^n\\psi(n)$ in the Eq.(1), the energy would change a sign, i.e., $E\\rightarrow -E$.\nDue to the chiral (sublattice) symmetry, the energy $E_n$ and $-E_n$ appear in pairs \\cite{Cheraghchi2005}.\nIn addition, the number of eigenenergies is same with the lattice site number. If the total lattice site number $N$ is an odd number, then there would be one zero energy state at least.\nIn the following, we find that when $N$ is even, usually there is no zero-energy eigenstates. In this section, we assume the total lattice number $N$ is odd, then the zero-energy state always exists.\n\n\n\nFurthermore, we assume the lattice sites of system are labeled with number $i=1,2,3,...,2m, N=2m+1$, where $m$ is a positive integer.\nStarting from wave functions of left-hand end site $\\psi(i=1)=1$ [and $\\psi(i=0)=0$], by Eq.(1),\nthe wave function of zero-energy state can be written as\n\\begin{align}\n&\\psi(N=2m+1)=\\frac{V_{2m,2m-1}V_{2m-2,2m-3}...V_{4,3}V_{2,1}}{V_{2N,2N+1}V_{2m-2,2m-1}...V_{4,5}V_{2,3}}\\psi(i=1),\\notag\\\\\n&=\\frac{v(2m-1)v(2m-3)...v(3)v(1)}{v(2m)v(2m-2)...v(4)v(2)}.\n\\end{align}\nIn the above equation, we set $v(i)\\equiv v(i,\\phi)\\equiv V_{i,i+1}$ and use the relation $V_{i+1,i}=V_{i,i+1}$.\nThe average growth\/decreasing ratio of wave function  is\n\\begin{align}\n&f=\\lim_{m\\rightarrow\\infty}\\frac{1}{2m}ln(|\\frac{\\psi(N=2m+1)}{\\psi(i=1)}|)\\notag\\\\\n&=\\lim_{m\\rightarrow\\infty}\\frac{1}{2m}ln(|\\frac{v(2m-1)v(2m-3)...v(3)v(1)}{v(2m)v(2m-2)...v(4)v(2)}|).\n\\end{align}\nIf $f\\geq0$,  $f$ would be the Lyapunov exponent $\\gamma(E)$ (see Sec. \\textbf{III}).\n\n\n\n\n\n\n\n\n\n\\subsection{$\\kappa$ is a positive even integer}\n When $\\kappa$ is a positive even integer, we can assume $N=n\\kappa+1$, where $n$ is an integer.\nBased on  Eqs. (1) and (5), due to the ergodicity of the map $\\phi\\longrightarrow 2\\pi\\beta i+\\phi$, the average growth\/decreasing rate of wave function can be reduced into\n\\begin{align}\n&f=\\lim_{n\\rightarrow\\infty}\\frac{1}{n\\kappa}ln(\\frac{|v(\\kappa-1)v(2\\kappa-1)...v(n\\kappa-1)|}{|v(\\kappa)v(2\\kappa)...v(n\\kappa)|}),\\notag\\\\\n&=\\frac{-1}{\\kappa\\times 2\\pi}[\\int_{0}^{2\\pi}d\\phi ln(|v(\\kappa,\\phi)|)],\\notag\\\\\n&=\\frac{-1}{\\kappa}ln(\\frac{2|\\lambda\/t|}{1+\\sqrt{1-\\tau}}).\n\\end{align}\n\n\nIt is shown that when $\\kappa$ is a positive even integer, for a generic  $\\lambda$, $f$ is usually not zero. Then the zero-energy state would be localized states which may situate at right-hand edge ($f>0$) or left-hand edge ($f<0$) of the lattices (see Figs.1 and 2). So for general parameters, the zero-energy state would be a localized edge state.\nOnly  when $f$ is exactly vanishing, i.e., $f=0$,\n\\begin{align}\n&\\rightarrow f=\\frac{-1}{\\kappa}ln(\\frac{2|\\lambda\/t|}{1+\\sqrt{1-\\tau}})=0\\notag\\\\\n&\\rightarrow |\\lambda\/t|=|\\lambda_c\/t|\\equiv\\frac{1+\\sqrt{1-\\tau}}{2},\n\\end{align}\n the zero-energy state would be  a critical state (see Figs.1 and 2).\nThe average growth\/decreasing rate $f$ for $\\kappa=2$ and $\\tau=-2$ is reported in Fig.1. At the critical strength $\\lambda_c=\\pm 1.366t$, $f=0$.\nWhen $\\lambda$ approaches the critical $\\lambda_c$, the localization  length can be arbitrarily large, i.e.,  $\\xi\\equiv 1\/|f|\\propto 1\/|\\lambda-\\lambda_c|\\rightarrow\\infty$.\n\n\n\n\nIn order to investigate the properties of zero-energy states, we also numerically solve Eq.(1) for $\\kappa=2$, $\\tau=-2$, and lattice size $N=2\\times500+1$.\n  Several typical wave functions for localized states and critical states are reported in Fig.2.\n  We know that the wave function of extended state usually extends all over the whole lattices, while localized state only occupies finite lattice sites.\nThe critical state consists of several disconnected patches which interpolates between the localized and extended states \\cite{yicai,Liu2022}.\nFrom Fig.2, we see the zero energy wave function of $f=0$ is critical state. The wave functions with non-vanishing $f$ correspond to localized edge states.\nWhen $f>0$, the state is at right-hand end edge, while for $f<0$, it is at left-hand end edge.\n\n\n\n\n\\subsection{$\\kappa$ is a positive odd integer}\nWhen $\\kappa$ is a positive odd integer, we can assume $N=2n\\kappa+1$, where $n$ is an integer.\n  Similarly, $f$ can be written as\n\\begin{align}\n&f=\\lim_{n\\rightarrow\\infty}\\frac{1}{2n\\kappa}ln(\\frac{|v(\\kappa)v(3\\kappa)...v((2n-1)\\kappa)|}{|v(2\\kappa)v(4\\kappa)...v(2n\\kappa)|}),\\notag\\\\\n&=\\frac{1}{2\\kappa\\times 2\\pi}[\\int_{0}^{2\\pi}d\\phi ln(|v(\\kappa,\\phi)|)-\\int_{0}^{2\\pi}d\\phi ln(|v(2\\kappa,\\phi)|)],\\notag\\\\\n&=0.\n\\end{align}\nIt is shown that when $\\kappa$ is a positive odd integer, the average growth\/decreasing ratio of zero-energy wave function is exactly zero.\nSo if $\\kappa$ is odd, all the zero-energy states are critical states.\n\n\n Some interesting even-odd effects of lattice site number $N$ have been investigated in the random off-diagonal disorder models.\nIt is found that there  exists a delocalization transition only when lattice size $N$ is odd \\cite{Brouwer1998}. In addition, the localization length of the zero-energy state depends sensitively boundary conditions \\cite{Brouwer2002}, and it can be an arbitrarily large value.\n\n\n\n\n The above discussions show that the parity of integer $\\kappa$ has important influences on the localization of zero-energy states.\n An odd integer $\\kappa$ results in a critical zero-energy state, while an even $\\kappa$ usually gives the localized edge states (see Fig.2).\n We notice that the above discussion can be also applied to other forms of quasiperiodic hopping with mosaic modulations.\n\n\nIn the following text, we will show that the above influences of parity of $\\kappa$ are also transmitted to other eigenstates near the zero energy.\nTo be specific, when $\\kappa$ is a positive odd integer, for a given hopping strength $\\lambda$,  the eigenstates  near zero energy are always critical (see Sec.IV). So if energy is sufficiently near the zero-energy, there is no Anderson localization transition for odd $\\kappa$.\nWhen $\\kappa$ is a positive even integer, the eigenstates near zero-energy would undergo  Anderson localization transition as the quasiperiodic hopping strength increases.\nThen the system would have localized states near the zero-energy.\n\n\n\n\n\\section{The  Lyapunov exponent  }\n When $E\\neq0$, the  localization properties of eigenstates  can be characterized by the Lyapunov exponent. In this section,  we calculate the Lyapunov exponent with  the transfer matrix method \\cite{Sorets1991,Davids1995}.\n\nFirst of all,  we assume the system is a half-infinite lattice system with left-hand end sites $i=0$ and $i=1$.\nFurther using Eq.(1), starting with $\\psi(0)$ and $\\psi(1)$ of left-hand end sites, the wave function can be obtained with relation\n\\begin{align}\n\\Psi(i)=T(i)T(i-1)...T(2)T(1)\\Psi(0)\n\\end{align}\nwhere transfer matrix\n\\begin{align}\\label{V}\nT(n)\\equiv\\left[\\begin{array}{ccc}\n\\frac{E}{V_{n,n+1}} &-\\frac{V_{n,n-1}}{V_{n,n+1}}  \\\\\n1&0\\\\\n  \\end{array}\\right].\n\\end{align}\nand\n\\begin{align}\n\\Psi(n)\\equiv\\left[\\begin{array}{ccc}\n\\psi(n+1)  \\\\\n\\psi(n)\\\\\n  \\end{array}\\right].\n\\end{align}\n\nFor a given parameter  $E$, with the increasing of $n$, we can assume that the wave function grows roughly according to an exponential law \\cite{Ishii,Furstenberg}, i.e.,\n\\begin{align}\n\\psi(n)\\sim e^{\\gamma(E) n}, &\\ as \\ n\\rightarrow \\infty,\n\\end{align}\nwhere $\\gamma(E)\\geq0$ is Lyapunov exponent which measures the average growth rate of wave function. If the parameter $E$ is not an eigen-energy of $H$, the Lyapunov exponent would be positive, i.e., $\\gamma(E)>0$ \\cite{Jonhnson1986}.\nWhen $E$ is an eigen-energy of system, the Lyapunov exponent can be zero or positive \\cite{yicai}.\nFor  critical states, the Lyapunov exponent $\\gamma(E)\\equiv0$ . While for localized states, the Lyapunov exponent $\\gamma(E)>0$.\n\nConsequently the Lyapunov exponent can be written as\n\\begin{align}\n&\\gamma(E)=\\lim_{L \\rightarrow \\infty }\\frac{\\log(|\\Psi(L)|\/|\\Psi(0)|)}{L}\\notag\\\\\n&=\\lim_{L\\rightarrow \\infty}\\frac{\\log(|T(L)T(L-1)...T(2)T(1)\\Psi(0)|\/|\\Psi(0)|)}{L}\n\\end{align}\nwhere $L$ is a positive integer and\n\\begin{align}\n|\\Psi(n)|=\\sqrt{|\\psi(n+1)|^2+|\\psi(n)|^2}.\n\\end{align}\n\n\n\nIn the following, we view the adjacent $\\kappa$ lattice sites as a ``super  unit cell\". Next we assume $L=m\\kappa+1$ ($m$ is an integer) and $|\\Psi(0)|\/|\\Psi(1)|$ is a finite non-zero real number, the Lyapunov exponent can be reduced into\n\\begin{align}\n&\\gamma(E)=\\lim_{L \\rightarrow \\infty }\\frac{\\log(|\\Psi(L)|\/|\\Psi(0)|)}{m\\kappa+1}\\notag\\\\\n&=\\lim_{m\\rightarrow \\infty}\\frac{\\log(|T(m\\kappa+1)T(m\\kappa)...T(2)\\Psi(1)|\/|\\Psi(0)|)}{m \\kappa}\\notag\\\\\n&=\\lim_{m\\rightarrow \\infty}\\frac{\\log(|T(m\\kappa+1)T(m\\kappa)...T(2)\\Psi(0)|\/|\\Psi(0)|)}{m \\kappa}\\notag\\\\\n&=\\frac{1}{\\kappa}\\lim_{m\\rightarrow \\infty}\\frac{\\log(|CT_m.CT(m-1)...CT_1\\Psi(0)|\/|\\Psi(0)|)}{m}\\notag\\\\\n\\end{align}\nwhere cluster transfer matrix $CT_n$  for $n-th$ ``super unit cell\"  is defined  as\n\\begin{align}\n&CT_n\\equiv T(n\\kappa+1)T(n\\kappa)...T((n-1)\\kappa+3)T((n-1)\\kappa+2)\\notag\\\\\n&=\\left[\\begin{array}{ccc}\nE &-v(n\\kappa) \\\\\n1&0\\\\\n  \\end{array}\\right]\\left[\\begin{array}{ccc}\n\\frac{E}{v(n\\kappa)} &-\\frac{1}{v(n\\kappa)} \\\\\n1&0\\\\\n  \\end{array}\\right]\\left[\\begin{array}{ccc}\nE &-1 \\\\\n1&0\\\\\n  \\end{array}\\right]^{\\kappa-2}\\notag\\\\\n  &=\\frac{\\left[\\begin{array}{ccc}\nE^2-(E^2\\tau+4\\lambda^2)cos^2(\\theta_n) &-E(1-\\tau cos^2(\\theta_n) ) \\\\\nE(1-\\tau cos^2(\\theta_n) )&-1+\\tau cos^2(\\theta_n) \\\\\n  \\end{array}\\right]}{2\\lambda cos(\\theta_n)\\sqrt{1-\\tau cos^2(\\theta_n)}}\\notag\\\\\n  &\\times \\left[\\begin{array}{ccc}\nE &-1 \\\\\n1&0\\\\\n  \\end{array}\\right]^{\\kappa-2}\n\\end{align}\nwhere $\\theta_n=2\\pi\\beta n\\kappa +\\phi$.\n\n\n\nThe cluster transfer matrix Eq.(16) can be further\nwritten as a product of two parts, i.e., $CT_n= A_nB_n$, where\n   \\begin{align}\n&A_n=\\frac{1}{2\\lambda cos(\\theta_n)\\sqrt{1-\\tau cos^2(\\theta_n)}},\\notag\\\\\n&B_n=\\left[\\begin{array}{ccc}\nB_{11} &B_{12} \\\\\nB_{21}&B_{22}\\\\\n  \\end{array}\\right]\\left[\\begin{array}{ccc}\nE &-1 \\\\\n1&0\\\\\n  \\end{array}\\right]^{\\kappa-2},\n\\end{align}\nwith $B_{11}=E^2-(E^2\\tau+4\\lambda^2)cos^2(\\theta_n)$, $B_{21}=-B_{12}=E(1-\\tau cos^2(\\theta_n))$ and $B_{22}=-1+\\tau cos^2(\\theta_n)$.\nNow  the Lyapunov exponent is\n\\begin{align}\n\\gamma(E)=\\frac{1}{\\kappa}[\\gamma_A(E)+\\gamma_B(E)],\n\\end{align}\nwhere\n\\begin{align}\n&\\gamma_A(E)=\\lim_{m\\rightarrow \\infty}\\frac{\\log(|A(m)A(m-1)...A(2)A(1)|)}{m}.\n\\end{align}\nand $\\gamma_B(E)$ are given by\n\\begin{align}\n&\\gamma_B(E)=\\lim_{m\\rightarrow \\infty}\\frac{\\log(|B(m)B(m-1)...B(2)B(1)\\Psi(0)|\/|\\Psi(0)|)}{m}.\n\\end{align}\n\n\nIn the following,  we would use Avila's global theory \\cite{Avila2015} to get the Lyapunov exponent and the Avila's acceleration (see next section).\nFollowing Refs.\\cite{Liu2021,YONGJIAN1}, first of all, we  complexify the phase $\\phi\\rightarrow \\phi+i \\epsilon$ with $\\epsilon >0$ , e.g., $A_n=\\frac{1}{2\\lambda cos(2\\pi \\beta n\\kappa +\\phi+i\\epsilon)\\sqrt{1-\\tau cos^2(2\\pi\\beta n\\kappa +\\phi+i\\epsilon)}}$.\n In addition, due to the ergodicity of the map $\\phi\\longrightarrow 2\\pi\\beta n+\\phi$, we can write  $\\gamma_A(E)$ as an integral over phase $\\phi$ \\cite{Longhi2019}, consequently\n\\begin{align}\n&\\gamma_A(E,\\epsilon)\\notag\\\\\n&=\\frac{1}{2\\pi}\\int_{0}^{2\\pi} d\\phi \\ln(|\\frac{1}{2\\lambda cos(\\phi+i\\epsilon)\\sqrt{1-\\tau cos^2(\\phi+i\\epsilon)}}|)\\notag\\\\\n&=-\\epsilon+\\ln(|\\frac{2}{\\lambda(1+\\sqrt{1-\\tau})}|),\n\\end{align}\nfor $\\epsilon< \\ln |\\frac{2+2\\sqrt{1-\\tau}-\\tau}{\\tau}|$.\n\n\n\nNext we take $\\epsilon\\rightarrow\\infty$\n \\begin{align}\n&B_n=\\frac{e^{-i(4\\pi\\beta n \\kappa+\\phi)+2\\epsilon}}{4}\\left[\\begin{array}{ccc}\n-( E^2 \\tau+4\\lambda^2) &E\\tau \\\\\n-E\\tau & \\tau\\\\\n  \\end{array}\\right]\\left[\\begin{array}{ccc}\nE & -1 \\\\\n1& 0\\\\\n  \\end{array}\\right]^{\\kappa-2}\\notag\\\\\n  &+O(1).\n\\end{align}\nThen for large $\\epsilon$, i.e.,  $\\epsilon\\gg1$, $\\gamma_B(E,\\epsilon)$ is determined by the largest eigenvalue (in absolute value) of $B_n$, i.e.,\n\\begin{align}\n\\gamma_B(E,\\epsilon)=2\\epsilon+\\ln(|\\frac{|P|+\\sqrt{P^2+16\\lambda^2\\tau}}{8}|),\n\\end{align}\nwhere\n\\begin{align}\nP=(\\tau E^2+4\\lambda^2)a_{\\kappa}+\\tau a_{\\kappa-2}-2\\tau E a_{\\kappa-1}.\n\\end{align}\nand $a_{\\kappa}$, is given by\n\\begin{align}\na_\\kappa=\\frac{1}{\\sqrt{E^2-4}}[(\\frac{E+\\sqrt{E^2-4}}{2})^{\\kappa-1}-(\\frac{E-\\sqrt{E^2-4}}{2})^{\\kappa-1}].\n\\end{align}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F211.eps}\n\\end{center}\n\\caption{ Lyapunov exponents for $\\kappa=3$,  $\\tau=-2$, and $\\lambda\/t=0.5,1.5, 2.5$.  The discrete points are the numerical results for all the eigenenergies. The solid lines are given by Eq.(27). The mobility edges for $\\lambda\/t=2.5$ are indicated by black arrows.  Near mobility edges of the localized-critical transition, the Lyapunov exponent $\\gamma(E)\\propto |E-E_c|$ approaches zero (as $E\\rightarrow E_c$). The critical index of the localization length $\\nu=1$.  }\n\\end{figure}\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F221.eps}\n\\end{center}\n\\caption{ Lyapunov exponents for $\\kappa=3$ and  $\\tau=1\/2$, and $\\lambda\/t=0.5,1.0, 1.5$.  The discrete points are the numerical results for all the eigenenergies. The solid lines are given by Eq.(27). The mobility edges for $\\lambda\/t=0.5$ are indicated by black arrows.  Near mobility edges of the localized-critical transition (e.g., $E_c\/t\\simeq\\pm \\sqrt{2}$ for $\\lambda\/t=0.5$), the Lyapunov exponent $\\gamma(E)\\propto |E-E_c|$ approaches zero (as $E\\rightarrow E_c$). The critical index of the localization length $\\nu=1$.}\n\\end{figure}\nWhen $\\epsilon$ is very small, using the facts that $\\gamma(E,\\epsilon)\\geq0$  and $\\gamma_B(E,\\epsilon)$ is  a convex and piecewise linear function of $\\epsilon$ \\cite{Avila2015,YONGJIAN1},  one can get\n\\begin{align}\n&\\gamma(E,\\epsilon)=Max\\{0,\\gamma_A(E,\\epsilon)+\\gamma_B(E,\\epsilon)\\},\\notag\\\\\n&=\n\\frac{1}{\\kappa}Max\\{0,\\epsilon+\\ln(|\\frac{|P|+\\sqrt{P^2+16\\lambda^2\\tau}}{4\\lambda(1+\\sqrt{1-\\tau})}|)\\}\n\\end{align}\nFurthermore, when $\\epsilon=0$, the Lyapunov exponent $\\gamma(E)\\equiv\\gamma(E,\\epsilon=0) $ is\n\\begin{align}\n\\gamma(E)=\\frac{1}{\\kappa}Max\\{\\ln|\\frac{|P(E)|+\\sqrt{P^2(E)+16\\lambda^2\\tau}}{4\\lambda(1+\\sqrt{1-\\tau})}|,0  \\}\n\\end{align}\nwhere\n\\begin{align}\nP(E)=(\\tau E^2+4\\lambda^2)a_{\\kappa}+\\tau a_{\\kappa-2}-2\\tau E a_{\\kappa-1}\n\\end{align}\nand\n\\begin{align}\na_\\kappa=\\frac{1}{\\sqrt{E^2-4}}[(\\frac{E+\\sqrt{E^2-4}}{2})^{\\kappa-1}-(\\frac{E-\\sqrt{E^2-4}}{2})^{\\kappa-1}].\n\\end{align}\n\nWhen $\\tau=0$, then $P(E)=4\\lambda^2a_{\\kappa}$, and\n\\begin{align}\n\\gamma(E)=\\frac{1}{\\kappa}Max\\{\\ln|\\lambda a_{\\kappa}|,0  \\}.\n\\end{align}\n\n\n\n\n\n\n\nThe above  formula Eq.(27)  has been  verified by our numerical results (see Figs.3 and 4).\nIn our numerical calculations, in order to get the correct Lyapunov exponents, on the one hand, the integer $L$ should be sufficiently large.\nOn the other hand, $L$ should be also much smaller than the system size $N$, i.e., $1\\ll L\\ll N$.\n\n To be specific, taking $\\kappa=3$, $\\tau=-2,1\/2$,  system size $N=3\\times1000$, we get the $N=3\\times1000$ eigenenergies and eigenstates.\n  Then we calculate the Lyapunov exponents numerically for all the eigenenergies [see the several sets of discrete points in  Figs.3 and 4].\n In our numerical calculation, we take $L=200$,  phase $\\phi=0$, $\\psi(0)=0$ and $\\psi(1)=1$ in Eq.(13).\n  The solid lines of Figs.3 and 4 are given by Eq.(27) with same parameters. It is shown that most of all discrete points fall onto the solid lines.\n\nHowever, we also note that there are some discrete points of localized states which are not on the solid lines. This is because these localized wave functions  are too near the left-hand boundary of system.\n\n\n\\section{mobility edge and critical region }\nIn this section, based on the Lyapunov exponent formula Eq.(27), we determine the mobility edges and the critical region.\n By the Eq.(27), the mobility edges $E_c$ which separate the localized states from the critical states, is determined by\n\\begin{align}\n\\gamma(E=E_c)=\\frac{1}{\\kappa}\\ln|\\frac{|P(E)|+\\sqrt{P^2(E)+16\\lambda^2\\tau}}{4\\lambda(1+\\sqrt{1-\\tau})}|=0\n\\end{align}\nthen\n\\begin{align}\n|P(E=E_c)|=4|\\lambda|\\sqrt{1-\\tau},\n\\end{align}\nThe critical regions which consist of critical states is given by\n\\begin{align}\n|P(E)|<4|\\lambda|\\sqrt{1-\\tau}.\n\\end{align}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F11.eps}\n\\end{center}\n\\caption{ Phase diagram in $(\\lambda, E)$ plane for $\\kappa=2$ and $\\tau=-2$. When $E$ is near zero, there exists localized-critical transitions. The blue solid lines are the phase boundaries (mobility edges $E_c$), which  are given by Eq.(32).\n Standard deviations are represented with different colors. }\n\\end{figure}\n\n\nBy expanding the Lyapunov exponent near the mobility edges $E_c$, we get\n\\begin{align}\n\\gamma(E)\\propto |E-E_c|\\rightarrow0, \\ as \\ E \\rightarrow E_c.\n\\end{align}\nThen the localization length is\n\\begin{align}\n\\xi(E)\\equiv1\/\\gamma(E)\\propto |E-E_c|^{-1}\\rightarrow\\infty,\\ as \\ E \\rightarrow E_c.\n\\end{align}\nIts critical index is $1$ [see the finite slopes of solid lines near $E_c$ in Figs.3 and 4].\n\n\n\nIn order to further distinguish the localized states from the  critical states, we also numerically calculate standard deviation of coordinates of eigenstates \\cite{Boers2007}\n\\begin{align}\\label{37}\n&\\sigma=\\sqrt{\\sum_{i}(i-\\bar{i})^2|\\psi(i)|^2},\n\\end{align}\nwhere the average value of coordinate $\\bar{i}$ is\n\\begin{align}\n\\bar{i}=\\sum_{i}i|\\psi(i)|^2.\n\\end{align}\nThe standard deviation $\\sigma$ describes the spatial extension of wave function in the lattices.\nThe phase diagram in $[\\lambda ( \\tau)- E]$ plane is reported in Figs.5,6,7 and 8.  In Figs.5,6,7 and 8, the standard deviations of coordinates are represented with different colors.\nFrom Figs.5,6,7 and 8, we can see that when the states are localized, standard deviations of coordinates are very small. For critical states, the standard deviations are very large.\n\n\n\nFrom Figs. 5, 7, and 8, we see that when $\\kappa>1$, there are $\\kappa-1$ loops for small hopping strength $\\lambda\/t$ in the phase diagram.\nWithin the loops, the Lyapunov exponent is positive, i.e., $\\gamma(E)>0$. Then if there is some eigenstates in the loops, these states would be localized states.\nHowever, numerical results show that there are no eigenenergies fallen into the loops.\n\nWhen $\\lambda\/t=0$, we see the system has $\\kappa$ eigenenergies  with multitude degeneracies. This is because  when  $\\lambda\/t=0$, the system in fact is composed by a lot of identical independent unit cells with $\\kappa$ lattice sites. Within a unit cell, there are exactly $\\kappa$ eigenenergies. With the increasing of $\\lambda\/t$, the degeneracies are removed and the system enters into the critical regions.\nIn addition, we also find that  the parity of $\\kappa$ also has important effects on the phase diagram.\n\n\\subsection{$\\kappa$ is an even number}\nWhen $\\kappa$ is even number and the energy $E$ is very near zero, there exist Anderson localizations if potential hopping strength $\\lambda$ is sufficiently large (see Fig.5 for $\\kappa=2$).\nEspecially when $E\\rightarrow0$, we get two critical hopping strengths\n\\begin{align}\n\\lambda_{c1}=\\pm \\frac{1-\\sqrt{1-\\tau}}{2} \\ \\ \\& \\ \\ \\lambda_{c2}=\\pm \\frac{1+\\sqrt{1-\\tau}}{2},\n\\end{align}\nwhich are independent of the $\\kappa$.\nHere $\\lambda_{c1}$ ($\\lambda_{c2}$) corresponds the point $A$ ($B$) in Fig.5.\nIt is noticed that $\\lambda_{c2}$ coincides the critical $\\lambda_c$ in Sec.II, where the average growth rate of zero-energy wave function is zero.\n\nFrom Fig.5, we can see that when the hopping $\\lambda$ is in the interval $|\\lambda_{c1}|<\\lambda<|\\lambda_{c2}|$, the eigenstates are critical states. Outside the interval, the eigenstates\nbecome localized states. So there exist Anderson localization transitions for even $\\kappa$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F1.eps}\n\\end{center}\n\\caption{Phase diagram for $\\kappa=1$. When energy is near zero-energy, the states are always critical states with vanishing Lyapunov exponent.}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F21.eps}\n\\end{center}\n\\caption{ Phase diagram in $(\\lambda, E)$ plane for $\\kappa=3$ and $\\tau=-2$. When $E$ is near zero, there are no localized-critical transitions. The blue solid lines are the phase boundaries (mobility edges $E_c$), which  are given by Eq.(32).\n Standard deviations are represented with different colors. }\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F22.eps}\n\\end{center}\n\\caption{ Phase diagram in $(\\lambda, E)$ plane for $\\kappa=3$ and $\\tau=1\/2$. There exists localized-critical transitions for nonzero energy states. The blue solid lines are the phase boundaries (mobility edges $E_c$), which  are given by Eq.(32).\n Standard deviations are represented with different colors.}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{$\\kappa$ is an odd number}\nWhen $\\kappa=1$, the system has no  the energy scale $t$ , then we would use the units of $\\lambda=1$. In Fig.6,  we report the phase diagram of  the $\\tau-E$ plane.\nFrom Fig.6, we see if $\\tau<0$, all the eigenstates are in the critical region. Only when $\\tau$ is positive and sufficiently large, the system has localized states.\n When $\\tau$ gets nearer and nearer to $1$, i.e., $\\tau\\rightarrow 1^{-}$, the range of  energy spectrum becomes larger and larger, and eventually diverges. This is  because when $\\tau=1$, the Hamiltonian defined by the Eq.(1) would be an unbounded operator, e.g, $\\frac{2\\lambda \\cos(2\\pi \\beta i+\\phi)}{\\sqrt{1-\\tau \\cos^2(2\\pi \\beta i+\\phi)}}=\\frac{2\\lambda \\cos(2\\pi \\beta i+\\phi)}{|\\sin(2\\pi \\beta i+\\phi)|}$ diverges for some lattice site index $i$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F5.eps}\n\\end{center}\n\\caption{ Standard deviations of localized states and critical states for parameters $\\kappa=3$, $\\tau=-2$ and $\\lambda\/t=1.5$. The eigenenergy $E_n$ increases gradually as eigenstate index $n$ runs from $1$ to $3000$ (along the black dashed line of Fig.7).}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\nWhen $\\kappa>1$  and the energy $E$ is very near zero, there are no Anderson localizations for a given potential strength $\\lambda$ (see Figs.7, and 8).\nIf $\\lambda$ is very large, i.e, $\\lambda\\rightarrow\\pm\\infty$, the mobility  edge can be obtained by Eq.(32), i.e.,\n\\begin{align}\n&E_c=\\pm\\frac{2\\sqrt{1-\\tau}}{(\\kappa-1)|\\lambda|}, \\ as \\ \\lambda\\rightarrow\\pm\\infty .\n\\end{align}\nIt is shown that for an arbitrarily strong quasiperiodic hopping strength $\\lambda$, there always exists an energy window, i.e., $-\\frac{2\\sqrt{1-\\tau}}{(\\kappa-1)|\\lambda|}<E<\\frac{2\\sqrt{1-\\tau}}{(\\kappa-1)|\\lambda|}$, in which there exist critical states. With the increasing of $\\lambda$, the energy window of critical regions gets smaller and smaller.\n\n\n\nFor given parameters $\\kappa=3$, $\\tau=-2$, $\\lambda\/t=1.5$, we report the standard deviations of all the eigenstates in Fig.9. It is shown that in comparison with localized states, the critical states have much larger fluctuation of standard deviations.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\columnwidth]{F7.eps}\n\\end{center}\n\\caption{ The inverse\nparticipation ratio $IPR_n$ of all the eigenstates for system size $N=3\\times250,3\\times500, 3\\times1000$ and $ N=3\\times2000$. The eigenenergy $E_n$ increases gradually as eigenstate index $n$ runs from $1$ to $N$ (along the black dashed line of Fig.8).}\n\\end{figure}\n\n\n\nIn order to investigate the properties of  the wave functions of critical states, we also numerically calculate the inverse participation ratio $IPR_n$ of  all eigenstates for different system sizes $N=3\\times 250,3\\times 500,3\\times1000$ and $N=3\\times2000$ \\cite{Xiaopeng,Deng}, i.e.,\n\\begin{align}\n&IPR_n=\\sum_{i}|\\psi_n(i)|^4.\n\\end{align}\nwhere $\\psi_n(i)$ is the normalized wave function for $n-th$ eigenstate.\nThe results are reported in Fig.10.\nWe find that the IPRs of localized states are basically same for different system sizes $N$,  while the IPRs of critical states have much larger fluctuation.\n\nOn the whole,   the IPR of critical states decreases with the increasing of system size $N$.\n The decreasing law can be captured by a function\n\\begin{align}\\label{H0}\n\\overline{IPR}\\propto 1\/N^x,\n\\end{align}\nwhere $\\overline{IPR}$ is an average value of the  $IPR$ within a typical energy interval.\n It is believed  that, for localized states, the scaling exponent $x=0$.  While for extended states (like plane wave states),  the scaling exponent $x=1$. For critical states, the exponent should be $0<x<1$.\nFor different system sizes $N$, due to randomness of $IPR$ of critical states (see Fig.10),  it is difficult to get a definite scaling exponent $x$.\nHere we find that for the critical states in the energy interval $-0.12<E\/t<-0.12$, its average value $\\bar{x}\\simeq0.47$.\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.1\\columnwidth]{F9.eps}\n\\end{center}\n\\caption{ Lyapunov exponents and Avila's accelerations. (a): Lyapunov exponents for $\\kappa=3$, $\\tau=1\/2$ and $\\lambda\/t=0.5$ (along the blue dashed line of Fig.8).   (b): Avila's acceleration  for for $\\kappa=3$, $\\tau=1\/2$ and $\\lambda\/t=0.5$. The mobility edges $E_c=\\pm\\sqrt{2}t$ are indicated by black arrows.  }\n\\end{figure}\n\n\n\\subsection{Avila acceleration}\nIn addition, for the bounded quasi-periodic potentials, Avila also defined the acceleration  $\\omega(E)$ by \\cite{Avila2015}\n\\begin{align}\n\\kappa\\omega(E)=lim_{\\epsilon\\rightarrow 0_+}\\frac{\\gamma(E,\\epsilon)-\\gamma(E,0)}{\\epsilon}.\n\\end{align}\nUsing Eqs.(26), when real number $E$ is an eigenvalue,  we get the Avila acceleration\n\\begin{align}\n&\\kappa \\omega(E)\n\\left\\{\\begin{array}{cccc}\n1, \\ for  \\ energy \\ of \\ localized \\ state \\\\\n0, \\ for  \\  energy \\ of \\ critical \\ state\n  \\end{array}\\right.\n\\end{align}\nThe above results are verified by our numerical calculations [see Fig.11].\nTo be specific, taking $\\tau=1\/2$, $\\lambda\/t=0.5$ and $\\epsilon=0,0.1,0.2$, we calculate the Lyapunov exponents  with Eq.(13) (taking the complexified phase $\\phi\\rightarrow \\phi+i\\epsilon=i\\epsilon$) for interval $-2.2\\leq E\\leq2.2$ [see the three solid lines in panel (a) of Fig.11]. In our calculation, we take $L=200$, $\\psi(0)=0$ and $\\psi(1)=1$ in Eq.(13).\nAt the same time, we calculate the Lyapunov exponents for all the eigenenergies with same parameters [see the three sets of discrete points in panel (a) of Fig.11].\nWe can find that\nif $E$ is an eigenenergy of critical state [$\\gamma(E)=0$], the Lyapunov exponents are the same for all three different  $\\epsilon=0,0.1,0.2$.\nWhile when $E$ is an eigenenergy of localized state [$\\gamma(E)>0$], the Lyapunov exponents are different for three different  $\\epsilon=0,0.1,0.2$. Their differences are linearly proportional to $\\Delta\\epsilon=0.1$ in Fig. 11.\n\nBy taking $\\epsilon=0.1$, we also  approximately calculate the Avila's  acceleration $\\omega(E)$ by\n\\begin{align}\n\\kappa\\omega(E)\\simeq\\frac{\\gamma(E,\\epsilon)-\\gamma(E,0)}{\\epsilon},\n\\end{align}\n[see panel (b) of Fig.11]. It shows that when $E$ is an eigenenergy of localized state [$\\gamma(E)>0$], the Avila's  acceleration is 1. When $E$ is  an eigenenergy of critical  state [$\\gamma(E)=0$],  the Avila's  acceleration is 0.\nWe also notice that if $E$  is not an eigenenergy, the Avila's  acceleration is $-1$.\n\n\n\n Further combining Eq.(27) and Eq.(43),\nthen one can classify systems with different real parameter $E$ (different phases) by Lyapunov exponent and the quantized acceleration, i.e.,\n\\begin{align}\n &(a): \\gamma(E)>0 \\ \\ \\& \\ \\kappa\\omega(E)=-1,  \\ if \\ E \\ is \\ not\\ an\\ eigenvalue \\notag\\\\\n &(b): \\gamma(E)>0 \\ \\ \\& \\ \\kappa\\omega(E)=1,  \\ for \\ localized \\ state\\notag\\\\\n &(c): \\gamma(E)=0 \\ \\ \\& \\ \\kappa\\omega(E)=0,  \\ for \\  critical \\ state.\n\n\\end{align}\n\n\n\n\n\n\n\n\n\n\n\\section{summary}\nIn conclusion, we investigate the localization properties of the one-dimensional lattice model with off-diagonal  mosaic quasiperiodic hopping.\n The parity of mosaic periodic has important effects on the localization of zero-energy states.\n When the mosaic period is odd, there always exists an energy window for critical states regardless hopping strength.\n While for even period, the states near zero-energy would become localized edge states for a sufficiently large hopping strength.\n It is found that there exist mobility edges which separate the localized states from critical states.\n Within the critical region, the spatial extensions of eigenstates have large fluctuation.\n\n The Lyapunov exponents and  mobility edges are  exactly obtained with Avila's theory.\n Furthermore, it is found that the critical index of localization length $\\nu=1$.\nFor $\\kappa=3$ and $\\tau=1\/2$, the numerical results show that the scaling exponent of inverse participation ratio (IPR) of critical states $x\\simeq0.47$. It is shown that these states indeed are critical states.\nIn addition,  it is shown that the Lyapunov exponent and Avila's acceleration can be used to classify the systems with different $E$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nThis work was supported by the NSFC under Grants Nos.\n11874127, 12061031, 12171039, 11871007, the Joint Fund with\nGuangzhou Municipality under No.\n202201020137, and the Starting Research Fund from\nGuangzhou University under Grant No.\nRQ 2020083.\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\thispagestyle{empty}\nA common problem faced in machine learning is the lack of sufficient training data. For colonoscopy the majority of readily-available public image data is limited to individual frames or short sequences for benchmarking CAD-based polyp detection. Public colonoscopy videos of the entire colon structure are limited to rather low-quality capsule endoscopy video footage. The lack of ground truth camera poses further hampers the training of models for applications different from polyp detection such as: anatomical segment classification, visual place recognition (VPR), simultaneous localization and mapping (SLAM) and structure from motion (SfM). These applications require high-quality colonoscopy videos of the entire examinations covering all phases of the intervention.\n\\begin{center}\n    \\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{domain.jpeg}\n    \\caption{Examples of synthetic colonoscopy images.}\n    \\label{fig:synthetic_images}\n\\end{figure}\n\\end{center}\n\nA common solution to this is the rendering of virtual endoscopy (VE) videos based on CT colonography data. VE provides both, image sequences and ground truth poses of varying anatomy, but (without further investigation) differs substantially from the visual appearance of real colonoscopy images. This entails gaps that have to be addressed by proper domain adaptation methods as demonstrated in \\cite{mathew2020augmenting}. This, however, implies that synthesized images resemble colonoscopy images (and their anatomical locations) of small datasets which likely do not generalize well to unseen or less observed colon regions. \nDomain randomization, in contrast, utilizes a large amount of data which is randomly sampled over the entire configuration space with the variables being carefully predefined. It is important to note that domain randomization is practically applicable to only simulated data as some of the parameters such as textures, materials, occlusions and coat masks have to be properly controlled in a simulated environment have to be more elaborated than for generating VE images in order to enable visual appearance close to real colonoscopy images (see Fig. \\ref{fig:synthetic_images}). Powerful engines such as \\textit{Unity} have gained particular interest in the computer vision and robotics communities \\cite{borkman2021unity,tremblay2018training}, but have been rarely investigated in medical imaging \\cite{incetan2020vrcaps,billot2021synthseg}. \n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{Architecture.png}\n    \\caption{Overview of the utilized processing pipeline for generating synthetic images.}\n    \\label{fig:architecture}\n\\end{figure*}\n\nGiven sufficient capabilities of simulation, models can solely be trained on domain-randomized data while still achieving high generalization performance for inference on real-world test data.\n\nThis paper presents an exemplary implementation of domain randomization for colonoscopy with all required algorithmic components. It is built up on prior work \\cite{incetan2020vrcaps} and supplements the latter by automated domain-randomized video recording through following waypoints along the interior colon's centerline.\n\n\n\\section{Material and Methods} \n\n\\subsection{Colon segmentation}\nAt first, a CT colonography (CT with radiocontrast material) obtained from TCIA is imported in \\textit{3D Slicer} for semi-automatic colon segmentation which is carried out as follows. A ROI around the colon is set manually with its image content being  thresholded. Subsequently we apply region-based segmentation on the (thresholded) mask to further delineate the colon structure. The segmentation mask is manually curated to ensure optimal results for successive steps.\n\n\\subsection{Centerline extraction}\nFor automated image collection we require an appropriate camera path through the interior colon. For this purpose, we estimate the centerline within the colon structure based on the prior work of \\cite{wan2002automatic}. The key idea is to plan an obstacle-free (w.r.t colon wall) path from the anus (colon entry) to the caecum. Since the intuitive approach based on the shortest path estimation tends to get too close to corners in turns, Wan et al. propose to explicitly incorporate the inversed map of distances to the colon wall \\cite{wan2002automatic} which was demonstrated to achieve optimal results with paths being exactly centered. Subsequently we sample equidistant waypoints along the extracted centerline which will be utilized within the simulation. Currently, we manually pick start and end points of the centerline extraction which, however, could be replaced by automatic anatomical landmark prediction through heatmap regression.\n\n\\subsection{3D model preparation}\nNext, the colon segmentation is imported in \\textit{Blender} for UV editing. Generally, a mesh is created surrounding the organ that can be edited along the vertices of the object. This mesh allows \\textit{UV mapping} which is a method for projecting a 3D model surface onto a 2D plane for texture mapping. An UV editing tool as part of \\textit{Blender} offers the possibility of unwrapping the 3D object onto a 2D plane where textures can be applied seamlessly throughout the region of the colon. This texture gives a realistic pattern to the object. Default shaders in \\textit{Blender} enable to change material properties corresponding to colon such as surface IOR, secular tint and anisotropy to further enhance the realism. \n\n\\subsection{Photorealistic rendering}\nThe 3D model prepared in \\textit{Blender} is subsequently imported in \\textit{Unity} which provides high definition render pipelines for our simulation environment that can produce photorealistic colonoscopy images. This virtual engine is commonly used for game development and has drawn particular interest in computer vision research due to its powerful graphical simulation platform for generating synthetic images. Using \\textit{Unity} we are able to synthesize images where parameters such as lighting, materials, occlusions, transparency and coat mask are altered to give it a more realistic appearance. These parameters are carefully selected such that real-world characteristics are optimally mimicked.  As a starting base we utilize parts of the \\textit{VR-Caps} project simulating a capsule endoscopic camera within \\textit{Unity} \\cite{incetan2020vrcaps}. A 3D model of this capsule with predefined attributes of an attached camera is placed inside the colon which is used for data collection. Adjusting these parameters is crucial for both mimicking real endoscopy and augmenting the data. The table below shows the camera parameters and post-processing effects required to achieve a fully synthetic model of the colon. For potential navigation tasks it is possible to additionally store corresponding depth images. \n\n\\begin{table}[!h]\n\\label{tab:fonts}\n\\centering\n\\begin{tabular}{|c|c| }\n\\hline\nAttributes &  Values\\\\\n\\hline\nSurface Metallic \t\t\t\t& 0.3 \\\\\nSurface Smoothness\t\t\t& 0.7 \\\\\nLens Intensity \t\t\t& 0.1\\\\\nChromic Abberation\t\t\t& 0.5 \\\\\nCoat Mask\t\t\t& 0.435\t \\\\\nCamera's Field of View \t\t\t& 91.375\t \\\\\nFocal length\t\t\t\t& 159.45\\\\\nISO \t\t& 200 \\\\\nAperture \t\t\t\t& 16\\\\\nAnisotropy \t\t\t& 1 \\\\\n\\hline\n\\end{tabular}\n\\caption{Camera parameters and Post-processing Effects}\n\\end{table}\n\n  \n \n   \n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{DR-Samples_Colon.png}\n    \\caption{Synthesized, domain-randomized images captured at the same pose inside the colon. Textures are obtained as from random patterns as well as synthetic patterns mimicking mucosa appearances.}\n    \\label{fig:dr-samples}\n\\end{figure*}\n\n\\subsection{Automated Video Rendering}\nManually collecting data for endoscopy becomes highly time-consuming when creating synthetic datasets consisting of all the required variations and diversity. For domain randomization we need to record sequences of images each time with different textures and materials which entails substantial individual setup. Thus, an approach for automating the process of data collection is introduced, which allows us to collect numerous samples inside the colon with different parameters. For this purpose we make use of the \\textit{scripting API} offered by \\textit{Unity} which gives access to the simulation environment and interactive components via executable scripts. Firstly, the simulated capsule is introduced to the colon and then automatically steered along the waypoints of the centerline (see Fig. \\ref{fig:waypoints}). The \\textit{Unity engine} is setup appropriately such that it enables smooth camera motion while following the waypoints.  \nOur path following script consist of two parts:  an \\textit{initialization} function which runs all required initial setups (parameter setup, initial capsule pose) and an \\textit{update} function which constantly controls the movement of the capsule (along the waypoints) and triggers actions such as changing parameters (e.g. lightning, texture). All images captured by the camera of the capsule are recorded. The parameters can either be adjusted on the fly allowing to capture images at the same pose with varying conditions or alternatively it is possible to alter the parameter set only for entire traversals. \\textit{Unity} also enables to configure the capsule's speed and camera's field of view and targeted frame rate (FPS). \n\n\\section{Results and Discussion} \n\nWe evaluate our simulation qualitatively based on image renderings for varying parameters which becomes particularly apparent when randomizing surface material and textural patterns. This is illustrated by Fig.\\ref{fig:dr-samples} which shows different renderings from the same captured from the same camera pose inside the colon. \nFig. \\ref{fig:waypoints} shows an example of an extracted centerline and generated waypoints being followed for automated video recording. For comparison, Fig. \\ref{fig:real_samples} shows exemplary real and synthetic images respectively.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{cenerline.png}\n    \\caption{Path following the centerline of the colon. The green line visualize the path and red circles waypoints being traced by the simulated capsule.}\n    \\label{fig:waypoints}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{RE-VE-comparison.jpeg}\n    \\caption{Comparison of synthetic (top) and real (bottom) colonoscopy images.}\n    \\label{fig:real_samples}\n\\end{figure}\n\\section{Conclusion}\nThis paper presented a pipeline for generating synthetic colonoscopy videos that can be used to improve training of deep learning models. By controlling environment (e.g. texture, reflectance) as well as virtual camera (e.g. lightning) properties we are able to simulate conditions being observed in inference but hardly ever presented in training data which is particularly the case for small-scale datasets. Inspired by substantive improvements reported on computer vision and robotics applications and limited prior work (VR-Caps)  \\cite{incetan2020vrcaps} we motivate to utilize domain-randomized synthesization for video colonoscopy. In our future work, we will incorporate this additional data for training deep learning-based approaches to SfM, SLAM and 3D reconstruction. In order to further simplify the variation in patient anatomy, we will investigate (fully) automatic segmentation of the colon in CT scans as well as an alternative to the 3D model preparation in \\textit{Blender}. \n\\pagebreak\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n\n\n\nConsider the one-dimensional Schr\\\"odinger equation\n\\begin{equation}\\label{eq_Schrodinger}\n {\\rm i}\\partial_t u=\\frac{\\nu(E)}{2}H_0u + W(E,\\omega t, x, -{\\rm i} \\partial_x)u,\\qquad x\\in{\\mathbb R} ,\n\\end{equation}\nwhere, we assume that\n\\begin{itemize}\n  \\item the frequencies $\\omega\\in {\\mathbb R}^d$, $d\\geq 1$, satisfy the {\\it Diophantine} condition (denoted by $\\omega\\in {\\rm DC}_d(\\gamma,\\tau)$ for $\\gamma>0$, $\\tau>d-1$):\n$$\\inf_{j\\in{\\mathbb Z}}|\\langle n,\\omega\\rangle-j|>\\frac{\\gamma}{|n|^\\tau},\\qquad \\forall \\ n\\in{\\mathbb Z}^d\\setminus\\{0\\},$$\n  \\item the parameter $E\\in {{\\mathcal I}}$, an interval $\\subset {\\mathbb R}$, and $\\nu\\in C^2({{\\mathcal I}},{\\mathbb R})$ satisfies\n  $$|\\nu'(E)|\\geq l_1,\\quad |\\nu''(E)|\\leq l_2,\\qquad \\forall \\  E\\in{{\\mathcal I}},$$\nfor some $l_1,l_2>0$,\n  \\item $H_0$ is the {\\it one-dimensional quantum harmonic operator}, i.e.\n$$(H_0u)(x):=-(\\partial_{x}^2 u)(x)+x^2\\cdot u(x),\\qquad \\forall \\ u\\in L^2({\\mathbb R}),$$\n  \\item $W(E,\\theta, x,\\xi)$ is a quadratic form of $(x,\\xi)$:\n$$W(E, \\theta, x,\\xi)=\\frac12 \\big(a(E,\\theta) x^2+2b(E,\\theta) x\\cdot\\xi+c(E,\\theta) \\xi^2\\big),$$\nwith $a,b,c:{{\\mathcal I}}\\times {\\mathbb T}^d\\to {\\mathbb R}$, all of which are $C^2$ w.r.t. $E\\in{{\\mathcal I}}$ and $C^\\omega$ w.r.t. $\\theta\\in{\\mathbb T}^d:=({\\mathbb R}\/{\\mathbb Z})^d$, and for every $E\\in{{\\mathcal I}}$, for $m=0,1,2$,\n$|\\partial^m_E a(E,\\cdot)|_r:=\\sup_{|\\Im z|<r}|\\partial^m_E a(E,z)|$, $|\\partial^m_E b(E,\\cdot)|_r$, $|\\partial^m_E c(E,\\cdot)|_r$ are small enough.\n\\end{itemize}\nWe will prove that, for almost every $E$ in the interval ${{\\mathcal I}}$, Eq. (\\ref{eq_Schrodinger}) is reducible, i.e., via a unitary transformation, Eq. (\\ref{eq_Schrodinger}) is conjugated to an equation which is independent of time (while the transformation depends on time in an analytic quasi-periodic way).\nAccording to the reducibility, we deduce the behavior of Sobolev norms for the solutions to Eq. (\\ref{eq_Schrodinger}).\n\n\n\\subsection{Reducibility for harmonic oscillators}\n\nOur main result is the following:\n\n\\begin{Theorem}\\label{thm_Schro}\nThere exists $\\varepsilon_*=\\varepsilon_*(\\gamma,\\tau,r,d,l_1,l_2)>0$ such that if\n$$\\max_{m=0,1,2}\\left\\{\\left|\\partial_E^m a\\right|_r, \\ \\left|\\partial_E^m b\\right|_r, \\ \\left|\\partial_E^m c\\right|_r\\right\\}=:\\varepsilon_0\\leq \\varepsilon_*,\\qquad \\forall \\ E\\in{{\\mathcal I}},$$\nthen for a.e. $E\\in{{\\mathcal I}}$, Eq. (\\ref{eq_Schrodinger}) is reducible, i.e., there exists a time quasi-periodic transformation $U(\\omega t)$, unitary in $L^2$ and analytically depending on $t$, such that\nEq. (\\ref{eq_Schrodinger}) is conjugated to\n${\\rm i}\\partial_t v= G v$ by the transformation $u=U(\\omega t) \\, v$, with $G$ a linear operator independent of $t$.\n\nMore precisely,\nthere exists a subset\n$${{\\mathcal O}}_{\\varepsilon_0}=\\bigcup_{j\\in{\\mathbb N}}\\Lambda_j\\subset \\overline{{\\mathcal I}}$$ with $\\Lambda_j$'s being closed intervals \\footnote{In this paper, the ``closed interval\" is interpreted in a more general sense, i.e., it can be degenerated to a point instead of a positive-measure subset of ${\\mathbb R}$.}\nand ${\\rm Leb}({{\\mathcal O}}_{\\varepsilon_0})<\\varepsilon_0^{\\frac{1}{40}}$,\nsuch that the following holds.\n\\begin{enumerate}\n\\item For a.e. $E\\in{{\\mathcal I}}\\setminus{{\\mathcal O}}_{\\varepsilon_0}$, $G$ is unitary equivalent to $\\varrho H_0$ for some $\\varrho=\\varrho_E\\geq 0$;\n\\item If ${\\rm Leb}(\\Lambda_j)>0$, then\n\\begin{itemize}\n  \\item for $E\\in{\\rm int}\\Lambda_j$, $G$ is unitary equivalent to $-\\frac{\\lambda {\\rm i}}{2}(x\\cdot \\partial_x+ \\partial_x \\cdot x)$ for some $\\lambda=\\lambda_E> 0$;\n  \\item for $E\\in\\partial\\Lambda_j\\setminus\\partial{{\\mathcal I}}$, $G$ is unitary equivalent to $-\\frac{\\kappa}2 x^2$ for some $\\kappa=\\kappa_E\\in{\\mathbb R}\\setminus\\{0\\}$.\n\\end{itemize}\nIf ${\\rm Leb}(\\Lambda_j)=0$, then $G=0$ for $E\\in\\Lambda_j$.\n\\end{enumerate}\n\\end{Theorem}\n\nBefore giving its application on the growth of Sobolev norm, let us first make a review on previous works about the reducibility on harmonic oscillators as well as the relative KAM theory.\n\nFor 1-d harmonic oscillators with time periodic smooth perturbations, Combescure \\cite{Com87} firstly showed the pure point nature of Floquet operator (see also \\cite{DLSV2002, EV83, Kuk1993}).\nFor 1-d harmonic oscillators with time quasi-periodic bounded perturbations, we can refer to \\cite{GreTho11, Wang08, WLiang17} for the reducibility and the pure point spectrum of Floquet operator.\nFor 1-d harmonic oscillators with unbounded time quasi-periodic perturbations, similar results can be found in \\cite{Bam2018, Bam2017, BM2018, Liangluo19}.\nIn investigating the reducibility problems, KAM theory for 1-d PDEs has been well developed by Bambusi-Graffi \\cite{BG2001} and Liu-Yuan \\cite{LY2010} in order to deal with unbounded perturbations.\n\nReducibility for PDEs in higher-dimensional case was initiated by Eliasson-Kuksin \\cite{EK2009}, based on their KAM theory \\cite{EK2010}.\nWe refer to \\cite{GrePat16} and \\cite{LiangWang19} for any dimensional harmonic oscillator with bounded potential. \nWe mention that some higher-dimensional results with unbounded perturbations have been recently obtained\n\\cite{BLM18, FGMP19, FG19, FGN19, Mon19}. \nHowever, a general KAM theorem for higher-dimensional PDEs with unbounded perturbations is far from success.\n\n\nRecently, Bambusi-Gr\\'ebert-Maspero-Robert \\cite{BGMR2018} built a reducibility result for the harmonic oscillators on ${\\mathbb R}^n$, ,$n\\geq 1$, in which the perturbation is a polynomial of degree at most two in $x$ and $-{\\rm i}\\partial_x$ with coefficients quasi-periodically depending on time.\nThe proof in \\cite{BGMR2018} exploits the fact that for polynomial Hamiltonians of degree at most $2$, there is an exact correspondence between classical and quantum mechanics, so that the result can be proved by exact quantization of the classical KAM theory which ensures reducibility of the classical Hamiltonian system.\nThe exact correspondence between classical and quantum dynamics of quadratic Hamiltonians was already exploited in the paper \\cite{HLS86} to prove stability and instability results for one degree of freedom time periodic quadratic Hamiltonians.\nTo prove our main result, we use the same strategy as \\cite{BGMR2018} and the reducibility result for the classical Hamiltonian by Eliasson \\cite{Eli1992}.\n\n\n\n\\subsection{Growth of Sobolev norms}\n\nBesides reducibility, the construction of unbounded solutions in Sobolev space for Schr\\\"odinger equations  attracts even more attentions.\n\nAs an application of Theorem \\ref{thm_Schro}, we can study the long time behaviour of its solution $u(t)$ to Eq. (\\ref{eq_Schrodinger}) in Sobolev space.\nFor $s\\geq 0$, we define Sobolev space\n$${{\\mathcal H}}^s:=\\left\\{\\psi\\in L^2({\\mathbb R}):H_0^{\\frac{s}2}\\psi \\in L^2({\\mathbb R})\\right\\}$$\nand Sobolev norm\n$\\|\\psi\\|_{s}:=\\|H_0^{\\frac{s}2} \\psi\\|_{L^2({\\mathbb R})}$.\nIt is well known that, for $s\\in {\\mathbb N}$, the above definition of norm is equivalent to\n$$\n\\sum\\limits_{\\alpha+\\beta\\leq s\\atop{\\alpha,\\beta\\in{\\mathbb N}} }\\|x^{\\alpha}\\cdot\\partial^{\\beta} \\psi\\|_{L^2({\\mathbb R})}.\n$$\n\\begin{remark}\\label{remark_norm_equiv}\n In view of Remark 2.2 of \\cite{BM2018}, we get that, for a given $\\psi\\in {{\\mathcal H}}^s$,\n\\begin{equation}\\label{norm_equiv}\n\\|\\psi\\|_{s}\\simeq \\|\\psi\\|_{H^s}+ \\|x^{s} \\psi\\|_{L^2},\n\\end{equation}\nreplacing $K_0=H_0$ in that remark by $K_0=H_0^{\\frac12}$, where $H^s$ means the standard Sobolev space and $\\|\\cdot\\|_{H^s}$ is the corresponding norm. Hence, to calculate the norm $\\|\\psi\\|_s$, $s\\geq 0$, it is sufficient to focus on $\\|x^{s} \\psi\\|_{L^2}$ for $s\\geq0$ and $\\|\\psi^{(s)}\\|_{L^2}$ for $s\\in{\\mathbb N}$.\n\\end{remark}\n\n\nFor different types of reduced systems, Sobolev norm of solution exhibits different behaviors.\n\n\n\n\\begin{Theorem}\\label{thm_Schro_sobolev} Under the assumption of Theorem \\ref{thm_Schro}, for any $s\\geq 0$, and any non-vanishing initial condition $u(0)\\in {{\\mathcal H}}^s$, the following holds true for the solution $u(t)$ to Eq. (\\ref{eq_Schrodinger}) for $t\\geq0$.\n\\begin{enumerate}\n\\item For a.e. $E\\in{{\\mathcal I}}\\setminus{{\\mathcal O}}_{\\varepsilon_0}$,\n$\nc \\leq \\|u(t)\\|_{s}\\leq C\n$.\n\\item If ${\\rm Leb}(\\Lambda_j)>0$, then\n\\begin{itemize}\n  \\item for $E\\in{\\rm int}\\Lambda_j$,\n$c  e^{\\lambda st}  \\leq \\|u(t)\\|_{s} \\leq C  e^{\\lambda st}$,\n  \\item for $E\\in\\partial \\Lambda_j\\setminus \\partial{{\\mathcal I}}$,\n$\nc |\\kappa|^s t^s \\leq \\|u(t)\\|_{s}\\leq C |\\kappa| (1+ t^2)^{\\frac{s}2}\n$.\n\\end{itemize}\nIf ${\\rm Leb}(\\Lambda_j)=0$, then for $E\\in \\Lambda_j$, $c  e^{\\lambda st}  \\leq \\|u(t)\\|_{s} \\leq C  e^{\\lambda st}$.\n\\end{enumerate}\nHere $\\lambda=\\lambda_E$ and $\\kappa=\\kappa_E$ are the same with Theorem \\ref{thm_Schro} and $c, \\, C>0$ are two constants depending on $s$, $E$ and $u(0)$.\n\\end{Theorem}\n\n\nLet us make more comments on constructing solutions growing with time in Schr\\\"odinger equations.\nBourgain \\cite{Bou99} built logarithmic lower and upper growth bounds for linear Schr\\\"odinger equation on ${\\mathbb T}$ by\nexploiting  resonance effects.  And the optimal polynomial growth example was given by Delort \\cite{Del2014} for 1-d harmonic oscillator with a time periodic order zero perturbation. Maspero \\cite{Mas2018} reproved the result of Delort by exploiting the idea in \\cite{GY00}. In \\cite{BGMR2018}, the authors also considered the higher-dimensional harmonic oscillator with a linear perturbation in $x$ and $-{\\rm i}\\partial_x$ with time quasi-periodic coefficients. Under the Diophantine condition of frequencies, the time-dependent equation can be reduced to a special ``normal form\" independent of time (see Theorem 3.3 of \\cite{BGMR2018}), which implies the polynomial growth of Sobolev norm.   There are also many literatures, e.g., \\cite{BGMR2019, MR2017}, which are relative to the upper growth bound of the solution in Sobolev space.\n\nFrom the above mentioned literatures, we can see that almost all the growth results of lower growth bound of the solution are closely related to the resonance phenomenon. However, it is not clear to us which kind of parameter set is connected to the growth of Sobolev norm.\nComparing with all the above results, we introduce the parameter set $\\bigcup_{j\\in{\\mathbb N}}\\Lambda_j$ following \\cite{Eli1992}, in which the solutions has exponential lower and upper growth bounds, while on the boundaries of this set the solutions has polynomial lower and upper growth bounds.  In the following, we will present several concrete examples to show that the set $\\bigcup_{j\\in{\\mathbb N}}\\Lambda_j$ is of positive measure.\n\n\n\\subsection{Examples with ${\\rm Leb}({{\\mathcal O}}_{\\varepsilon_0})>0$}\n\nIn view of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev}, the growth of Sobolev norm can be obtained via the reducibility if ${\\rm Leb}({{\\mathcal O}}_{\\varepsilon_0})>0$.\nWe need to point that the time-dependent quadratic perturbation $W(E,\\omega t, x, -{\\rm i} \\partial_x)$ with ${\\rm Leb}({{\\mathcal O}}_{\\varepsilon_0})>0$ exists universally. In other words, it is a quite ``extreme\" case that\n$${\\rm Leb}(\\Lambda_j)=0, \\qquad \\forall \\ j\\in{\\mathbb N}.$$\nWe have the following concrete examples.\n\n\\\n\nFor ${{\\mathcal I}}={\\mathbb R}$, $\\nu(E)=E$, the equation\n\\begin{equation}\\label{example_1}\n {\\rm i}\\partial_t u=\\frac{E}{2}H_0u + \\left(\\frac{a(\\omega t)}{2} x^2-\\frac{b(\\omega t)}2\\left(x\\cdot{\\rm i}\\partial_x+{\\rm i}\\partial_x\\cdot x\\right)-\\frac{c(\\omega t)}2 \\partial_x^2 \\right) u,\n\\end{equation}\nsatisfies the assumptions of Theorem \\ref{thm_Schro} if $a,b,c\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$ are small enough. Hence, for Eq. (\\ref{example_1}), the reducibility and the behaviors of ${{\\mathcal H}}^s$ norm of solutions described in Theorem \\ref{thm_Schro_sobolev} can be obtained.\n\\begin{Theorem}\\label{thm_example_1}\nFor generic $a,b,c\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$ with $|a|_r, |b|_r, |c|_r$ small enough (depending on $r,\\gamma,\\tau,d$), the conclusions of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev} hold for Eq. (\\ref{example_1}) for ${{\\mathcal I}}={\\mathbb R}$ with ${\\rm Leb}({{\\mathcal O}}_{\\varepsilon_0})>0$.\n\\end{Theorem}\n\n\\\n\n\n\nFor $\\nu(E)=\\sqrt{E}$, consider the equation\n\\begin{equation}\\label{eq_Schrodinger-example}\n {\\rm i}\\partial_t u=\\frac{\\sqrt{E}}{2} H_0 u -\\frac{q(\\omega t)}{2\\sqrt{E}}\\left(x^2-x\\cdot{\\rm i}\\partial_x-{\\rm i}\\partial_x\\cdot x-\\partial^2_x \\right)u.\n\\end{equation}\nwith $q\\in C_r^\\omega({\\mathbb T}^d,{\\mathbb R})$. The equation is important, since as we will show later, it is closely related to quasi-periodic Schr\\\"odinger operator.\n\\begin{Theorem}\\label{thm_example_schro}\nFor generic $q\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$, the conclusions of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev} hold for Eq. (\\ref{eq_Schrodinger-example}) for ${{\\mathcal I}}=[E_0,E_1]$ with ${\\rm Leb}(\\Lambda_j)>0$ for infinitely many $j$'s, where $E_0>0$ is large enough (depending on $|q|_r$) and $E_1<\\infty$.\n\\end{Theorem}\n\n\\\n\n\nTheorem \\ref{thm_example_1} gives the example that ${\\rm Leb}(\\Lambda_j)>0$ for at least one $j$, while Theorem \\ref{thm_example_schro} gives the  example that  ${\\rm Leb}(\\Lambda_j)>0$ for infinitely many $j$'s. Indeed, if the dimension of the frequency $d=2$, we could even gives ${\\rm Leb}(\\Lambda_j)>0$ for every $j$'s.  To construct such an example, we consider\n \\begin{equation}\\label{eq_AMO}\n  {\\rm i}\\partial_t u=\\frac{\\nu(E)}{2} H_0 u+ \\left(\\frac{a(E,\\omega t)}{2} x^2-\\frac{b(E,\\omega t)}2\\left(x\\cdot{\\rm i}\\partial_x+{\\rm i}\\partial_x\\cdot x\\right)-\\frac{c(E,\\omega t)}2 \\partial_x^2 \\right) u.\n \\end{equation}\nwhere $\\nu(E)=\\cos^{-1}(-\\frac{E}{2})$,  ${{\\mathcal I}}\\subset[-2+\\delta,2-\\delta]$ with $\\delta$ a small numerical constant (e.g., $\\delta=10^{-6}$). Then our result is the following:\n\n\n\n\n \\begin{Theorem}\\label{thm_AMO} There exist a sub-interval ${{\\mathcal I}}\\subset[-2+\\delta,2-\\delta]$ and $a,b,c:{{\\mathcal I}}\\times {\\mathbb T}^2\\to{\\mathbb R}$ with $a(E,\\cdot), \\, b(E,\\cdot), \\, c(E,\\cdot)\\in C^{\\omega}({\\mathbb T}^2,{\\mathbb R})$ for every $E\\in{{\\mathcal I}}$,\n\n such that the conclusions of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev} hold for Eq. (\\ref{eq_AMO}). Moreover,  ${\\rm Leb}(\\Lambda_j)> 0$ for every $j\\in{\\mathbb N}$.\n \\end{Theorem}\n\n\n\n\\begin{remark} One can even further get precise size of ${\\rm Leb}(\\Lambda_j)$ according to \\cite{LYZZ}.\n\\end{remark}\n\n\\\n\nThe rest of paper will be organised as follows. In Section \\ref{sec_weyl}, which serves as a preliminary section, we recall the definition of Weyl quantization and some known results on the relation between classical Hamiltonian to quantum Hamiltonian.\nWe give an abstract theorem in Section \\ref{sec_abstract} on the reducibility for quantum Hamiltonian, provided that the reducibility for the corresponding classical Hamiltonian is known.\nBy applying this abstract theorem, we exploit the connection between reducibility and property of Sobolev norm.\nThe abstract theorem is proved in Section \\ref{sec_reduc}.\nIn Section \\ref{sec_proof}, we prove the main result just by verifying the hypothesis of abstract theorem.\nIn Section \\ref{sec_pr_examples}, the proofs of Theorem \\ref{thm_example_1} -- \\ref{thm_AMO} are given.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Classical Hamiltonian and quantum Hamiltonian}\\label{sec_weyl}\n\nTo give some preliminary knowledge,\nlet us recall the definition of Weyl quantization, which relates the classical and quantum mechanics, and its properties. The conclusions listed in this section can also be found in \\cite{BGMR2018}.\n\n\nThe Weyl quantization is the operator ${\\rm Op}^W:f\\mapsto f^W$ for any symbol $f=f(x,\\xi)$, with $x,\\xi\\in{\\mathbb R}^n$, where $f^{W}$ is the Weyl operator of $f$:\n$$\\left(f^{W} u\\right)(x)=\\frac{1}{(2\\pi)^n}\\int_{y, \\, \\xi\\in{\\mathbb R}^n} f\\left(\\frac{x+y}{2},\\xi\\right) u(y) \\, dy  \\, d\\xi,\\qquad \\forall \\ u\\in L^2({\\mathbb R}^n).$$\nIn particular, if $f$ is a polynomial of degree at most $2$ in $(x,\\xi)$, then $f^W$ is exactly $f(x,-{\\rm i}\\partial_x)$.\n\n\n\nFor the $1-$parameter family of Hamiltonian $\\chi(t, x, \\xi )$, with $t$ an external parameter, let $\\phi^\\tau(t,x,\\xi)$ be the time $\\tau-$flow it generates, precisely the\nsolution of\n$$\\frac{dx}{d\\tau}=\\frac{\\partial\\chi}{\\partial\\xi}(t, x, \\xi ),\\qquad \\frac{d\\xi}{d\\tau}=-\\frac{\\partial\\chi}{\\partial x}(t, x, \\xi).$$\nThe time-dependent coordinate transformation\n\\begin{equation}\\label{time1}\n(x,\\xi)=\\phi^1\\left(t,\\tilde x,{\\tilde\\xi}\\right)=\\left.\\phi^{\\tau}\\left(t,\\tilde x,{\\tilde\\xi}\\right)\\right|_{\\tau=1}\n\\end{equation}\ntransforms a Hamiltonian system with\nHamiltonian $h$ into a system with Hamiltonian $g$ given by\n$$g(t,\\tilde x,\\tilde\\xi)=h(\\phi^1(t,\\tilde x,\\tilde\\xi))-\\int_0^1 \\frac{\\partial\\chi}{\\partial t}(t,\\phi^{\\tau}(t,\\tilde x,\\tilde\\xi) ) d\\tau.$$\n\n\n\\begin{Lemma} [Remark 2.6 of \\cite{BGMR2018}] If the Weyl operator $\\chi^W(t, x, -{\\rm i}\\partial_x)$ is self-adjoint for any fixed $t$, then the transformation\n\\begin{equation}\\label{tran}\n\\psi=e^{{\\rm i}\\chi^W(t, x, -{\\rm i}\\partial_x)}\\tilde\\psi\n\\end{equation}\ntransforms the equation ${\\rm i}\\partial_t\\psi=H\\psi$ into ${\\rm i}\\partial_t\\tilde\\psi=G\\tilde\\psi$ with\n\\begin{eqnarray*}\nG&:=&e^{{\\rm i}\\chi^W(t, x, -{\\rm i}\\partial_x)}He^{-{\\rm i}\\chi^W(t, x, -{\\rm i}\\partial_x)}\\\\\n& & - \\, \\int_0^1 e^{{\\rm i}\\tau\\chi^W(t, x, -{\\rm i}\\partial_x)}\\left(\\partial_t \\chi^W(t, x, -{\\rm i}\\partial_x)\\right)e^{-{\\rm i}\\tau\\chi^W(t, x, -{\\rm i}\\partial_x)}d\\tau.\n\\end{eqnarray*}\n\n\\end{Lemma}\n\n\\begin{Proposition} [Proposition 2.9 of \\cite{BGMR2018}]\\label{Prop_hami} Let $\\chi(t, x, \\xi )$ be a polynomial of degree at most $2$ in $(x,\\xi)$ with smooth time-dependent coefficients.\nIf the transformation (\\ref{time1}) transforms a classical system with Hamiltonian $h$ into\na system with Hamiltonian $g$, then the transformation (\\ref{tran}) transforms the quantum Hamiltonian system\n$h^W$ into $g^W$.\n\\end{Proposition}\n\n\nNow, let us focus on the case $n=1$.\n\n\\begin{Lemma} [Lemma 2.8 of \\cite{BGMR2018}]\\label{lem_Sobolev}\nLet $\\chi(\\theta,x,\\xi)$ be a polynomial of degree at most $2$ in $(x,\\xi)$ with real coefficients depending in a $C^\\infty-$way on $\\theta\\in {\\mathbb T}^d$.\nFor every $\\theta\\in {\\mathbb T}^d$, the Weyl operator $\\chi^W(\\theta,x, -{\\rm i}\\partial_x)$ is self-adjoint in $L^2({\\mathbb R})$ and $e^{-{\\rm i}\\tau\\chi^W(\\theta,x, -{\\rm i}\\partial_x)}$ is unitary in $L^2({\\mathbb R}^n)$ for every $\\tau\\in{\\mathbb R}$.\nFurthermore, if the coefficients of $\\chi(\\theta,x,\\xi)$ are uniformly bounded w.r.t. $\\theta\\in {\\mathbb T}^d$, then for any $s\\geq 0$, there exist $c'$, $C' > 0$ depending on $\\|[H_0^s,\\chi^W(\\theta,x, -{\\rm i}\\partial_x)]H_0^{-s}\\|_{L^2\\mapsto L^2}$ and $s$, such that\n   \\begin{equation}\\label{change_Sobolevnorm}\n      c'\\|\\psi\\|_{s}\\leq \\|e^{-{\\rm i}\\tau\\chi^W(\\theta,x,-{\\rm i}\\partial_x)}\\psi\\|_{s}\\leq C'\\|\\psi\\|_{s},\\qquad \\tau\\in [0,1], \\quad \\theta\\in{\\mathbb T}^d.\n    \\end{equation}\n\\end{Lemma}\n\n\n\n\n\\section{Reducibility and growth of Sobolev norm}\\label{sec_abstract}\n\n\n\n\\subsection{An abstract theorem on reducibility}\n\n\nConsider the 1-d time-dependent equation\n\\begin{equation}\\label{eq_abs}\n {\\rm i}\\partial_t u=L^{W}(\\omega t, x, -{\\rm i} \\partial_x)u,\\qquad x\\in{\\mathbb R} ,\n\\end{equation}\nwhere $L^{W}(\\omega t, x, -{\\rm i} \\partial_x)$ is a linear differential operator, $\\omega\\in{\\mathbb T}^d$, $d\\geq 1$, and the symbol $L(\\theta, x,\\xi)$ is a quadratic form of $(x,\\xi)$ with coefficients\nanalytically depending on $\\theta\\in{\\mathbb T}^d$. More precisely, we assume that\n\\begin{equation}\\label{op_L}\nL(\\theta, x,\\xi)=\\frac12 \\big(a(\\theta) x^2+ b(\\theta) x\\cdot \\xi + b(\\theta) \\xi\\cdot  x + c(\\theta)  \\xi^2\\big),\n\\end{equation}\nwith coefficients $a,b,c\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$.\n\n\n\nThrough Weyl quantization, the reducibility for the time-dependent PDE can be related to the reducibility for the ${\\rm sl}(2,{\\mathbb R})-$linear system $(\\omega, \\, A(\\cdot))$:\n$$X'=A(\\omega t)X,\\qquad A\\in C^{\\omega}({\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R})).$$\nGiven $A_1, A_2  \\in C^{\\omega}({\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$, if there exists $Y\\in C^{\\omega}(2{\\mathbb T}^d,{\\rm SL}(2,{\\mathbb R}))$ such that\n$$\\frac{d}{dt}Y(\\omega t)=A_1(\\omega t)Y(\\omega t)-Y(\\omega t)A_2(\\omega t),$$\n we say that $(\\omega, \\, A_1(\\cdot))$ is conjugated to $(\\omega, \\, A_2(\\cdot))$ by $Y$.\nIf $(\\omega, \\, A(\\cdot))$ can be conjugated to $(\\omega, \\, B)$ with $B\\in{\\rm sl}(2,{\\mathbb R})$, we say that $(\\omega, \\, A(\\cdot))$ is {\\it reducible}.\n\n\\smallskip\n\nNow let $A(\\cdot):=\\left(\\begin{array}{cc}\n            b(\\cdot) & c(\\cdot) \\\\[1mm]\n           -a(\\cdot) & -b(\\cdot)\n          \\end{array}\\right) \\in C^{\\omega}({\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R})) $ with $a,b,c$ coefficients given in (\\ref{op_L}).\n\\begin{Theorem}\\label{thm_redu}\nAssume that there exist $B\\in{\\rm sl}(2,{\\mathbb R})$ and $Z_j\\in C^\\omega(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$, $j=0, \\cdots,K$,\nsuch that $(\\omega, \\, A(\\cdot))$ is conjugated to $(\\omega, \\, B)$ by $\\prod_{j=0}^K e^{Z_j}$. Then Eq. (\\ref{eq_abs}) is reducible, i.e., there exists a time quasi-periodic map $U(\\omega t)$, unitary in $L^2$ and analytic on $t$, satisfying\n\\begin{equation}\\label{norm_U}\nc'\\|\\psi\\|_{s}\\leq \\|U(\\omega t)\\psi\\|_{s}\\leq C'\\|\\psi\\|_{s},  \\quad \\forall \\ \\psi\\in{{\\mathcal H}}^s,\n\\end{equation}\nfor constants $c', \\, C'>0$ depending on $s$ and $\\psi$, such that\nEq. (\\ref{eq_abs}) is conjugated to\n\\begin{equation}\\label{Ham_G}\n{\\rm i}\\partial_t v= G v\n\\end{equation}\nby the transformation $u=U(\\omega t) v$, with $G$ an operator independent of time.\n\n\nMore precisely,\n\\begin{itemize}\n\\item [(\\uppercase\\expandafter{\\romannumeral1})] $G$ is unitary equivalent to $\\frac{\\sqrt{{\\rm det} B}}{2} H_0$ if\n\\begin{equation}\\label{type1}\n{\\rm det}B>0 \\  or  \\ B=\\left(\\begin{array}{cc}\n           0 & 0 \\\\[1mm]\n           0 & 0\n          \\end{array}\n\\right).\n\\end{equation}\n\n\n\\item [(\\uppercase\\expandafter{\\romannumeral2})]\n$G$ is unitary equivalent to $-\\frac{{\\rm i}\\sqrt{-{\\rm det}B}}{2}(x\\cdot \\partial_x+ \\partial_x \\cdot x)$ if\n\\begin{equation}\\label{type2}\n{\\rm det}B<0.\\end{equation}\n\n\\item [(\\uppercase\\expandafter{\\romannumeral3})]\n$G$ is unitary equivalent to $-\\frac{\\kappa}{2} x^2$ if\n\\begin{equation}\\label{type3}\nB \\ is \\  similar  \\  to \\  \\left(\\begin{array}{cc}\n            0 & 0\\\\[1mm]\n            \\kappa & 0\n          \\end{array}\n\\right) \\  with \\ \\kappa\\neq 0.\n\\end{equation}\n\\end{itemize}\n\\end{Theorem}\n\n\n\n\n\n\\subsection{Growth of Sobolev norm via reducibility}\n\nAs an corollary of Theorem \\ref{thm_redu}, we have:\n\n\\begin{Theorem}\\label{thm_sobo} Under the assumption of Theorem \\ref{thm_redu}, we consider the solution $u(t)=u(t,\\cdot)$ to Eq. (\\ref{eq_abs}) with the non-vanishing initial condition $u(0)\\in {{\\mathcal H}}^s$, $s\\geq 0$. There exists $c, C>0$, depending on $s$ and $u(0)$, such that, for any $t\\geq 0$,\n\\begin{itemize}\n\\item If (\\ref{type1}) holds, then\n$c \\leq \\|u(t)\\|_{s}\\leq C$.\n\\item If (\\ref{type2}) holds, then\n$c  e^{\\sqrt{-{\\rm det}B}st}  \\leq \\|u(t)\\|_{s}\\leq C  e^{\\sqrt{-{\\rm det}B}st}$.\n\\item If (\\ref{type3}) holds, then\n$\nc |\\kappa|^s t^s \\leq \\|u(t)\\|_{s}\\leq C   |\\kappa|^s \\sqrt{1+t^2}^{\\frac{s}2}\n$.\n\\end{itemize}\n\\end{Theorem}\n\n\n\nAccording to (\\ref{norm_U}), to precise the growth of Sobolev norms for the solution to Eq. (\\ref{eq_abs}), it is sufficient to study the reduced quantum Hamiltonian $G(x, -{\\rm i}\\partial_x)$ obtained in (\\ref{Ham_G}), or more simply, the unitary equivalent forms of types (\\uppercase\\expandafter{\\romannumeral1})$-$(\\uppercase\\expandafter{\\romannumeral3}) listed in Theorem \\ref{thm_redu}.\n\n\\smallskip\n\nIf (\\ref{type1}) holds,  then $G$ is unitary equivalent to $\\frac{\\sqrt{{\\rm det}B}}{2} H_0$. Since the ${{\\mathcal H}}^s-$norm of $e^{-{\\rm i}t\\frac{\\sqrt{{\\rm det}B}}{2} H_0}\\psi_0$ is conserved for any $\\psi_0\\in{{\\mathcal H}}^s$, the boundedness of Sobolev norm is shown.\nWe focus on the cases where (\\ref{type2}) and (\\ref{type3}) hold, in which the growth of Sobolev norm occurs.\n\n\n\\begin{Proposition} \\label{prop6}\nFor the equation\n\\begin{equation}\\label{eq_hyper}\n\\partial_t v(t,x)=-\\frac\\lambda2 x\\cdot\\partial_x v (t,x)-\\frac\\lambda2 \\partial_x(x\\cdot v (t,x)), \\qquad \\lambda>0,\n\\end{equation}\n with non-vanishing initial condition $ v (0, \\cdot)= v_0\\in {{\\mathcal H}}^s$, $s\\geq 0$, there exist two constants $\\tilde c, \\, \\tilde C>0$, depending on $s$, $\\lambda$ and $ v _0$, such that the solution satisfies\n\\begin{equation}\\label{bounds_hyper}\n\\tilde c e^{\\lambda st} \\leq \\|\\psi(t,\\cdot )\\|_{s}\\leq   \\tilde C e^{\\lambda st}, \\qquad \\forall \\ t\\geq0.\n\\end{equation}\n\\end{Proposition}\n\n\\begin{remark}\nThis conclusion is also given in Remark 1.4 of \\cite{MR2017}.\n\\end{remark}\n\n\n\n\\proof Through a straightforward computation, we can verify that, for the initial condition $ v (0,\\cdot)= v _0(\\cdot)\\in {{\\mathcal H}}^s$, the solution to Eq. (\\ref{eq_hyper}) satisfies\n$$ v (t,x)=e^{-\\frac\\lambda2 t}  v _0(e^{-\\lambda t} x).$$\nFor any $s\\geq 0$,\n\\begin{eqnarray}\n\\int_{\\mathbb R} x^{2s } |  v (t,x)|^2 \\, dx &=& \\int_{\\mathbb R}  x^{2s }   | v _0(e^{-\\lambda t} x)|^2 \\, d (e^{-\\lambda t}x) \\nonumber\\\\\n&=& e^{2\\lambda s t}\\int_{\\mathbb R} (e^{-\\lambda t} x)^{2s}| v _0(e^{-\\lambda t} x)|^2 \\, d (e^{-\\lambda t}x)\\nonumber\\\\\n&=& e^{2\\lambda s t}\\int_{\\mathbb R} x^{2s} | v _0(x)|^2 \\, dx.\\label{sobolev_hyper}\n\\end{eqnarray}\nand for $s\\in{\\mathbb N}$,\n\\begin{equation}\\label{sobolev_hyper-ds}\n\\int_{\\mathbb R} |\\partial_x^s  v (t,x)|^2 \\, dx = e^{-2\\lambda s t} \\int_{\\mathbb R}  | v _0^{(s)}(e^{-\\lambda t} x)|^2 \\, d (e^{-\\lambda t}x)\n= e^{-2\\lambda s t}\\int_{\\mathbb R} | v _0^{(s)}(x)|^2 \\, dx.\n\\end{equation}\nIn view of the equivalent definition (\\ref{norm_equiv}) of the ${{\\mathcal H}}^s-$norm given in Remark \\ref{remark_norm_equiv}, we get (\\ref{bounds_hyper}) by combining (\\ref{sobolev_hyper}) and (\\ref{sobolev_hyper-ds}).\\qed\n\n\n\n\n\n\n\n\\begin{Proposition}\\label{prop_para}\nFor the equation\n\\begin{equation}\\label{eq_para}\n{\\rm i}\\partial_t v (t,x)=-\\frac{\\kappa}{2} x^2\\cdot v (t,x), \\qquad \\kappa\\in{\\mathbb R},\n\\end{equation}\nwith non-vanishing initial condition $ v _0\\in {{\\mathcal H}}^s$, $s\\geq 0$, there exists constants $\\tilde c, \\tilde C>0$, depending on $s$, $\\kappa$ and $ v _0$, such that the solution satisfies\n\\begin{equation}\\label{sobo_para}\n\\tilde c |\\kappa|^s |t|^s \\leq \\| v (t,\\cdot)\\|_{s}\\leq \\tilde C  |\\kappa|^s (1+ t^2)^\\frac{s}2,\\qquad \\forall \\  t\\in{\\mathbb R}.\n\\end{equation}\n\\end{Proposition}\n\\proof\nWith the initial condition $ v (0,\\cdot)= v _0(\\cdot)\\in {{\\mathcal H}}^s$, the solution to Eq. (\\ref{eq_para}) is\n$$ v (t,x)=e^{{\\rm i}\\frac{\\kappa}{2} x^2 t} v _0(x).$$\nFor any $s\\geq 0$,\n$$\\|x^s v (t,x)\\|_{L^2}=\\|x^s e^{{\\rm i}\\frac{\\kappa}{2} x^2 t} v _0(x)\\|_{L^2}=\\|x^s  v _0(x)\\|_{L^2},$$\nand for $s\\in{\\mathbb N}$,\n\\begin{eqnarray*}\n\\partial_{x}^{s}( v (t,x))\n&=&\\partial_{x}^{s}(e^{{\\rm i}\\frac{\\kappa}{2} x^2 t} v _0(x))\\\\\n&=& \\sum_{\\alpha=0}^{s} C_s^\\alpha (e^{{\\rm i}\\frac{\\kappa}{2} x^2 t})^{(\\alpha)} v _0^{(s-\\alpha)}(x)\\\\\n&=& e^{{\\rm i}\\frac{\\kappa}{2} x^2 t}  \\sum_{\\alpha=0}^{s} C_s^\\alpha \\left(({\\rm i}\\kappa t)^{\\alpha} x^{\\alpha}+P_{\\alpha}({\\rm i}\\kappa t,x)\\right)  v _0^{(s-\\alpha)}(x) \\\\\n&=&({\\rm i} \\kappa t)^{s}  x^{s}  e^{{\\rm i}\\frac{\\kappa}{2} x^2 t}\\cdot  v _0(x) +P_{s}({\\rm i}\\kappa t,x)e^{{\\rm i}\\frac{\\kappa}{2} x^2 t}\\cdot  v _0(x)\\\\\n& &+ \\, x^{\\alpha} e^{{\\rm i}\\frac{\\kappa}{2} x^2 t}  \\sum_{\\alpha=0}^{s-1} C_s^\\alpha \\left(({\\rm i}\\kappa t)^{\\alpha} x^{\\alpha}+P_{\\alpha}({\\rm i}\\kappa t,x)\\right)  v _0^{(s-\\alpha)}(x),\n\\end{eqnarray*}\nwhere, for $\\alpha\\geq 2$, $P_{\\alpha}({\\rm i}\\kappa t,x)$ is a polynomial of degree $\\alpha-2$ of $x$, with the coefficients being monomials of ${\\rm i}\\kappa t$ of degree $\\leq \\alpha-1$ and $P_{1}=P_0=0$. Then, there exists a constant $D>0$ such that\n$$\\left|\\|\\partial_{x}^{s}( v (t,x))\\|_{L^2}-|\\kappa t |^{s}\\|x^{s}   v _0(x)\\|_{L^2}\\right|\\leq D |\\kappa t |^{s-1} \\| v _0(x)\\|_s.$$\nIn view of the equivalent definition (\\ref{norm_equiv}) of norm in Remark \\ref{remark_norm_equiv},\nwe get (\\ref{sobo_para}). \\qed\n\n\\smallskip\n\n\\noindent{\\bf Proof of Theorem \\ref{thm_sobo}.}\nFrom Theorem \\ref{thm_redu}, we know that\nEq. (\\ref{eq_abs}) is conjugated to\n${\\rm i}\\partial_t v= G v$ by the transformation $u=U(\\omega t) v$, with $G=G(x,-{\\rm i}\\partial_x)$ the operator independent of $t$ given in (\\ref{Ham_G_pr}).\n\nRecall Proposition \\ref{prop6} and \\ref{prop_para}.\nGiven $s\\geq 0$, for any non-vanishing $v_0\\in{{\\mathcal H}}^s$, for the three types of unitary equivalence of $G$, there are three different behaviours of the solution to the equation ${\\rm i}\\partial_t v= G v$ as $t\\to  \\infty$.\n\\begin{itemize}\n\\item If $G$ is unitary equivalent to $\\frac{\\sqrt{{\\rm det} B}}{2} H_0$ (under (\\ref{type1})), then\n$\\|e^{-{\\rm i}Gt}v_0\\|_s=O(1)$.\n\\item If $G$ is unitary equivalent to $-\\frac{{\\rm i}\\sqrt{-{\\rm det} B}}{2} (x\\cdot \\partial_x +\\partial_x\\cdot x)$ (under (\\ref{type2})), then\n$\\|e^{-{\\rm i}Gt}v_0\\|_s=O(e^{\\sqrt{-{\\rm det} B} s t}).$\n\\item If $G$ is unitary equivalent to $-\\frac{\\kappa}{2} x^2$ (under (\\ref{type3})), then $\\|e^{-{\\rm i}Gt}v_0\\|_s=O(|\\kappa|^st^s)$.\n\\end{itemize}\nMoreover, according to (\\ref{norm_U}), for $s\\geq 0$, there exist constants $c', C'>0$ such that\n$$  c'\\|v\\|_{s}\\leq \\|U(\\omega t)v\\|_{s}\\leq C'\\|v\\|_{s},\\qquad  \\forall \\  v\\in {{\\mathcal H}}^s.$$\nHence Theorem \\ref{thm_sobo} is shown.\\qed\n\n\n\\section{Reducibility in classical Hamiltonian system and Proof of Theorem \\ref{thm_redu}}\\label{sec_reduc}\n\n\\subsection{Conjugation between classical hamiltonians}\n\nGiven two quadratic classical Hamiltonians\n$$h_j(\\omega t, x, \\xi)=\\frac12 \\big(a_j(\\omega t)x^2+ 2b_j(\\omega t) x\\cdot \\xi+ c_j(\\omega t)  \\xi^2 \\big), \\qquad j=1,2,$$\nwhich can be presented as\n$$h_j(\\omega t, x,\\xi)=\\frac12\\left(\n                             \\begin{array}{c}\n                               x \\\\\n                               \\xi \\\\\n                             \\end{array}\n                           \\right)^{\\top}J A_j(\\omega t)\\left(\n                             \\begin{array}{c}\n                               x \\\\\n                               \\xi \\\\\n                             \\end{array}\n                           \\right), \\qquad j=1,2$$\nwith $J:=\\left(\\begin{array}{cc}\n0 & -1 \\\\1 & 0 \\end{array}\\right)$ and\n$A_j(\\cdot)=\\left(\\begin{array}{cc}\nb_j(\\cdot) & c_j(\\cdot) \\\\ -a_j(\\cdot) & -b_j(\\cdot) \\end{array}\\right)\\in C^{\\omega}({\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$.\nThe corresponding equations of motion are  given by\n$$x'=\\frac{\\partial h_j}{\\partial\\xi},\\quad \\xi'=-\\frac{\\partial h_j}{\\partial x},\\qquad j=1,2,$$\nwhich are the linear systems $(\\omega, \\, A_j)$:\n$$\\left(\\begin{array}{c}\n          x(t) \\\\\n          \\xi(t)\n        \\end{array}\n\\right)'=A_j(\\omega t)\\left(\\begin{array}{c}\n          x(t) \\\\\n          \\xi(t)\n        \\end{array}\n\\right).$$\n\n\n\n\\begin{Proposition}\\label{prop_ham_cl}\nIf the linear system $(\\omega, \\, A_1(\\cdot))$ is conjugated to $(\\omega, \\, A_2(\\cdot))$ by a time quasi-periodic ${\\rm SL}(2,{\\mathbb R})-$transformation, i.e.,\n\\begin{equation}\\label{conj_ode}\n\\frac{d}{dt} e^{Z(\\omega t)}=A_1(\\omega t)e^{Z(\\omega t)}-e^{Z(\\omega t)} A_2(\\omega t),\\qquad Z \\in C^\\omega(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R})),\n\\end{equation}\nthen the classical Hamiltonian $h_1(\\omega t,x,\\xi)$ is conjugated to $h_2(\\omega t,x,\\xi)$ via the time$-1$ flow $\\phi_{\\chi}^1(t,x,\\xi)$\n generated by the Hamiltonian\n \\begin{equation}\\label{chi_eZ}\n \\chi(\\omega t,x,\\xi)=\\frac12\\left(\n                             \\begin{array}{c}\n                               x \\\\\n                               \\xi \\\\\n                             \\end{array}\n                           \\right)^{\\top}J Z(\\omega t)\\left(\n                             \\begin{array}{c}\n                               x \\\\\n                               \\xi \\\\\n                             \\end{array}\n                           \\right).\n \\end{equation}\n\\end{Proposition}\n\n\\proof Note that the equation of motion of the classical Hamiltonian $h_1$ is the linear system $(\\omega, \\, A_1(\\cdot))$:\n$$ \\left(\\begin{array}{c}\n   x \\\\\n    \\xi\n  \\end{array}\\right)'=A_1(\\omega t)\\left(\\begin{array}{c}\n   x \\\\\n    \\xi\n  \\end{array}\\right).$$\nIn view of (\\ref{conj_ode}), the transformation\n\\begin{equation}\\label{tramSL}\n  \\left(\\begin{array}{c}\n   x \\\\\n    \\xi\n  \\end{array}\\right)= e^{Z(\\omega t)}\\left(\\begin{array}{c}\n    \\tilde x \\\\\n   \\tilde \\xi\n  \\end{array}\\right),\\qquad Z\\in C^{\\omega}(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R})),\n\\end{equation}\nconjugates $(\\omega, \\, A_1(\\cdot))$ to $(\\omega, \\, A_2(\\cdot))$. More precisely,\n\\begin{eqnarray*}\n\\left(\\begin{array}{c}\n\\tilde x \\\\\n\\tilde \\xi\n\\end{array}\\right)'&=&e^{-Z(\\omega t)}A_1(\\omega t)\\left(\\begin{array}{c}\nx \\\\\n\\xi\n\\end{array}\\right)-e^{-Z(\\omega t)}\\frac{d}{dt}e^{Z(\\omega t)}\\left(\\begin{array}{c}\n\\tilde x\\\\\n\\tilde \\xi\n\\end{array}\\right)  \\\\\n&=& e^{-Z(\\omega t)}A_1(\\omega t)e^{Z(\\omega t)}\\left(\\begin{array}{c}\n{\\tilde x} \\\\\n{\\tilde \\xi}\n\\end{array}\\right)-e^{-Z(\\omega t)}\\frac{d}{dt}e^{Z(\\omega t)}\\left(\\begin{array}{c}\n\\tilde x\\\\\n\\tilde \\xi\n\\end{array}\\right)\\\\\n&=&A_2(\\omega t)\\left(\\begin{array}{c}\n\\tilde x\\\\\n\\tilde \\xi\n\\end{array}\\right),\n\\end{eqnarray*}\nfor which the corresponding Hamiltonian is $h_2(\\omega t,\\tilde x,\\tilde\\xi)$.\nAs in (3-35) of \\cite{BGMR2018}, the time$-1$ map between the two Hamiltonians is generated by (\\ref{chi_eZ}) since there is only quadratic terms in the Hamiltonian in our case.\\qed\n\n\n\n\n\\subsection{Proof of Theorem \\ref{thm_redu}}\n\n\nWe consider the classical Hamiltonian\n\\begin{eqnarray*}\nL(\\omega t,x,\\xi)&=&\\frac{a(\\omega t)}{2}x^2+\\frac{b(\\omega t)}{2}(x\\cdot\\xi+\\xi \\cdot x)+\\frac{c(\\omega t)}{2}\\xi^2 \\nonumber \\\\\n&=&\\frac12X^{\\top}J A(\\omega t)X,\\qquad X:= \\left(\n                             \\begin{array}{c}\n                               x \\\\\n                               \\xi \\\\\n                             \\end{array}\n                           \\right).\n\\end{eqnarray*}\nwith $a,b,c\\in C^{\\omega}({\\mathbb T}^d)$ given in Eq. (\\ref{eq_abs}), and $A:=\\left(\\begin{array}{cc}\n            b & c \\\\\n           -a & -b\n          \\end{array}\n\\right)\\in C^{\\omega}({\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$.\n\n\n\nBy the hypothesis of Theorem \\ref{thm_redu}, the linear system $(\\omega, \\, A(\\cdot))$ can be reduced to the constant system $(\\omega, \\, B)$, with $B=\\left(\\begin{array}{cc}\n            B_{11} & B_{12} \\\\\n            -B_{21} & -B_{11}\n          \\end{array}\n\\right)\\in{\\rm sl}(2,{\\mathbb R})$, via finitely many transformations $(e^{Z_j})_{j=0}^K$ with $Z_j\\in C^{\\omega}(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$. Hence the reduced classical Hamiltonian is\n$$g(x,\\xi)=\\frac12X^{\\top}J B X= \\frac{B_{21}}2 x^2+\\frac{B_{11}}{2}(x\\cdot \\xi+ \\xi\\cdot x)+\\frac{B_{12}}2 \\xi^2.$$\nBy Proposition \\ref{Prop_hami}, we see that $L^W(\\omega t, x, -{\\rm i}\\partial_x)$ is conjugated to\n\\begin{equation}\\label{Ham_G_pr}\nG(x, -{\\rm i}\\partial_x):=g^W(x, -{\\rm i}\\partial_x)=\\frac{B_{21}}2 x^2-\\frac{B_{11}}{2}(x\\cdot{\\rm i}\\partial_x+{\\rm i}\\partial_x\\cdot x)-\\frac{B_{12}}2 \\partial_x^2\n\\end{equation}\nvia the product of unitary (in $L^2({\\mathbb R})$) transformations\n$$\nU(\\omega t):= \\prod_{j=0}^K e^{-{\\rm i}\\chi^W_j(\\omega t,x,-{\\rm i}\\partial_x)}\n$$\nwhere $\\chi^W_j$ is the Weyl quantization of\n$$\\chi_j(\\omega t,x,\\xi)=\\frac12X^{\\top} J Z_j(\\omega t)X.$$\nThen (\\ref{norm_U}) is deduced from (\\ref{change_Sobolevnorm}) in Lemma \\ref{lem_Sobolev}.\nThe following diagram gives a straightforward explanation for the above proof.\n$$\n\\begin{array}{rcccl}\n& X'=A(\\omega t)X &\\stackrel{\\prod_{j=0}^K e^{Z_j(\\omega t)}}{\\longrightarrow} & X'=BX &  \\  \\  Z_j\\in C^\\omega(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))  \\\\\n  &   &  & &  \\\\\n  &  \\big\\updownarrow &  &  \\big\\updownarrow & \\\\\n  &  &  &  & \\\\\n & L(\\omega t)=\\frac12X^{\\top} J A(\\omega t)X & \\stackrel{ \\Phi^1_{\\chi_0(\\omega t)}\\circ \\cdots \\circ  \\Phi^1_{\\chi_K(\\omega t)}}{\\longrightarrow} & g=\\frac12X^{\\top} J B X & \\ \\ \\chi_j =\\frac12X^{\\top}J Z_j X\\\\\n  &  &  & &  \\\\\n   &  \\big\\updownarrow &  &  \\big\\updownarrow & \\\\\n  &  &  & &  \\\\\n & {\\rm i}\\partial_t u=L^W(\\omega t)u &  \\stackrel{\\prod_{j=0}^{K} e^{-{\\rm i} \\chi_j^W(\\omega t)}}{\\longrightarrow} &  {\\rm i}\\partial_t u = g^W u &\n  \\end{array}\n$$\n\nIf (\\ref{type1}) holds, i.e., ${\\rm det}B>0$ or $B=\\left(\\begin{array}{cc}\n            0 & 0 \\\\\n            0 & 0\n          \\end{array}\\right)$,\nthen there exists $C_B\\in {\\rm sl}(2,{\\mathbb R})$ such that\n\\begin{equation}\\label{elliptic}\nB=e^{C_B}\\left(\\begin{array}{cc}\n            0 & \\sqrt{{\\rm det}B} \\\\\n            -\\sqrt{{\\rm det}B} & 0\n          \\end{array}\\right)e^{-C_B}.\n\\end{equation}\nIf (\\ref{type2}) holds,  i.e., ${\\rm det}B<0$, then there exists $C_B\\in {\\rm sl}(2,{\\mathbb R})$ such that\n\\begin{equation}\\label{hyerbolic}\nB=e^{C_B}\\left(\\begin{array}{cc}\n            \\sqrt{-{\\rm det}B} & 0 \\\\\n            0 & -\\sqrt{-{\\rm det}B}\n          \\end{array}\\right)e^{-C_B}.\n\\end{equation}\nIf (\\ref{type3}) holds,\nthen there exists $C_B\\in {\\rm sl}(2,{\\mathbb R})$ such that\n\\begin{equation}\\label{parapolic}\nB=e^{C_B}\\left(\\begin{array}{cc}\n            0 & 0 \\\\\n            \\kappa & 0\n          \\end{array}\\right)e^{-C_B}.\n\\end{equation}\nTherefore, for Eq. (\\ref{eq_abs}), the three types of unitary equivalence of $G=G(x,-{\\rm i}\\partial_x)$ are shown by (\\ref{elliptic})$-$(\\ref{parapolic}) respectively. \\qed\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev}}\\label{sec_proof}\n\n\nIn view of Theorem \\ref{thm_redu}, to show the reducibility of Eq. (\\ref{eq_Schrodinger}), it is sufficient to show the reducibility of the corresponding ${\\rm sl}(2,{\\mathbb R})-$linear system.\n\nFor $E\\in{{\\mathcal I}}$, the symbol of the quantum Hamiltonian (\\ref{eq_Schrodinger}) is\n$$h_E(\\omega t, x,\\xi)=\\frac{\\nu(E)}{2}(\\xi^2+x^2)+W(E,\\omega t,x,\\xi)$$\nwhich corresponds the quasi-periodic linear system $(\\omega, \\,  A_0+F_0)$\n\\begin{equation}\\label{linear_system_pr}\n\\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right)'=\\left[\\left(\\begin{array}{cc}\n             0 & \\nu(E) \\\\\n             -\\nu(E) & 0\n           \\end{array}\\right)\n           +\\left(\\begin{array}{cc}\n             b(E,\\omega t) & c(E,\\omega t) \\\\\n             -a(E,\\omega t) & -b(E,\\omega t)\n           \\end{array}\\right)\\right] \\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right),\n\\end{equation}\nwhere, for every $E\\in{{\\mathcal I}}$,\n\\begin{align*}\nA_0(E):= & \\left(\\begin{array}{cc}\n             0 & \\nu(E) \\\\\n             -\\nu(E) & 0\n           \\end{array}\\right)\\in {\\rm sl}(2,{\\mathbb R}), \\\\\n F_0(E,\\cdot):=& \\left(\\begin{array}{cc}\n             b(E,\\cdot) & c(E,\\cdot) \\\\\n             -a(E,\\cdot) & -b(E,\\cdot)\n           \\end{array}\\right)\\in C^{\\omega}_r({\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))\n\\end{align*}\nwith $|\\partial_E^m F_0|_r<\\varepsilon_0$, $m=0,1,2$, sufficiently small.\n\n\nThe reducibility of linear system (\\ref{linear_system_pr}) is exploited by Eliasson \\cite{Eli1992} (see also \\cite{HA} for results about ${\\rm SL}(2,{\\mathbb R})$-cocycles). We summarise the needed results in the following proposition. To make the paper as self-contained as possible, we give a short proof without adding too many details on known facts.\nSince every quantity depends on $E$, we do not always write this dependence explicitly in the statement of proposition.\n\n\nBefore stating the precise result, we introduce the concept of rotation number. The {\\it rotation number}\nof quasi-periodic ${\\rm sl}(2,{\\mathbb R})-$linear system (\\ref{linear_system_pr})\n is defined as\n$$\\rho(E)=\\rho(\\omega, \\, A_0(E)+F(E,\\omega t))=\\lim_{t\\to\\infty}\\frac{\\arg(\\Phi_E^t X)}{t},\\qquad \\forall \\ X\\in \\mathbb{R}^2\\setminus\\{0\\}$$\nwhere $\\Phi_E^t$ is the basic\nmatrix solution and $\\arg$ denotes the angle. The rotation number\n$\\rho$ is well-defined and it does not depend on $X$\n\\cite{JM82}.\n\n\n\\begin{Proposition}\\label{prop_eliasson} There exists $\\varepsilon_*=\\varepsilon_*(r,\\gamma,\\tau,d,l_1,l_2)>0$ such that if\n\\begin{equation}\\label{small_F_0}\n\\max_{m=0,1,2}|\\partial_E^m F_0|_r=:\\varepsilon_0<\\varepsilon_*,\n\\end{equation}\nthen the following holds for the quasi-periodic linear system $(\\omega, \\, A_0+F_0)$.\n\\begin{enumerate}\n  \\item [(1)] For a.e. $E\\in{{\\mathcal I}}$, $(\\omega, \\, A_0+F_0(\\cdot))$ is reducible. More precisely,\n there exist $B\\in{\\rm sl}(2,{\\mathbb R})$ and  $Z_j\\in C^\\omega(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$, $j=0,1,\\cdots,K$, such that\n \\begin{equation}\\label{reducibility_sl2R}\n   \\frac{d}{dt}\\left(\\prod_{j=0}^K e^{Z_j(\\omega t)}\\right)=\\left(A_0+F_0(\\omega t)\\right)\\left(\\prod_{j=0}^K e^{Z_j(\\omega t)}\\right)-\\left(\\prod_{j=0}^K e^{Z_j(\\omega t)}\\right)B.\n \\end{equation}\n  \\item [(2)] The rotation number $\\rho=\\rho(E)$ is monotonic on ${\\mathcal I}$. For any $k\\in{\\mathbb Z}^d$,\n  $$\\tilde\\Lambda_k:=\\left\\{E\\in\\overline{{\\mathcal I}}:\\rho(E)=\\frac{\\langle k,\\omega\\rangle}{2}\\right\\}  \\   \\footnote{$\\tilde\\Lambda_k$ can be empty for some $k\\in{\\mathbb Z}^d$ if the closed interval $\\rho^{-1}\\left(\\frac{\\langle k,\\omega\\rangle}{2}\\right)$ does not intersect ${{\\mathcal I}}$.} $$ is a closed interval, and we have\n\\begin{equation}\\label{measure_esti}\n\\sum_{k\\in{\\mathbb Z}^d}{\\rm Leb}(\\tilde\\Lambda_k)<\\varepsilon_0^{\\frac{1}{40}}.\n\\end{equation}\n  \\item [(3)] For every $E\\in \\tilde\\Lambda_k=:[a_k,b_k]$, $(\\omega, \\, A_0+F_0(\\cdot))$ is reducible and the matrix $B\\in {\\rm sl}(2,{\\mathbb R})$ in (\\ref{reducibility_sl2R}) satisfies\n \\begin{itemize}\n   \\item if $a_k=b_k$, then $B=\\left(\\begin{array}{cc}\n                                       0 & 0 \\\\\n                                       0 & 0\n                                     \\end{array}\\right)$;\n   \\item if $a_k<b_k$, then\n\\begin{itemize}\n  \\item ${\\rm det} B<0$ for $E\\in(a_k,b_k)$,\n  \\item ${\\rm det} B=0$ for $E=a_k,b_k$ and $E\\not\\in\\partial {{\\mathcal I}}$.\n\\end{itemize}\n \\end{itemize}\n   \\item [(4)] For a.e. $E\\in{{\\mathcal I}}\\setminus\\bigcup_k \\tilde\\Lambda_k$, $(\\omega, \\, A_0+F_0(\\cdot))$ is reducible and the matrix $B\\in {\\rm sl}(2,{\\mathbb R})$ in (\\ref{reducibility_sl2R}) satisfies ${\\rm det} B>0$.\n\\end{enumerate}\n\\end{Proposition}\n\n\\proof Since $\\nu$ is a strictly monotonic real-valued function of $E\\in{{\\mathcal I}}$ and $|\\nu'|\\geq l_1$, $|\\nu''|\\leq l_2$, (\\ref{small_F_0}) implies that\n$|\\partial^m_E F_0(\\nu^{-1}(E),\\cdot)|_r$, $m=0,1,2$, is also small enough.\n Hence, to prove the above arguments, we can simply consider the case where $\\nu(E)=E\\in {{\\mathcal I}}= {\\mathbb R}$ and then obtain Proposition \\ref{prop_eliasson} by replacing $E$ by $\\nu(E)$.\n\n\n\n\n\n\n\n\n\n\\smallskip\n\n\\noindent {\\it Proof of (1).}\nThe almost reducibility has already been shown by Eliasson \\cite{Eli1992} for every $E\\in{\\mathbb R}$.\nIndeed, if $\\max_{m=0,1,2}|\\partial_E^m F_0|_r$ is small enough (depending on $r,\\gamma,\\tau,d$), then there exists sequences $(Y_j)_{j\\in{\\mathbb N}}\\subset C^\\omega(2{\\mathbb T}^d, {\\rm SL}(2,{\\mathbb R}))$, $(A_j)_{j\\in{\\mathbb N}}\\subset {\\rm sl}(2,{\\mathbb R})$, and $(F_j)_{j\\in{\\mathbb N}}\\subset C^\\omega(2{\\mathbb T}^d, {\\rm sl}(2,{\\mathbb R}))$, all of which are piecewise $C^2$ w.r.t. $E$,\n with $\\max_{m=0,1,2}|\\partial^m_E F_j|_{{\\mathbb T}^d}<\\varepsilon_j:=\\varepsilon_0^{(1+\\sigma)^j}$ for $\\sigma=\\frac1{33}$, such that\n$$\\frac{d}{dt}Y_{j}(\\omega t)=\\left(A_j+F_j(\\omega t)\\right)Y_{j}(\\omega t)-Y_{j}(\\omega t) \\left(A_{j+1}+F_{j+1}(\\omega t)\\right).$$\nMore precisely, at the $j-$th step, for $\\pm{\\rm i}\\xi_j\\in{\\mathbb R}\\cup{\\rm i}{\\mathbb R}$, the two eigenvalues of $A_j$, and\n$$N_j:=\\frac{2\\sigma}{r_j-r_{j+1}}\\ln\\left(\\frac{1}{\\varepsilon_j}\\right)$$\nwith $(r_j)_{j\\in{\\mathbb N}}$ a decreasing sequence of positive numbers such that $r_j-r_{j+1}\\geq 2^{-(j+1)}r$ for each $j$,\n\\begin{itemize}\n  \\item (non-resonant case) if for every $n\\in{\\mathbb Z}^d$ with $0<|n|\\leq N_j$, we have\n  \\begin{equation}\\label{non_resonant}\n  \\left|2\\xi_j-\\langle n,\\omega\\rangle\\right|\\geq \\varepsilon_j^{\\sigma},\n  \\end{equation}\n  then $Y_{j}=e^{\\tilde Z_{j}}$ for some $\\tilde Z_{j}\\in C^{\\omega}(2{\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$ with $|\\tilde Z_{j}|_{2{\\mathbb T}^d}<\\varepsilon_j^{\\frac23}$, and $|A_{j+1}-A_j|<\\varepsilon_j^{\\frac23}$;\n  \\item (resonant) if for some $n_j\\in{\\mathbb Z}^d$ with $0<|n_j|\\leq N_j$, we have\n  \\begin{equation}\\label{resonant}\n  \\left|2\\xi_j-\\langle n_j,\\omega\\rangle\\right|< \\varepsilon_j^{\\sigma},\n  \\end{equation}\n  then $Y_{j+1}(\\cdot)=e^{\\frac{\\langle n_j ,\\cdot\\rangle}{2\\xi_j}A_j} e^{\\tilde Z_{j+1}}$ for some $\\tilde Z_{j}\\in C^{\\omega}(2{\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$ with $|\\tilde Z_{j}|_{2{\\mathbb T}^d}<\\varepsilon_j^{\\frac23}$ and $|A_{j+1}|<\\varepsilon_j^{\\frac{\\sigma}2}$.\n\\end{itemize}\nAs $j$ goes to $\\infty$, the time-dependent part $F_{j}$ tends to vanish. Hence $(\\omega,\\, A_0(E)+F_0)$ is almost reducible. For the detailed proof, we can refer to Lemma 2 of \\cite{Eli1992} and its proof.\n\nIn view of Lemma 3 b) of \\cite{Eli1992}, if the rotation number $\\rho(E)$ of $(\\omega, \\, A_0(E)+F_0)$ is Diophantine or rational w.r.t. $\\omega$, which corresponds to a.e. $E\\in{\\mathbb R}$, then the resonant case occurs for only finitely many times.\nTherefore, for a.e. $E\\in{\\mathbb R}$, there exists a large enough $J_*\\in{\\mathbb N}^*$, depending on $E$, such that\n\\begin{equation}\\label{J_large}\nY_{j}=e^{\\tilde Z_{j}} \\  \\ {\\rm with} \\   \\  |\\tilde Z_{j}|_{2{\\mathbb T}^d}<\\varepsilon^{\\frac23}_{j},\\qquad \\forall \\ j\\geq J_*.\n\\end{equation}\nThis implies that $\\prod_{j=0}^\\infty|Y_j|_{2{\\mathbb T}^d}$ is convergent.\nAs explained in the proof of Lemma 3.5 of \\cite{BGMR2018}, (\\ref{J_large}) also implies that there exists $S\\in C^{\\omega}(2{\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$ such that\n$\\prod_{j=J_*}^\\infty Y_{j}=e^S$,\nsince $\\varepsilon_0$ is sufficiently small.\nHence (\\ref{reducibility_sl2R}) is shown, i.e., the reducibility is realized via finitely many transformations of the form $e^{Z_j(\\omega t)}$ with $Z_j\\in C^{\\omega}(2{\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$.\n\n\n\n\n\n\n\n\\\n\n\\noindent {\\it Proof of (2).}\nFor $k\\in{\\mathbb Z}^d$, $\\tilde\\Lambda_k$ is obtained after several resonant KAM-steps, saying $j_1$, $\\cdots$, $j_L$, where $n_{j_i}\\in{\\mathbb Z}^d$ with $0<|n_{j_i}|\\leq N_{j_i}$, $i=1,\\cdots,L$, satisfies\n$$ \\left|2\\xi_{j_i}-\\langle n_{j_i},\\omega\\rangle\\right|< \\varepsilon_{j_i}^{\\sigma},$$\nand $k=n_{j_1}+\\cdots + n_{j_L}$. We will show that\n\\begin{equation}\\label{k_n_j_L}\n\\frac{10|n_{j_L}|}{11}\\leq |k|\\leq  \\frac{12|n_{j_L}|}{11}.\n\\end{equation}\n Assume that $L\\geq 2$ (otherwise we have already $k=n_{j_L}$). After the $(j_{i-1}+1)-$th step, $i=2,\\cdots,L$, the eigenvalues $\\pm{\\rm i}\\xi_{j_{i-1}+1}$ satisfies $|\\xi_{j_{i-1}+1}|<2\\varepsilon_{j_{i-1}}^{\\frac\\sigma2}$. On the other hand, before the $(j_{i}+1)-$th step, the resonant condition (\\ref{resonant}) implies that the eigenvalues $\\pm{\\rm i}\\xi_{j_{L}}$ satisfy that\n$$|2\\xi_{j_{i}}-\\langle n_{j_i},\\omega\\rangle|\\leq \\varepsilon_{j_{i}}^\\sigma.$$\nSince the steps between these two successive resonant steps are all non-resonant, and $\\omega\\in {\\rm DC}_{d}(\\gamma,\\tau)$,\nwe have that\n$$\\frac{\\gamma}{|n_{j_i}|^\\tau}\\leq|\\langle n_{j_i},\\omega\\rangle|\\leq 2|\\xi_{j_{i-1}+1}|+2\\varepsilon^{\\frac13}_{j_{i-1}+1}+\\varepsilon_{j_{i}}^\\sigma<3\\varepsilon_{j_{i-1}}^{\\frac\\sigma2},$$\nwhich implies that\n$$|n_{j_i}|>\\left(\\frac\\gamma3\\right)^{\\frac{1}{\\tau}}\\varepsilon^{-\\frac{\\sigma}{2\\tau}}_{j_{i-1}}>12|N_{j_{i-1}}|\\geq 12|n_{j_{i-1}}|.$$\nHence, we get (\\ref{k_n_j_L}).\n\n\n\n$\\tilde\\Lambda_k$ is firstly formed at the $j_L-$th step, with the initial measure smaller than $\\varepsilon_{j_L}^{2\\sigma}$.\nSince all the succedent steps are non-resonant, the measure of $\\tilde\\Lambda_k$ varies up to $\\varepsilon_{j_L}^{2\\sigma}$. Then, for $\\varsigma:=\\frac{\\ln(1+\\sigma)}{\\ln(8+8\\sigma)}$, we have\n$$\n{\\rm Leb}(\\tilde\\Lambda_k)< 2\\varepsilon_{j_L}^{2\\sigma}< 2\\varepsilon_0^{\\sigma} e^{-\\left(\\frac{12}{11}\\right)^\\varsigma N_{j_L}^\\varsigma}\\leq 2\\varepsilon_0^{\\sigma} e^{-\\left(\\frac{12}{11}\\right)^\\varsigma |n_{j_L}|^\\varsigma}.\n$$\nIndeed, recalling that $r_j-r_{j+1}\\geq 2^{-(j+1)}r$ for every $j$, we have\n\\begin{eqnarray*}\n\\varepsilon_{j_L} &=& \\exp\\{-|\\ln\\varepsilon_0|(1+\\sigma)^{j_L}\\} \\\\\n &=& \\exp\\left\\{- \\frac{|\\ln\\varepsilon_0|^{1-\\varsigma}(1+\\sigma)^{j_L(1- \\varsigma)}(r_{j_L}-r_{j_L+1})^{\\varsigma}}{(2\\sigma)^{\\varsigma}} N_{j_L}^\\varsigma\\right\\}\\\\\n &\\leq&\\exp\\left\\{- \\frac{|\\ln\\varepsilon_0|^{1-\\varsigma} r^\\varsigma}{(4\\sigma)^{\\varsigma}} \\left(\\frac{(1+\\sigma)^{1- \\varsigma}}{2^\\varsigma}\\right)^{j_L} N_{j_L}^\\varsigma\\right\\}\\\\\n &<&\\exp\\left\\{-\\left(\\frac{12}{11}\\right)^\\varsigma \\frac{N_{j_L}^\\varsigma}{\\sigma}\\right\\},\n\\end{eqnarray*}\nsince $\\varepsilon_0$ is small enough and\n$$\\frac{(1+\\sigma)^{1- \\varsigma}}{2^\\varsigma}=\\exp\\left\\{\\frac{\\ln(1+\\sigma)}{\\ln(8+8\\sigma)}\\left(\\ln 8-\\ln 2\\right)\\right\\}>1.$$\nTherefore, by (\\ref{k_n_j_L}), we get\n${\\rm Leb}(\\tilde\\Lambda_k)<2\\varepsilon_0^{\\sigma} e^{- |k|^\\varsigma}$,\nwhich implies (\\ref{measure_esti}). For detailed proof of the measure estimate of $\\tilde\\Lambda_k$, we can also refer to Corollary 1 of \\cite{HA}.\n\n\\\n\n\n\\noindent {\\it Proof of (3) and (4).} It can be deduced from Lemma 5 of \\cite{Eli1992}. \\qed\n\n\n\n\\\n\n\\noindent{\\bf Proof of Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev}.} Theorem \\ref{thm_Schro_sobolev} can be seen as a corollary of Theorem \\ref{thm_sobo}.\nAccording to Theorem \\ref{thm_redu}, the reducibility of Eq. (\\ref{eq_Schrodinger}) for a.e. $E\\in{{\\mathcal I}}$ is deduced from Proposition \\ref{prop_eliasson}-(1).\nLet $\\{\\Lambda_j\\}_{j\\in{\\mathbb N}}$ be the intervals $\\tilde\\Lambda_k$'s intersecting ${{\\mathcal I}}$ and let\n$${{\\mathcal O}}_{\\varepsilon_0}:=\\bigcup_{j\\in{\\mathbb N}}\\Lambda_j=\\bigcup_{k\\in{\\mathbb Z}^d}\\tilde\\Lambda_k.$$\nProposition \\ref{prop_eliasson}-(2) gives the measure estimate of ${{\\mathcal O}}_{\\varepsilon_0}$.\nThe unitary equivalences of the reduced quantum Hamiltonian follow from Proposition \\ref{prop_eliasson}-(3) and (4). Hence Theorem \\ref{thm_Schro} is shown. \\qed\n\n\\section{Proof of Theorem \\ref{thm_example_1} -- \\ref{thm_AMO}}\\label{sec_pr_examples}\n\n\nIn this section, we show that the measure of the subset ${{\\mathcal O}}_{\\varepsilon_0}$ is positive for the equations (\\ref{example_1}) -- (\\ref{eq_AMO}),\nwhich implies the growths of Sobolev norm.\n\n\\subsection{Proof of Theorem \\ref{thm_example_1}}\n\nFor Eq. (\\ref{example_1}), $E\\in{\\mathbb R}$, the corresponding linear system is\n$$\n\\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right)'=\\left[\\left(\\begin{array}{cc}\n             0 & E \\\\\n             -E & 0\n           \\end{array}\\right)\n           +\\left(\\begin{array}{cc}\n             b(\\omega t) & c(\\omega t) \\\\\n             -a(\\omega t) & -b(\\omega t)\n           \\end{array}\\right)\\right] \\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right).$$\nIn view of Lemma 5 of \\cite{Eli1992}, for ``generic\" $a,b,c\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$, there is at least one non-degenerate $\\tilde\\Lambda_k$, $k\\in{\\mathbb Z}^d$.\nMore precisely, at the resonant step of KAM scheme described in the proof of Proposition \\ref{prop_eliasson}-(1),\nthe condition (\\ref{resonant}) defines a resonant interval of $E$, on which the two eigenvalues $\\pm{\\rm i}\\xi_j$ of $A_j$ are purely imaginary since $\\xi_j$ is bounded frow below. After this resonant step, the two new eigenvalues $\\pm{\\rm i}\\xi_{j+1}$ of $A_{j+1}$ can be real or still purely imaginary for $E$ in this resonant interval, since $|\\xi_{j+1}|$ is close to zero.\nWe say that $a,b,c\\in C^\\omega({\\mathbb T}^d,{\\mathbb R})$ are {\\it generic} if, for at least one resonant step in the KAM scheme, the two new eigenvalues $\\pm{\\rm i}\\xi_{j+1}$ become real on a sub-interval of the resonant interval.\n\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{thm_example_schro}}\n\n\n\nFor Eq. (\\ref{eq_Schrodinger-example}) with $E\\in{{\\mathcal I}}=[E_0,E_1]$ with $E_0>0$ large enough, and $E_1<\\infty$, Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev} hold.\nThe corresponding linear system $(\\omega, \\, A_0+F_0)$ of Eq. (\\ref{eq_Schrodinger-example}) is\n$$\n\\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right)'=\\left[\\left(\\begin{array}{cc}\n             0 & \\sqrt{E} \\\\\n             -\\sqrt{E} & 0\n           \\end{array}\\right)\n           +\\frac{q(\\omega t)}{2\\sqrt{E}}\\left(\\begin{array}{cc}\n             -1 & -1 \\\\\n             1 & 1\n           \\end{array}\\right)\\right] \\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right).$$\nThen, through the change of variables\n$$\\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\\right)=\\frac{1}{2\\sqrt{E}}\\left(\\begin{array}{cc}\n           \\sqrt{E} & -1 \\\\\n           \\sqrt{E} & 1\n           \\end{array}\\right)\\left(\\begin{array}{c}\n          \\tilde x \\\\\n          \\tilde\\xi\n        \\end{array}\\right),$$\n$(\\omega, \\, A_0+F_0)$ is conjugated to\n$$\n\\left(\\begin{array}{c}\n          \\tilde x \\\\\n          \\tilde\\xi\n        \\end{array}\\right)'=C^E_q(\\omega t)\\left(\\begin{array}{c}\n          \\tilde x \\\\\n          \\tilde\\xi\n        \\end{array}\\right):=\\left(\\begin{array}{cc}\n             0 & 1 \\\\\n             -E+q(\\omega t) & 0\n           \\end{array}\\right)\\left(\\begin{array}{c}\n          \\tilde x \\\\\n          \\tilde\\xi\n        \\end{array}\\right).\n$$\nThe quasi-periodic linear system $(\\omega, \\, C^E_q(\\cdot))$ corresponds exactly to the eigenvalue problem of the quasi-periodic continuous Schr\\\"odinger operator ${{\\mathcal L}}_{\\omega, q}$:\n $$({{\\mathcal L}}_{\\omega, q}y)(t)=-y''(t)+q(\\omega t) y(t).$$\nBy Gap labeling Theorem \\cite{JM82}, if $\\tilde\\Lambda_k$ is not empty for $k\\in{\\mathbb Z}^d$, then it is indeed a ``spectral gap\" of ${{\\mathcal L}}_{\\omega, q}$ intersecting $[E_0,E_1]$, i.e., a connected component of $[E_0,E_1]\\setminus\\Sigma_{\\omega, q}$ with $\\Sigma_{\\omega, q}$ denoting the spectrum of ${{\\mathcal L}}_{\\omega, q}$.\nIn view of Theorem C of \\cite{Eli1992}, for a generic potential $q$ (in the $|q|_r$-topology), for $E_0>0$ large enough, $[E_0,\\infty[ \\, \\cap \\, \\Sigma_{\\omega, q}$ is a Cantor set.\nHence there are infinitely many $\\tilde\\Lambda_k$'s satisfying ${\\rm Leb}(\\tilde\\Lambda_k)>0$.\n\n\n\\subsection{Proof of Theorem \\ref{thm_AMO}}\n\nFor Eq. (\\ref{eq_AMO}) with $\\nu(E)=\\cos^{-1}(-\\frac{E}{2})$, $E\\in[-2+\\delta,2-\\delta]$ with $\\delta>0$ a sufficiently small numerical constant (e.g. $\\delta:=10^{-6}$), we can apply Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev}.\n if $a,b,c:[-2+\\delta,2-\\delta]\\times{\\mathbb T}^2\\to{\\rm  sl}(2,{\\mathbb R})$ are small enough as assumed in Theorem \\ref{thm_Schro}.\n\n\n\nFor the quasi-periodic Schr\\\"odinger cocycle $(\\alpha, \\, S_E^\\lambda)$\n$$\nX_{n+1} =S_E^\\lambda(\\theta+n\\alpha) X_n= \\left[\\left(\\begin{array}{cc}\n             -E & -1 \\\\\n             1 & 0\n           \\end{array}\\right)+\\left(\\begin{array}{cc}\n             2\\lambda \\cos(\\theta+n\\alpha) & 0 \\\\\n             0 & 0\n           \\end{array}\\right)\\right] X_n,\n$$\nwith $\\alpha\\in{\\rm DC}_1(\\gamma,\\tau)$, $|\\lambda|$ small enough,\nit can be written as\n$$X_{n+1}=e^{B(E)}e^{G(E,\\theta+n\\alpha)} X_n,$$\nfor $e^{B(E)}:=\\left(\\begin{array}{cc}\n             -E & -1 \\\\\n             1 & 0\n           \\end{array}\\right)$ and some $G(E,\\cdot)\\in{\\rm sl}(2,{\\mathbb R})$.\nThis cocycle is related to the almost-Mathieu operator $H_{\\lambda,\\alpha,\\theta}$ on $\\ell^2({\\mathbb Z})$:\n$$(H_{\\lambda,\\alpha,\\theta}\\psi)_n=-(\\psi_{n+1}+\\psi_{n-1})+2\\lambda \\cos(\\theta+n\\alpha)\\psi_n,\\qquad n\\in{\\mathbb Z}.$$\nIt is known that its spectrum, denoted by $\\Sigma_{\\lambda,\\alpha}$, is a Cantor set \\cite{AvilaJito1},  which is well-known as Ten Martini Problem. In fact, Avila-Jitomirskaya\n \\cite{AvilaJito} further show that  all spectral gaps are ``open\" , which means that, for every $k\\in{\\mathbb Z}$,\n$$\\tilde\\Lambda_k:=\\left\\{E\\in{\\mathbb R}: \\tilde\\rho_{(\\alpha, \\, S_E^\\lambda)}=\\frac{k\\alpha}{2} \\mod {\\mathbb Z} \\right\\}$$\nhas positive measure. Indeed, the size of $\\tilde\\Lambda_k$ decays exponentially with respect to $|k|$, as was shown in \\cite{LYZZ}.\nHere, $\\tilde\\rho_{(\\alpha, \\, S_E^\\lambda)}$ is the fibered rotation number of cocycle $(\\alpha, \\, S_E^\\lambda)$.\nRecall that for any  $A:{\\mathbb T}^d \\to {\\rm SL}(2,{\\mathbb R})$ is continuous and homotopic to the identity,   \\textit{fibered rotation number} of\n$(\\alpha,A)$ is defined as\n\\begin{equation*}\n\\tilde\\rho(\\alpha,A)=\\int \\psi \\, d \\tilde{\\mu} \\mod {\\mathbb Z}\n\\end{equation*}\nwhere  $\\psi:{\\mathbb T}^{d+1} \\to {\\mathbb R}$ is  lift of $A$ such that\n$$\nA(x) \\cdot \\left (\\begin{matrix} \\cos 2 \\pi y \\\\ \\sin 2 \\pi y \\end{matrix} \\right )=u(x,y)\n\\left (\\begin{matrix} \\cos 2 \\pi (y+\\psi(x,y)) \\\\ \\sin 2 \\pi (y+\\psi(x,y)) \\end{matrix} \\right),\n$$\nand $\\tilde{\\mu}$ is\ninvariant probability  measure of $(x,y) \\mapsto (x+\\alpha,y+\\psi(x,y))$  (according to \\cite{Her}, it does not depend on the choices of $\\psi, \\tilde\\mu$).\n\n\nNote that $(\\alpha, \\, S_E^\\lambda)$ is a discrete dynamical system, however, with the help of Local Embedding Theorem (Theorem \\ref{localemb-sl}), we can embed the cocycle $(\\alpha, \\, S_E^\\lambda)$ into a quasi-periodic linear system $(\\omega, \\, B(E)+F(E,\\cdot))$. \nFor an individual cocycle, the Local Embedding Theorem was already shown in \\cite{YZ2013}.\nNevertheless, the crucial point here is that we really need a parameterized version of Local Embedding Theorem, that means the embedded system   $(\\omega, \\, B(E)+F(E,\\cdot))$ should have smooth dependence on $E$.\n\n\n\nTo show the parameterized version of Local Embedding Theorem, let us first introduce more notations.\n Given $f \\in C^2({\\mathcal I})$, define\n$$|f|_{*}= \\sum_\n{0\\leq m\\leq 2} \\sup_{E\\in{\\mathcal I}}|f^{(m)}|.$$\nFor any $ f(E,\\theta)=\\sum_{k\\in {\\mathbb Z}^d}\\widehat f_k(E)e^{2\\pi  {\\rm i}\n\\langle k,\\theta\\rangle}$ which is $C^2$ w.r.t. $E\\in{{\\mathcal I}}$, $C^\\omega$ w.r.t. $\\theta\\in{\\mathbb T}^d$, denote\n$$\\|f\\|_h:=\\sum_{k\\in {\\mathbb Z}^d}|\\widehat f_k(E)|_{*}e^{2\\pi |k|h}<\\infty,$$\nand we denote by  $C_h^\\omega( {\\mathcal I} \\times {\\mathbb T}^d,{\\mathbb C})$ all these functions with $\\|f\\|_h<\\infty$. Then our result is the following:\n\n\n\n\\begin{Theorem}\\label{localemb-sl}[Local Embedding Theorem]\nGiven $d\\geq 2$, $h>0$ and $G\\in C^\\omega_{h}({{\\mathcal I}} \\times {\\mathbb T}^{d-1}, {\\rm sl}(2,{\\mathbb R}))$, suppose that $\\mu\\in {\\mathbb T}^{d-1}$ such that $(1,\\mu)$ is rationally independent. Then, for any $\\nu\\in C^2({\\mathcal I})$ satisfying\n \\begin{equation}\\label{varition}\n \\sup_{E\\in{\\mathcal I}}|\\nu'(E)|\\cdot |{\\mathcal I}|< \\frac{1}{6},\n \\end{equation}\nthere exist $\\epsilon=\\epsilon(|\\nu|_{*},h,|\\mu|)>0,$\n$c=c(|\\nu|_*,h,|\\mu|)>0,$ and $F\\in\nC^\\omega_{\\frac{h}{1+|\\mu|}}(  {\\mathcal I} \\times {\\mathbb T}^d,{\\rm sl}(2,{\\mathbb R}))$ such that the cocycle $(\\mu,e^{2\\pi \\nu J}\ne^{G(\\cdot)})$ is the Poincar\\'e map of linear system\n\\begin{eqnarray}\\label{al-ref1}\n\\left(\\begin{array}{c}x\\\\ \\xi\n \\end{array}\n \\right)'=\\left(\\nu J+F(\\omega t)\\right)\\left(\\begin{array}{c}x\\\\ \\xi\n \\end{array}\n \\right) , \\qquad \\omega=(1,\\mu)\n\\end{eqnarray}\nprovided that $\\|G\\|_{h}<\\epsilon.$ Moreover, we have $\\|F\\|_{\\frac{h}{1+|\\mu|}}\\leq 2c \\|G\\|_{h}$.\n\\end{Theorem}\nWe postpone the proof of Theorem \\ref{localemb-sl} to Appendix \\ref{app_proof}.\n\n\\smallskip\n\nNow let us show how we can apply Theorem \\ref{localemb-sl} to finish the proof of Theorem \\ref{thm_AMO}. First note the constant matrix $e^{B}$ can be rewritten as\n\n$$e^{B}:=\\left(\\begin{array}{cc}\n             -E & -1 \\\\\n             1 & 0\n           \\end{array}\\right) = M\\left(\\begin{array}{cc}\n            \\cos(\\nu) & -\\sin(\\nu) \\\\\n             \\sin(\\nu) & \\cos(\\nu)\n           \\end{array}\\right)M^{-1} , $$\nwhere\n$$M:=\\frac{1}{\\sqrt{\\sin(\\nu)}}\\left(\\begin{array}{cc}\n             \\cos(\\nu) & -\\sin(\\nu) \\\\\n             1 & 0\n           \\end{array}\\right), $$\nrecalling that\n$$\\cos(\\nu(E))=-\\frac{E}{2},\\quad \\sin(\\nu(E))=\\frac{\\sqrt{4-E^2}}{2},\\qquad E\\in[-2+\\delta,\\ 2-\\delta].$$\nHence, by noting\n$$\\left(\\begin{array}{cc}\n            \\cos(\\nu) & -\\sin(\\nu) \\\\\n             \\sin(\\nu) & \\cos(\\nu)\n           \\end{array}\\right)=\\exp\\left\\{\\left(\\begin{array}{cc}\n            0 & -\\nu \\\\\n             \\nu & 0\n           \\end{array}\\right)\\right\\},$$\n          we see that $B$ can be written as\n$B=M  \\cdot ( \\nu J )  \\cdot  M^{-1}.$\n\n\n\n\\smallskip\n\nFor $\\nu(E)=\\cos^{-1}(-\\frac{E}{2})$, there exists ${{\\mathcal I}}\\subset [-2+\\delta, 2-\\delta]$ such that (\\ref{varition}) is satisfied.\nFor example, we can take ${{\\mathcal I}}= ]-\\frac{2}{\\sqrt{37}},\\frac{2}{\\sqrt{37}}[$.\nTherefore, according to Theorem \\ref{localemb-sl}, for $\\omega\\in (1,\\alpha)$, we have a quasi-periodic linear system $(\\omega, \\, B(E)+F(E,\\cdot))$ from  the quasi-periodic Schr\\\"odinger cocycle $(\\alpha, \\, S_E^\\lambda)$:\n\\begin{equation}\\label{Schrodinger_cocycle-conti}\n\\left(\\begin{array}{c}\n          x \\\\\n         \\xi\n        \\end{array}\n\\right)'=(B(E)+F(E,\\omega t)) \\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\n\\right),\n\\end{equation}\nThrough the change of variables\n$$\\left(\\begin{array}{c}\n          x \\\\\n          \\xi\n        \\end{array}\\right)=M\\left(\\begin{array}{c}\n          \\tilde x \\\\\n          \\tilde\\xi\n        \\end{array}\\right),$$\n$(\\omega, \\, B(E)+F(E,\\cdot))$ is conjugated to\n$$\\left(\\begin{array}{c}\n          \\tilde x \\\\\n         \\tilde\\xi\n        \\end{array}\n\\right)'=\\left(\\left(\\begin{array}{cc}\n            0 & -\\nu \\\\\n             \\nu & 0\n           \\end{array}\\right)+MF(E,\\omega t)M^{-1}\\right)\\left(\\begin{array}{c}\n          \\tilde x \\\\\n         \\tilde\\xi\n        \\end{array}\n\\right).$$\nThen by Theorem \\ref{thm_Schro} and \\ref{thm_Schro_sobolev},  Theorem \\ref{thm_AMO} is shown with\n$$\\left(\\begin{array}{cc}\n            b(E,\\cdot) & c(E,\\cdot) \\\\\n            -a(E,\\cdot) & -b(E,\\cdot)\n           \\end{array} \\right)=MF(E,\\cdot)M^{-1}.$$\n\nFinally we point out that $\\rho_{(\\omega, \\, B(E)+F(E,\\cdot))}=\\tilde\\rho_{(\\alpha, \\, S_E^\\lambda(\\cdot))}$, since  $(\\alpha, \\, S_E^\\lambda)$ is the Poincar\\'e map of linear system\n$(\\omega, \\, B(E)+F(E,\\cdot))$.  Let\n$$\\tilde\\Lambda_{(-p, k)}:=\\left\\{E\\in\\overline{{\\mathcal I}}:  \\rho_{(\\omega, \\, B(E)+F(E,\\cdot))} =   \\frac{ k\\alpha-p}{2} =    \\min_{j\\in {\\mathbb Z}}  \\left| \\frac{ k\\alpha}{2} -j\\right| \\right\\} ,$$\nthen by well-known result of Avila-Jitomirskaya  \\cite{AvilaJito},  ${\\rm Leb}(\\tilde\\Lambda_{(-p, k)})>0$, for every $k\\in{\\mathbb Z}$ such that  $\\tilde\\Lambda_k$ intersect with ${{\\mathcal I}}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\nIn a world in which many pressing global issues require large scale cooperation, understanding the group size effect on cooperative behavior is a topic of central importance. Yet, the nature of this effect remains largely unknown, with lab experiments insisting that it is either positive or negative or null, and field experiments suggesting that it is instead curvilinear. Here we shed light on this apparent contradiction by considering a novel class of public goods games inspired to the realistic scenario in which the natural output limits of the public good imply that the benefit of cooperation increases fast for early contributions and then decelerates. We report on a large lab experiment providing evidence that, in this case, group size has a curvilinear effect on cooperation, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups. In doing so, our findings help fill the gap between lab experiments and field experiments and suggest concrete ways to promote large scale cooperation among people.\n\n\n\n\n\\section*{Introduction}\n\nCooperation has played a fundamental role in the early evolution of our societies\\cite{KG,Tomasello14natural} and continues playing a major role still nowadays. From the individual level, where we cooperate with our romantic partner, friends, and co-workers in order to handle our individual problems, up to the global level where countries cooperate with other countries in order to handle global problems, our entire life is based on cooperation.\n\nGiven its importance, it is not surprising that cooperation has inspired an enormous amount of research across all biological and social sciences, spanning from theoretical accounts \\cite{Tr,Ax-Ha,FF03,nowak2006five,Perc10coevolutionary,press2012iterated,perc2013evolutionary,Ca,hilbe2013evolution,Ra-No,capraro2014translucent} to experimental studies \\cite{Andreoni1988why,Fischbacher2001people,milinski2002reputation,Frey2004social,Fischbacher2010social,traulsen2010human,apicella2012social,capraro2014heuristics,capraro2014benevolent,capraro2014good,hauser2014cooperating,gallo2015effects} and numerical simulations\\cite{Nowak92evolutionary,boyd2003evolution,santos2005scale,perc2008social,roca2009evolutionary,gardenes2012evolution,jiang2013spreading}.\n\nSince the resolution of many pressing global issues, such as global climate change and depletion of natural resources, requires cooperation among many actors, one of the most relevant questions about cooperation regards the effect of the size of the group on cooperative behavior. Indeed, since the influential work by Olson \\cite{olson1965logic}, scholars have recognized that the size of a group can have an effect on cooperative decision-making. However, the nature of this effect remains one of the most mysterious areas in the literature, with some scholars arguing that it is negative \\cite{olson1965logic,dawes1977behavior,komorita1982cooperative,baland1999ambiguous,ostrom2005understanding,grujic2012three,vilone2014partner,nosenzo2015cooperation}, others that it is positive \\cite{mcguire1974group,isaac1994group,haan2002free,agrawal2006explaining,masel2007bayesian,zhang2011group,szolnoki2011group}, and yet others that it is ambiguous \\cite{esteban2001collective,pecorino2008group,oliver1988paradox,chamberlin1974provision} or non-significant \\cite{todd1992collective,gautam2007group,rustagi2010conditional}. Interestingly, the majority of field experiments seem to agree on yet another possibility, that is, that group size has a curvilinear effect on cooperative behavior, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups \\cite{poteete2004heterogeneity,agrawal2001group,agrawal2000small,yang2013nonlinear,cinner2013looking}.\nThe emergence of a curvilinear effect of the group size on cooperation in real life situations is also supported by data concerning academic research, which in fact support the hypothesis that research quality of a research group is optimized for medium-sized groups \\cite{kenna2011critical,kenna2011critical2,kenna2012managing}.\n\nHere we aim at shedding light on this debate, by providing evidence that a single parameter can be responsible for all the different and apparently contradictory effects that have been reported in the literature. Specifically, we show that the effect of the size of the group on cooperative decision-making depends critically on a parameter taking into account different ways in which the notion of cooperation itself can be defined when there are more than two agents.\n\nIndeed, while in case of only two agents a cooperator can be simply defined as a person willing to pay a cost $c$ to give a greater benefit $b$ to the other person \\cite{nowak2006five}, the same definition, when transferred to situations where there are more than two agents, is subject to multiple interpretations. If cooperation, from the point of view of the cooperator, means paying a cost $c$ to create a benefit $b$, what does it mean from the point of view of the \\emph{other} player\\emph{s}? Does $b$ get earned by each of the other players or does it get shared among all other players, or none of them? In other words, what is the marginal return for cooperation?\n\nOf course, there is no general answer and, in fact, previous studies have considered different possibilities. For instance, in the standard Public Goods game it is assumed that $b$ gets earned by each player (including the cooperator); instead, in the N-person Prisoner's dilemma (as defined in \\cite{barcelo2015group}) it is assumed that $b$ gets shared among all players; yet, the Volunteer's dilemma \\cite{diekmann1985volunteer} and its variants using critical mass \\cite{szolnoki2010impact} rest somehow in between: one or more cooperators are needed to generate a benefit that gets earned by each player, but, after the critical mass is reached, new cooperators do not generate any more benefit; finally, it has been pointed out \\cite{marwell1993critical,heckathorn1996dynamics} that a number of realistic situations can be characterized by a marginal return which increases linearly for early contributions and then decelerates, reflecting the natural decrease of marginal returns that occurs when output limits are approached.\n\nIn order to take into account this variety of possibilities, we consider a class of \\emph{social dilemmas} parametrized by a function $\\beta=\\beta(\\Gamma,N)$ describing the marginal return for cooperation when $\\Gamma$ people cooperate in a group of size $N$. More precisely, our \\emph{general Public Goods game} is the N-person game in which N people have to simultaneously decide whether to cooperate (C) or defect (D). In presence of a total of $\\Gamma$ cooperators, the payoff of a cooperator is defined as $\\beta(\\Gamma,N)-c$ ($c>0$ represents the cost of cooperation) and the payoff of a defector is defined as $\\beta(\\Gamma,N)$. In order to have a social dilemma (i.e., a tension between individual benefit and the benefit of the group as a whole) we require that:\n\\begin{itemize}\n\\item Full cooperation pays more than full defection, that is, $\\beta(N,N) - c > \\beta(0,N)$, for all $N$; \n\\item Defecting is individually optimal, regardless of the number of cooperators, that is, for all $\\Gamma < N$, one has $\\beta(\\Gamma,N)-c < \\beta(\\Gamma-1,N)$.\n\\end{itemize}\n\nThe aim of this paper is to provide further evidence that the function $\\beta$ might be responsible for the confusion in the literature about group size effect on cooperation. In particular, we focus on the situation, inspired from realistic scenarios, in which the natural output limits of the public good imply that $\\beta(\\Gamma,N)$ increases fast for small $\\Gamma$'s and then stabilizes. \n\nIndeed, in our previous work \\cite{barcelo2015group}, we have shown that the size of the group has a positive effect on cooperation in the standard Public Goods game and has a negative effect on cooperation in the N-person Prisoner's dilemma. A reinterpretation of these results is that, if $\\beta(N,N)$ increases linearly with $N$ (standard Public Goods game), then the size of the group has a positive effect on cooperation; and, if $\\beta(N,N)$ is constant with $N$ (N-person Prisoner's dilemma), then the size of the group has a negative effect on cooperation. This reinterpretation suggests that, in the more realistic situations in which the benefit for full cooperation increases fast for early contributions and then decelerates once the output limits of the public good are approached, we may observe a curvilinear effect of the group size, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups.\n\nTo test this hypothesis, we have conducted a lab experiment using a general public goods game with a piecewise function $\\beta$, which increases linearly up to a certain number of cooperators, after which it remains constant. While it is likely that realistic scenarios would be better described by a smoother function, this is a good approximation of all those situations in which the natural output limits of a public good imply that the increase in the marginal return for cooperation tends to zero as the number of contributors grows very large. The upside of choosing a piecewise function $\\beta$ is that, in this way, we could present the instructions of the experiment in a very simple way, thus minimizing random noise due to participants not understanding the decision problem at hand (see Method).\n\nOur results support indeed the hypothesis of a curvilinear effect of the size of the group on cooperative decision-making. Taken together with our previous work \\cite{barcelo2015group}, our findings thus (i) shed light on the confusion regarding the group size effect on cooperation, by pointing out that different values of a single parameter might give rise to qualitatively different group size effects, including positive, negative, and even curvilinear; and (ii) they help fill the gap between lab experiments and field experiments. Indeed, while lab experiments use either the standard Public Goods game or the N-person Prisoner's dilemma, \\emph{real} public goods game are mostly characterized by a marginal return of cooperation that increases fast for early contributions and then approaches a constant function as the number of cooperators grows very large - and our results provide evidence that these three situations give rise to three different group size effects.\n\n\\section*{Method}\n\nWe have recruited participants through the online labour market Amazon Mechanical Turk (AMT) \\cite{paolacci2010running,horton2011online,mason2012conducting}. After entering their TurkID, participants were directed to the following instruction screen.\n\n\\emph{Welcome to this HIT.}\n \n\\emph{This HIT will take about 5 minutes and you will earn 20c for participating.} \n \n\\emph{This HIT consists of a decision problem followed by a few demographic questions.}  \n \n\\emph{You can earn an additional bonus depending on the decisions that you and the participants in your cohort will make.} \n\n\\emph{We will tell you the exact number of participants in your cohort later.} \n\n\\emph{Each one of you will have to decide to join either Group A or Group B.} \n \n\\emph{Your bonus depends on the group you decide to join and on the size of the two groups, A and B, as follows:}\n\\begin{itemize}\n\\item \\emph{If the size of Group A is 0 (that is, everybody chooses to join Group B), then everybody gets 10c}\n\\item \\emph{If the size of Group A is 1, then the person in Group A gets 5c and each person in Group B gets 15c}\n\\item \\emph{If the size of Group A is 2, then each person in Group A gets 10c and each person in Group B gets 20c}\n\\item \\emph{If the size of Group A is 3, then each person in Group A gets 15c and each person in Group B gets 25c}\n\\item \\emph{If the size of Group A is 4, then each person in Group A gets 20c and each person in Group B gets 30c}\n\\item \\emph{And so on, up to 10: If the size of Group A is 10, then each person in Group A gets 50c and each person in Group B gets 60c}\n\\item \\emph{However, if the size of Group A is larger than 10, then, independently of the size of the two groups, each person in group A will still get 50c and each person in group B will still get 60c.}\n\\end{itemize}\n\nAfter reading the instructions, participants were randomly assigned to one of 12 conditions, differing only on the size of the cohort ($N=3,5,10,15,20,25,30,40,50,60,80,100$). For instance, the decision screen for the participants in the condition where the size of the cohort is 3 was:\n\n\\emph{You are part of a cohort of 3 participants.}\n\n\\emph{Which group do you want to join?}\n\nBy using appropriate buttons, participants could select either Group A or Group B. \n\nWe opted for not asking any comprehension questions. We made this choice for two reasons. First, with the current design, it is impossible to ask general comprehension questions such as ``what is the strategy that benefits the group as a whole'', since this strategy depends on the strategy played by the other players. Second, we did not want to ask particular questions about the payoff structure since this may anchor the participants' reasoning on the examples presented. Of course, a downside of our choice is that we could not avoid random noise. However, as it will be discussed in the Results section, random noise cannot be responsible for our findings. Instead, our results would have been even cleaner, if we had not had random noise, since the initial increase of cooperation and its subsequent decline would have been more pronounced (see Results section for more details).\n\nAfter making their decisions, participants were asked to fill a standard demographic questionnaire (in which we asked for their age, gender, and level of education), after which they received the ``survey code'' needed to claim their payment. After collecting all the results, bonuses were computed and paid on top of the participation fee, that was \\$0.20. In case the number of participants in a particular condition was not divisible by the size of the cohort (it is virtually impossible, in AMT experiments, to decide the exact number of participants playing a particular condition), in order to compute the bonus of the remaining people we formed an additional cohort where these people where grouped with a random choice of people for which the bonus had been already computed.  Additionally, we anticipate that only 98 subjects participated in the condition with N=100. This does not generate deception in the computation of the bonuses since the payoff structure of the game does not depend on $N$ (as long as $N>10$). As a consequence of these observations, no deception was used in our experiment. \n\nAccording to the Dutch legislation, this is a non-WMO study, that is (i) it does not involve medical research and (ii) participants are not asked to follow rules of behavior. See http:\/\/www.ccmo.nl\/attachments\/files\/wmo- engelse-vertaling-29-7-2013-afkomstig-van-vws.pdf, Section 1, Article 1b, for an English translation of the Medical Research Act. Thus (see http:\/\/www.ccmo.nl\/en\/non-wmo- research) the only legislations which apply are the Agreement on Medical Treatment Act, from the Dutch Civil Code (Book 7, title 7, section 5), and the Personal Data Protection Act (a link to which can be found in the previous webpage). The current study conforms to both. In particular, anonymity was preserved because AMT ``requesters'' (i.e., the experimenters) have access only to the so-called TurkID of a participant, an anonymous ID that AMT assigns to a subject when he or she registers to AMT. Additionally, as demographic questions we only asked for age, gender, and level of education. \n\n\\section*{Results}\n\nA total of 1.195 \\emph{distinct} subjects located in the US participated in our experiment. \\emph{Distinct} subjects means that, in case two or more subjects were characterized by either the same TurkID or the same IP address, we kept only the first decision made by the corresponding participant and eliminated the rest. These multiple identities represent usually a minor problem in AMT experiments (only 2\\% of the participants in the current dataset). Participants were distributed across conditions as follows: 101 participants played with $N=3$, 99 with $N=5$, 102 with $N=10$, 101 with $N=15$, 98 with $N=20$, 103 with $N=25$, 97 with $N=30$, 99 with $N=40$, 97 with $N=50$, 101 with $N=60$, 99 with $N=80$, 98 with $N=100$.\n\nFig. 1 summarizes the main result. The rate of cooperation, that is the proportion of people opting for joining Group A, first increases as the size of the group increases from $N=3$ to $N=15$, then it starts decreasing. The figure suggests that the relation between the size of the group and the rate of cooperation is \\emph{not} quadratic: while the initial increase of cooperation is relatively fast, the subsequent decrease of cooperation seems extremely slow. This is confirmed by linear regression predicting rate of cooperation as a function of $N$ and $N^2$, which shows that neither the coefficient of $N$ nor that of $N^2$ are significant ($p=0.4692, p=0.2003$, resp.). For this reason we use a more flexible econometric model than the quadratic model, consisting of two linear regressions, one with a positive slope (for small $N$'s) and the other one with a negative slope (for large $N$'s). As a switching point, we use the $N=15$, corresponding to the size of the group which reached maximum cooperation. Doing so, we find that both the initial increase of cooperation and its subsequent decline are highly significant (from $N=3$ to $N=15$: coeff $= 0.0187553$, $p=0.00042$; from $N=15$ to $N=100$: coeff $= -0.00177618$, $p=0.00390$). \n\n\\begin{figure}\n   \\centering\n   \\includegraphics[scale=0.80]{Fig1.jpg} \n   \\label{fig:intermediate}\n   \\caption{\\emph{Proportion of cooperators (people choosing to join Group A) for each group size. Error bars represent the standard errors of the means. Group size has initially a positive effect on cooperation, which increases and reaches its maximum in groups of size 15, followed by a gradual decrease. Linear regression predicting cooperation using group size as independent variable confirms that both the initial increase of cooperation and its subsequent decline are highly significant (from $N=3$ to $N=15$: coeff $= 0.0187553$, $p=0.00042$; from $N=15$ to $N=100$: coeff $= -0.00177618$, $p=0.00390$).}}\n\\end{figure}\n\nWe conclude by observing that not only random noise cannot explain our results, but, without random noise, the effect would have been even stronger. Indeed, first we observe that there is no a priori worry that random noise would interact with any condition and so we can assume that it is randomly distributed across conditions. Then we observe that subtracting a binary distribution with average $0.5$ from a binary distribution with average $\\mu>0.5$, one would obtain a distribution with average $\\mu_0>\\mu$. Similarly, subtracting a binary distribution with average $0.5$ from a binary distribution with average $\\mu<0.5$ one would obtain a distribution with average $\\mu_0<\\mu$. Thus, if the $\\mu$'s are the averages that we have found (containing random noise) and the $\\mu_0$'s are the \\emph{true} averages (without random noise), the previous inequalities allow us to conclude that the initial increase of cooperation and its following decrease would have been stronger in absence of random noise.\n\n\\section*{Discussion}\n\nHere we have reported on a lab experiment providing evidence that the size of a group can have a curvilinear effect on cooperation in one-shot social dilemmas, with intermediate-size groups cooperating more than smaller groups and more than larger groups. Joining the current results with those of a previously published study of us \\cite{barcelo2015group}, we can conclude that group size can have qualitatively different effects on cooperation, ranging from positive, to negative and curvilinear, depending on the particular decision problem at hand. Interestingly, our findings suggest that different group size effects might be ultimately due to different values of a single parameter, the number $\\beta(N,N)$, describing the benefit for full cooperation. If $\\beta(N,N)$ is constant in $N$, then group size has a negative effect on cooperation; if $\\beta(N,N)$ increases linearly with $N$, then group size has a positive effect on cooperation; in the \\emph{middle}, all sorts of things may a priori happen. In particular, in the realistic situation in which $\\beta(N,N)$ is a piecewise function that increases linearly with $N$ up to a certain $N_0$ and then remains constant, then group size has a curvilinear effect, according to which intermediate-size groups cooperate more than smaller groups and more than larger groups. See Table 1.\n\n\\begin{center}\n\\begin{table}\n\\begin{tabular}{| l | c | c| }\n  \\hline                       \n  shape of $\\beta(N,N)$ & group size effect on cooperation & paper \\\\\n\\hline\n  linear & positive & Barcelo \\& Capraro (2015) \\\\\n  constant & negative & Barcelo \\& Capraro (2015) \\\\\n  linear-then-constant & curvilinear & this paper\\\\\n  \\hline  \n\\end{tabular}\n\\caption{Summary of the different group size effects on cooperation depending on how the benefit for full cooperation varies as a function of the group size.}\n\\end{table}\n\\end{center}\nTo the best of our knowledge, ours is the first study reporting a curvilinear effect of the group size on cooperation in an experiment conducted in the ideal setting of a lab, in which confounding factors are minimized. Previous studies reporting a qualitatively similar effect \\cite{poteete2004heterogeneity,agrawal2001group,agrawal2000small,yang2013nonlinear} used field experiments, in which it is difficult to isolate the effect of the group size from possibly confounding effects. In our case, the only possibly confounding factor is random noise due to a proportion of people that may have not understood the rules of the decision problem. As we have shown, our results cannot be driven by random noise and, in fact, the curvilinear effect would have been even stronger, without random noise. Moreover, since our experimental design was inspired by a tentative to mimic all those \\emph{real} public goods games in which the natural output limits of the public good imply that the increase of the marginal return for cooperation, when the number of cooperators diverges, tends to zero, our results might explain the apparent contradiction that field experiments tend to converge on the fact that the effect of the group size is curvilinear, while lab experiments tend to converge on either of the two linear effects.\n\nOur contribution is also conceptual, since we have provided evidence that a single parameter might be responsible for different group size effects: the parameter $\\beta(N,N)$, describing the way the benefit for full cooperation varies as a function of the size of the group. Of course, we do not pretend to say that this is the only ultimate explanation of why different group size effects have been reported in experimental studies. In particular, in real-life situations, which are typically repeated and in which communication among players is allowed, other factors, such as within-group enforcement, may favor the emergence of a curvilinear effect of the group size on cooperation, as highlighted in \\cite{yang2013nonlinear}. If anything, our results provide evidence that the curvilinear effect on cooperation goes beyond contingent factors and can be found also in the ideal setting of a lab experiment using one-shot anonymous games. We believe that this is a relevant contribution in light of possible applications of our work. Indeed, the difference between $\\beta(N,N)$ and the total cost of full cooperation $cN$ can be interpreted has the incentive that an institution needs to pay to the contributors in order to make them cooperate. Since institutions are interested in minimizing their costs and, at the same time, maximizing the number of cooperators, it is crucial to understand what is the ``lowest'' $\\beta$ such that the resulting effect of the group size on cooperation is positive. This seems to be an non-trivial question. For instance, does $\\beta(\\Gamma,N)=\\frac{\\Gamma}{N}\\log_2(N+1)$ give rise to a positive effect or is it still curvilinear or even negative? The technical difficulty here is that it is hard to design an experiment to test people's behavior in these situations, since one cannot expect that an average person would understand the rules of the game when presented using a logarithmic functions. \n\nIn terms of economic models, our results are consistent with utilitarian models such as the Charness \\& Rabin model \\cite{charness2002understanding} and the novel cooperative equilibrium model \\cite{Ca,capraro2013cooperative,barcelo2015group}. Both these models indeed predict that, in our experiment, cooperation initially (i.e., for $N\\leq10$) increases with $N$ (see \\cite{barcelo2015group} for the details), and then starts decreasing. This behavioral transition follows from the simple observation that free riding when there are more than 10 cooperators costs zero to each of the other players and benefits the free-rider. Thus, cooperation in larger groups is not supported by utilitarian models, which then predict a decrease in cooperative behavior whose speed depends on the particular parameters of the model, such as the extent to which people care about the group payoff versus their individual payoff, and people's beliefs about the behavior of the other players. Thus our results add to the growing body of literature showing that utilitarian models are qualitatively good descriptors of cooperative behavior in social dilemmas.\n\nHowever, we note that while theoretical models predict that the rate of cooperation should start decreasing at $N=10$, our results show that the rate of cooperation for $N=15$ is marginally significantly higher than the rate of cooperation for $N=10$ (Rank sum, $p=0.0588$). Although ours is a between-subjects experiment, this finding seems to hint at the fact that there is a proportion of subjects who would defect for $N=10$ and cooperate for $N=15$. This is not easy to explain: why should a subject cooperate with $N=15$ and defect with $N=10$? One possibility is that there is a proportion of ``inverse conditional cooperators'', who cooperate only if a small percentage of people cooperate: if these subjects believe that the rate of cooperation decreases quickly after $N=10$, they would be more motivated to cooperate for $N=15$ than for $N=10$. Another possibility, of course, is that this discrepancy is just a false positive. In any case, unfortunately our experiment is not powerful enough to detect the reason of this discrepancy between theoretical predictions and experimental results and thus we leave this interesting question for future research.\n\n\\section*{Acknowledgements}\n\nV.C. is supported by the Dutch Research Organization (NWO) Grant No. 612.001.352. This material is based upon work supported by the National Science Foundation under Grant No. 0932078000 while the first author was in residence at the Mathematical Science Research Institute in Berkeley, California, during the Spring 2015 semester.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n    In 1982, Feynman showed that a classical Turing machine would not be able to efficiently simulate quantum mechanical systems \\cite{feynman1982simulating}. Feynman went on to propose a model of computation based on quantum mechanics, which would not suffer the same limitations. Feynman's ideas were later refined by Deutsch who proposed a \\textit{universal quantum computer} \\cite{Deutsch97}. In this scheme, computation is performed by a series of \\textit{quantum gates}, which are the quantum analog to classical binary logic gates. A series of  gates is called a \\textit{quantum circuit} \\cite{nielsen-book}. Quantum gates act on \\textit{qubits} which is the quantum analog of a bit.\n    \\\\\n    \\\\\n    Lloyd later proved that a quantum computer would be able to simulate any quantum mechanical system efficiently \\cite{lloyd1996universal}. Equivalently, this can be stated as; given some special unitary operation \\( U \\in \\mathrm{SU}(2^n)\\), \\( U^{\\dagger} U = I\\), there exists some quantum circuit that  approximates \\(U\\), where \\(n\\) is the number of qubits.  One pertinent question that remains is how to find the circuit which implements this \\(U\\). In certain situations the circuit to implement \\(U\\) can be found exactly. However in general it is a difficult problem, and it is acceptable to approximate \\(U\\). Previously \\(U\\) has been found via expensive algebraic means \\cite{qcompiler,opt-qcompiler,cosine-sekigawa, Mottonen2004}.  Another novel attempt at finding an approximate \\(U\\) has been to use the tools of \\textit{Riemannian geometry}. \n    \\\\\n    \\\\\n    Nielsen originally proposed calculating special curves called \\textit{geodesics} between two points, \\(I\\) and \\(U\\) in \\(\\mathrm{SU}(2^n)\\). Geodesics are fixed points of the energy functional \\cite{wolfgang}. Nielsen claimed that when an energy minimising geodesic is discretised into a quantum circuit,  this would efficiently simulate \\(U\\) \\cite{nielsen-geom-1,nielsen-geom-2,nielsen-geom-3,nielsen-geom-4,nielsen-geom-5}. In practice however, finding the geodesics is a difficult task. Computing geodesics requires one to solve a boundary value problem in a high dimensional space. Furthermore, Nielsen originally formulated the problem on a Riemannian manifold equipped with a so called \\textit{penalty} metric, where the penalty was made large. This complicated solving the boundary value problem \\cite{brachistochrone}.\n    \\\\\n    \\\\\n    The Nielsen approach can be refined by considering subRiemannian geodesics. A subRiemannian geodesic is only allowed to evolve in directions from a \\textit{horizontal subspace} of the tangent space \\cite{montgomery}. This approach still involves solving a complicated boundary value problem. For a practical tool, a much faster methodology to synthesise a \\(U\\) is required. With recent advances in computing power, \\textit{neural networks} (NN) are an attractive option.\n    \\\\\n    \\\\\n    The problem is to find \\(U\\) approximately as a product of exponentials\n    \\begin{align} U \\approx \\,  &\\mathbf{E}(c) = \\exp(c^1_1 \\tau_{1}) \\dots \\exp( c^1_m \\tau_m ) \\nonumber \\\\\n                            &\\dots \\exp( c^N_1 \\tau_1 ) \\dots  \\exp( c^N_m \\tau_{m}), \\label{eqn:U} \\end{align}\n    where \\( \\mathbf{E} \\) we call the \\textit{embedding} function, \\( c = (c_1^1 ,\\dots, c^N_m )\\) and the \\(\\tau_i\\) are a basis for a \\textit{bracket generating} subset of the Lie algebra \\( \\Delta \\subset  \\mathfrak{su}(2^n)\\) of dimension \\(m\\). Bracket generating means that repeated Lie brackets of terms in \\(\\Delta \\) can generate any term in \\( \\mathfrak{su}(2^n)\\). Because products of matrix exponentials generate Lie bracket terms\n    \\[ \\exp(A) \\exp(B) = \\exp(A+ B+ \\frac{1}{2}[A,B] + \\dots ),\\]\nany \\(U \\in \\mathrm{SU}(2^n) \\) can be written as Equation (\\ref{eqn:U}) with sufficiently many products. We restrict ourselves to \\(U\\) which can be written as a product of a polynomial in \\(n\\) terms . An example of such a \\(\\Delta\\) could be the matrix logarithms of universal gates. For convenience it is easier to work with all permutations of Kronecker products of one and two Pauli matrices, so \n    \\[ \\Delta = \\mathrm{span} \\{\\frac{\\mathrm{i}}{\\sqrt{2^n}} \\sigma_i^j , \\frac{\\mathrm{i}}{\\sqrt{2^n}} \\sigma_i^k \\sigma_j^l \\} ,\\]\n    where \\( \\sigma^j_i \\) represents the \\(N\\) fold Kronecker product, \\( I \\otimes \\dots \\otimes \\sigma_i \\otimes \\dots \\otimes I\\), with a \\( \\sigma_i \\) inserted in the \\(j\\)-th slot and \\(I\\) representing the \\( 2 \\times 2 \\) identity matrix. Exponentials of these basis elements have very simple circuits, for more detail see Appendix A.\n    \\\\\n    \\\\\n    We propose that a neural network be trained to learn \\( \\mathbf{E}^{-1}\\). The neural network will try to find all the coefficients \\( c^k_{i} \\) so the product approximates \\(U\\). In this approach, the neural network takes a unitary matrix \\(U\\) as an input and returns a list \\(c\\) of \\( c^k_{i}\\). A segment is a product of \\( m \\) exponentials of each basis element. In total there are \\(N\\) segments. We only examine \\(U\\) which are implementable in a reasonable number of segments. We found that we required two neural networks to achieve this. The first is a Gated Recurrent Unit, GRU, network \\cite{gru_paper_1,gru_paper_2} which factors a \\(U\\) into a product of \\(U_j\\),\n    \\[U \\approx U_1 U_{2} \\dots U_j \\dots U_N, \\]\n    where each \\(U_j\\) is implementable in polynomially many gates, which we call \\textit{global decomposition}.\n    The second is simply several dense fully connected layers, which decomposes the \\(U_j\\) into products of exponentials\n    \\[ U_j \\approx \\exp( c^j_1 \\tau_1 ) \\dots \\exp( c^j_m \\tau_m ), \\]\n    which we term \\textit{local decomposition}. These procedures can be done with traditional optimisation methods. The lack of a good initial guess meant that it took an order of an hour in \\(\\mathrm{SU}(8)\\). While the output from the neural network may not implement \\(U\\) to a required tolerance, it does provide a good initial guess as the error will be small. The output from the neural network could be refined with another optimisation algorithm.\n\\section{Training data \\label{training_data}}\n    \\noindent To generate the training data, the \\(c\\) should not be chosen randomly. If there is no structure to how \\(c\\) is chosen, it will introduce extra redundancy. More seriously, \\(\\mathbf{E}^{-1} \\) will not be well defined. There are infinitely many ways to factor a \\(U\\) into some unordered product of matrix exponentials. Geometrically this could be visualised as taking any path from \\(I\\) to \\(U\\) on \\(\\mathrm{SU}(2^n)\\). Randomly generating data may give two different decompositions for a \\(U\\), and so \\(\\mathbf{E}\\) is not one to one. To ensure the training data is unique, we propose that these paths should be chosen to be, at least approximately, minimal normal subRiemannian geodesics. \n    \\\\\n    \\\\\n    The choice of using geodesics is not particularly special. Other types of curves could be used as long as it uniquely joins \\(I\\) and \\(U\\). This is so \\( \\mathbf{E}^{-1} \\) is well defined. Generating random geodesics can be done simply by generating random initial conditions. However the geodesics must also be minimal. The first way to try and ensure they are minimal is to bound the norms of the initial conditions. \n    \\\\\n    \\\\\n    The normal subRiemannian geodesics in \\(\\mathrm{SU}(2^n)\\) can be found via the Pontryagin Maximum Principle \\cite{pmp-intro,pmp-book} by minimising the energy functional\n    \\[ \\mathcal{E}[x] = \\int_0^1 dt \\langle \\dot{x}, \\dot{x} \\rangle, \\]\n    where \\( \\langle, \\rangle \\) is the restriction of the bi-invariant norm to \\(\\Delta \\subset \\mathfrak{su}(2^n)\\), and \\( x :[0,1] \\rightarrow \\mathrm{SU}(2^n) \\). See Chapter 7 of \\cite{opt-control-lie-group} for a review. The normal subRiemannian geodesic equations can be written as\n    \\begin{align*}\n        \\dot{x} &= u x, \\\\\n        \\dot{\\Lambda} &= [\\Lambda, u ],\\\\\n        u &= \\mathrm{proj}_{\\Delta}( \\Lambda),\n    \\end{align*}\n    where  \\( \\Lambda : [0,1] \\rightarrow \\mathfrak{su}(2^n) \\),  \\( u : [ 0,1] \\rightarrow \\Delta \\subset \\mathfrak{su}(2^n) \\) and \\( \\mathrm{proj}_{\\Delta} \\) is projection onto \\( \\Delta\\).\n    This can be re-written as the single equation\n    \\begin{equation} \\dot{x} = \\mathrm{proj}_{\\Delta}( x \\Lambda_0 x^{\\dagger} ) x, \\label{eqn:geod} \\end{equation}\n    where \\( \\Lambda_0 = \\Lambda(0) \\).\n    Choosing the \\( \\Lambda_0 \\) completely determines the geodesic. To generate the training data for the \\(U_j\\), first randomly choose a \\( \\Lambda_0\\).  The \\(U_j \\) are then matrices which forward solve the geodesic equations\n    \\[ x(t_{j+1} ) = U_j x(t_j),\\]\n    where \\( [0,1] \\) has been divided into \\(N\\) segments of width \\(h\\). For this paper we utilised the simple first order integrator\n    \\[ U_j = \\exp\\big( h \\, \\mathrm{proj}_{\\Delta}(x_j \\Lambda_0 x_j^{\\dagger}) \\big),  \\]\n    since approximating the geodesic is sufficient. There are infinitely many bi-invariant Riemannian geodesics joining \\(I \\) and \\(U\\), for the different branches of \\( \\log(U) \\). SubRiemannian geodesics are similarly behaved, but it varies on the norm of \\( \\Lambda_0\\). To generate the training data we bounded the norms by \\(  \\mathrm{dim}(\\Delta) = \\mathcal{O}(n^2) \\), to try and ensure the geodesics are unique. \n    \\\\\n    \\\\\n    Further, the norm \\( || \\mathrm{proj}_{\\Delta} (\\Lambda_0 ) || = || u_0 ||\\) determines the distance between \\(I \\) and a \\(U\\). Nielsen showed that the distance can be thought of as approximately the complexity to implement \\(U\\). Lemma (3) in \\cite{nielsen-geom-1} shows that a \\(U\\) further away from \\(I\\) requires more gates. The distance however is likely to scale exponentially. By bounding the norm by a polynomial, this ensures the training data only contains \\(U\\) which are reachable with a polynomial number of quantum gates.    \n\\section{Network Design - SU(8) }    \n    \\subsection{Global decomposition}\n    The neural network for the global decomposition takes an input of \\(U\\) and returns a list of \\(U_j\\). To do this \\(U\\) is decomposed into rows of length \\(2^n\\). This makes \\( 2^n\\) real vectors. Each row is treated as a single timestep in the GRU layer. The output \\(U_j\\) are also decomposed into their rows and these rows are treated as timesteps in the output. This gives \\( 2^n N \\) output vectors of length \\( 2^n\\). In particular we examined the \\(n=3\\) qubit case. For \\(\\mathrm{SU}(8) \\) we found \\(10\\) stacked GRU layers was sufficient to give reasonable results.  In \\(\\mathrm{SU}(8)\\) we chose \\(N=10\\) , so there were \\( 8 \\) input vectors of length \\( 8\\) and \\( 80 \\) output vectors of length \\( 8\\). The network was implemented in the Keras Python library \\cite{keras} with the TensorFlow backend, on a Nvidia GTX 1080.\n    \\subsection{Local decomposition}\n    For \\(\\mathrm{SU}(8)\\) a network with \\(2\\) fully connected dense hidden layers of \\(2000\\) neurons, with the ReLU activation function was found to be sufficient. The input layer took a vectorised \\(U_j\\), and outputted \\( \\dim(\\Delta)\\) values. The network was implemented in the Keras Python library with the TensorFlow backend, on a Nvidia GTX 1080.\n\\section{Results - SU(8)}\n   \\subsection{Global decomposition}\n    The global decomposition network was trained on \\(U_j\\) taken from \\(5000\\) randomly generated geodesics in \\(\\mathrm{SU}(8)\\). \\(500\\) were used for validation data. The loss function used was the standard Euclidean distance between the output vector and the desired output. After \\( 1500 \\) training epochs the validation loss reached \\( \\sim 0.9 \\) and did not decrease. This was found to be sufficient to generate \\(U_j\\) close to the training data. Figure (\\ref{fig:gruLoss}) shows the validation and training loss. Figure (\\ref{fig:outUi}) and figure (\\ref{fig:validUi}) shows a randomly chosen \\( U_j\\) from a list of \\(U_j\\) generated by the network, and from the training data respectively for some random \\(U\\). Most \\(U_j\\) appeared to be very similar. Figure (\\ref{fig:u34}) and figure (\\ref{fig:u25}) show the same entry in consecutive \\( U_i\\) for validation data. Again the network was able to output values very close to the values in the validation dataset. This similarity was typical. This shows the network is able to reasonably approximate the \\(U_j\\). \n    \\begin{figure}[h!]\n        \\centering\n        \\includegraphics[width=0.45\\textwidth]{GRU_loss.pdf}\n        \\caption{The loss and validation loss from training the global decomposition.}\n        \\label{fig:gruLoss}\n    \\end{figure}\n    \\begin{figure}[h!]\n        \\centering\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.4\\linewidth]{out.pdf}\n            \\caption{Real components of a \\(U_j\\) generated by the NN.}\n            \\label{fig:outUi}\n        \\end{subfigure}\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.4\\linewidth]{valid.pdf}\n            \\caption{The respective known real components of a \\(U_j\\) from the validation dataset.}\n            \\label{fig:validUi}\n        \\end{subfigure}\n        \\caption{A known \\(U_j\\) from the validation data and the \\(U_j\\) generated by the NN in \\(\\mathrm{SU}(8)\\) for global decomposition. Each \\(U_j\\) is close to the identity matrix. The shading from blue to orange represents \\( [-1,1] \\)}\n    \\end{figure}\n    \\begin{figure}[h!]\n        \\centering\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.6\\linewidth]{u14_25.pdf}\n            \\caption{The same real entry from the \\(10\\) \\(U_i\\) from the validation data set (blue), vs the predicted output (red).}\n            \\label{fig:u34}\n        \\end{subfigure}\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.6\\linewidth]{u34_40.pdf}\n            \\caption{The same real entry from the \\(10\\) \\(U_i\\) from the validation data set (blue), vs the predicted output (red). }\n            \\label{fig:u25}\n        \\end{subfigure}\n        \\caption{Real entries of validation \\(U_i\\) vs the \\(U_i\\) generated by the NN. Recall the \\(U_i\\) are not constant, and solve equation (\\ref{eqn:geod}). The behaviour displayed here was typical in other entries. }\n    \\end{figure}\n\n\n\\subsection{Local decomposition}\nThe network to implement the local decomposition was trained on \\(U_j\\) generated by choosing a random \\(m\\)-vector of the coefficients \\(c^j_i\\), where each \\(c^j_i \\) was order \\( 1\/N \\). In total there were \\(5000\\) pairs in the training set, and \\(500\\) in the validation set. Figure (\\ref{fig:denseLoss}) shows the validation and training loss. After \\( 500\\) epochs the network was able to sufficiently compute the local decomposition to reasonable error (on average 0.16).  Figures (\\ref{fig:localoutUi}) and (\\ref{fig:localvalidUi}) show a matrix generated by the neueral network and the target matrix.\n    \\begin{figure}[h!]\n        \\centering\n        \\includegraphics[width=0.45\\textwidth]{Dense_loss.pdf}\n        \\caption{The loss and the validation loss from training the local decomposition. There was no significant improvement after \\(500\\) epochs.}\n        \\label{fig:denseLoss}\n    \\end{figure}\n    \\begin{figure}[h!]\n        \\centering\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.4\\linewidth]{outMat.pdf}\n            \\caption{Real components of a \\(U_j\\) generated by the NN.}\n            \\label{fig:localoutUi}\n        \\end{subfigure}\n        \\begin{subfigure}{0.48\\textwidth}\n            \\centering\n            \\includegraphics[width=0.4\\linewidth]{localValidM.pdf}\n            \\caption{The respective known real components of a \\(U_j\\) from the validation dataset.}\n            \\label{fig:localvalidUi}\n        \\end{subfigure}\n        \\caption{A known \\(U_j\\) from the validation data and the \\(U_j\\) generated by the NN in \\(\\mathrm{SU}(8)\\). These figures are for the local decomposition network. The shading from blue to orange represents \\( [-1,1] \\)}\n    \\end{figure}\n\n\\section{Conclusion}\nTraining two neural networks to together decompose \\(U\\) into \\(c^j_i \\) via a two-step approach (global decomposition followed by local decomposition) was found to be successful, when restricting the set of training data generated to paths which approximate minimal normal subRiemannian geodesics.  This restriction limited the training data pairs to ones which were one-to-one, eliminating redundancy.  For the global decomposition, using a neural network consisting of stacked GRU layers allowed for efficient training of the network, with the validation loss of the network approaching its minimum at 500 epochs for \\(\\mathrm{SU}(8)\\).  A simple dense network with two hidden layers proved sufficient for the local decomposition.  In \\(\\mathrm{SU}(8)\\), the networks were small enough that both networks were able to be trained on a desktop machine with a single NVidia GTX 1080 GPU.  The two stage decomposition proved more successful than single-stage attempts to form a solution, with the decomposition of a given \\(U\\) into \\(U_j\\) being crucial for this increase in effectiveness.  This approach to the solution for this problem demonstrates a novel use of neural networks.\n\\\\\n\\\\\nAlthough this approach works well for systems with small numbers of qubits (such as the \\(\\mathrm{SU}(8)\\) case used as an example), the approach does not scale well with increasing number of qubits. This is because the size of the network scales by the number of entries in matrices in \\(\\mathrm{SU}(2^n)\\). Although this is not a significant problem for currently realisable quantum computers, or those in the near future, it will increasingly become problematic as quantum computing continues to advance.  To somewhat counteract this, the complexity of the problem can be decreased by restricting the set of \\(U\\) on which the neural network is trained. For example if the \\(U\\) are sparse, some savings in the size of the network may be made. Investigating this will be increasingly significant, as it will increase the practical usefulness of this approach.  \n\\\\\n\\\\\nAs noted in section \\ref{training_data}, the choice of using geodesics to restrict the training data is fairly arbitrary, and as such, there may be different ways of restricting the training data which, while still ensuring the input\/output is one-to-one, may produce a better dataset, improving the accuracy of the networks. This is heavily related to the nature of \\( \\Lambda_0 \\) which is currently not fully understood. Exploring this problem is a possible future avenue of investigation, which may improve the effectiveness of the approach described in this paper.  \n\\\\\n\\\\\nFinally note that training the network is the most computationally expensive part of this approach. Once the network is trained, propagating an input through through the network is much more efficient than the conventional optimisation techniques for compiling \\(U\\). \n\\\\\n\\\\\nAll data and programs used to produce this work can be found at \\href{https:\/\/github.com\/Swaddle\/nnQcompiler}{\\url{https:\/\/github.com\/Swaddle\/nnQcompiler}}. This work was supported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia \\footnote{\\url{https:\/\/www.pawsey.org.au\/}}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}