diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzkguf" "b/data_all_eng_slimpj/shuffled/split2/finalzzkguf" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzkguf" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\t\tLet $\\mathbb{D}$ be the open unit disc, and let $\\mathbb{T} \\colonequals \\partial \\mathbb{D}$ be the unit circle. The Hardy space $H^2$ consists of analytic functions $f$ on $\\mathbb{D}$ such that \n\t\t\\[\n\t\t\\| f\\| _{H^2} \\colonequals\\left(\\sup_{01$, if and only if $g$ is in the Besov space $B_p$ given by\n\t\t\\[\n\t\tB_p=\\left \\{ g \\in H^2:\\ \\displaystyle \\int _\\mathbb{D}\\left ( ( 1-|z|^2)|g'(z)| \\right )^p\\frac{dm(z)}{(1-|z|^2)^2} \\right \\},\n\t\t\\]\n\t\twhere $m$ denotes the normalized Lebesgue measure on $\\mathbb{D}$. They also proved that if ${I_g} \\in \\mathcal{S}_1$ then $g'=0$.\\\\\n\t\t\n\t\t\n\t\tThe weighted Bergman space ${\\mathcal{A}}^2_\\alpha$, with $\\alpha>-1$, is the space of analytic functions $f$ on $\\mathbb{D}$ for which\n\t\t\\[\n\t\t\\|f\\|_\\alpha \\colonequals \\left((\\alpha+1)\\displaystyle \\int_\\mathbb{D} |f(z)|^2(1-|z|^2)^\\alpha dm(z)\\right)^{1\/2} <\\infty.\n\t\t\\]\n\t\tIn \\cite{AS2}, Aleman and Siskakis considered the operator $I_g$ acting on weighted Bergman spaces. They proved that $I_g$ is bounded on $\\mathcal{A}^2_\\alpha$ (respectively compact) if and only if $g$ is in the Bloch space $\\mathcal{B}$ (respectively in the little Bloch space $\\mathcal{B}_0$), and that $I_g$ is in the class $\\mathcal{S}_p(\\mathcal{A}^2_\\alpha)$, for $p > 1$, if and only if $g$ is in the space $B_p$. For further information about \n\t\tthe spaces used in this paper, we refer the reader to the monograph \\cite{Zhu}.\\\\\n\t\t\n\t\tThis paper is devoted to the study of the asymptotic behavior of the singular values of compact integration operators acting on $H^2$ or on ${\\mathcal{A}}^2_\\alpha$. \n\t\n\t\tBefore stating our main results, we give some basic notations.\n\t\t\\\\\n\t\t\\\\\n\t\tThe Carleson window (or box) associated with an arc $I\\subset \\mathbb{T}$ is defined by\n\t\t\\[\n\t\tW(I)=\\left\\lbrace z\\in\\mathbb{D}:1-|z|\\leqslant | I | \/2\\pi,\\ z\/|z|\\in I\\right\\rbrace,\n\t\t\\]\n\t\twhere $|I|$ denotes the length of $I$. \n\t\tLet $R(I)$ denote the inner half of $W(I)$ which is given by \n\t\t\\[\n\t\tR(I)= \\left\\{ z\\in \\mathbb{D}:\\ |I|\/4\\pi <1-|z|\\leqslant | I | \/2\\pi ,z\/|z| \\in I \\right\\}.\n\t\t\\]\n\t\tThe family $\\left\\{ R(I_{n,k})\\right\\} _{n,k}$ constitutes a pairwise disjoint covering of $\\mathbb{D}$, where\n\t\t\\[\n\t\tI_{n,k} \\colonequals \\lbrace e^{i\\theta}:2\\pi k\/2^n\\leqslant \\theta <2\\pi(k+1)\/2^n\\rbrace,\\quad n\\geqslant 1\\ \\mbox{and}\\ 1\\leqslant k\\leqslant 2^n,\n\t\t\\]\n\t\tare the dyadic arcs.\\\\ \\\\\n\t\tThe {Bloch} space $\\mathcal{B}$ (respectively the little {Bloch} space $\\mathcal{B}_0$) consists of analytic functions $g$ on $\\mathbb{D}$ such that\n\t\n\t\t\\[\n\t\t\\displaystyle \\sup _{z\\in \\mathbb{D}}(1-|z|^2)|g'(z)| <\\infty \\quad (\\text{respectively } \\displaystyle \\lim _{|z| \\to 1}(1-|z|^2)|g'(z)|=0).\n\t\t\\]\n\t\n\t\n\t\n\t\n\t\n\t\t{Let $dm_g(z) = |g'(z)|^2 (1-|z|^2)^2dm(z)$}. It is established in \\cite{AS2} that $g$ is in the space $\\mathcal{B}$ if and only if\n\t\t\\[ \n\t\t\\frac{m_g(R(I_{n,k}))}{{m(R(I_{n,k}))}}=O(1),\n\t\t\\]\n\t\tand that $g$ is in the space $\\mathcal{B}_0$ if and only if \n\t\t\\begin{equation}\\label{00}\n\t\t\\displaystyle \\lim_{n\\to \\infty} \\sup _{1\\leqslant k \\leqslant 2^n}\\frac{m_g(R(I_{n,k}))}{{m(R(I_{n,k}))}}=0.\n\t\t\\end{equation}\n\t\tHence, $I_g$ is compact on ${\\mathcal{A}}^2_\\alpha$ if and only if $g$ satisfies assumption (\\ref{00}). In this case, {let $\\left(I_{n}\\right)_{n\\geqslant 1}$ be an enumeration of $\\left(I_{n,k}\\right)_{n,k}$ so that the sequence \n\t\t\t\\[a_n(m_g) \\colonequals \\frac{m_g(R(I_{n}))}{{m(R(I_{n}))}}\n\t\t\t\\] \n\t\t\tis nonincreasing.}\n\t\t\\\\\n\t\t\\\\\n\t\tThroughout this paper, the notation $A\\lesssim B$ stands for $A\\leqslant cB$ with $c$ is a certain positive constant, and the notation $A\\asymp B$ is used if both $A\\lesssim B$ and $B\\lesssim A$ hold.\n\t\t\\\\\n\t\t\n\t\tFirst, we state the main result of this paper in the case of Bergman spaces.\n\t\n\t\t\\begin{theo}\\label{Bergman}\n\t\t\tLet $g\\in \\mathcal{B}_0$. There exists an absolute integer $p$ such that\n\t\t\t\\begin{eqnarray}\n\t\t\t\\sqrt{a_{pn}(m_g)} \\lesssim s_{n}(I_g)\\lesssim \\sqrt{a_{[\\frac{n}{p}]}(m_g)},\\quad n\\geqslant 1,\n\t\t\t\\end{eqnarray}\n\t\t\twhere $[\\frac{n}{p}]$ is the integer part of $\\frac{n}{p}$.\n\t\t\\end{theo}\n\t\t\n\t\tAn immediate consequence of Theorem \\ref{Bergman} is given as follows.\n\t\n\t\t\\begin{cor}\\label{Cor_Bergman}\n\t\t\tLet $g\\in \\mathcal{B}_0$. If $\\eta=(\\eta(n))_n$ is a nonincreasing sequence of positive numbers such that $\\eta(2n)\\asymp \\eta(n)$, then\n\t\t\t\\[\n\t\t\ts_n^2(I_g) \\asymp \\eta(n) \\Longleftrightarrow a_n(m_g) \\asymp \\eta(n).\n\t\t\t\\]\n\t\t\\end{cor}\n\t\t\n\t\tThe situation in the Hardy space is more subtle. Indeed, the boundedness of $I_g$ on $H^2$ is equivalent to the membership of the symbol $g$ to $ BMOA$, that is,\n\t\t\\[\n\t\t\\| g\\|_{BMOA} \\colonequals |g(0)| + \\sup \\{\\| g\\circ\\varphi _w\\|_{H^2}:\\ w\\in \\mathbb{D}\\}<\\infty,\n\t\t\\]\n\t\twhere $\\varphi _w(z) {\\colonequals} \\frac{w-z}{1-\\bar{w}z}$ is the M\u00f6bius automorphism of the unit disc. Recall also that $I_g$ is compact on $H^2$ if and only if $g\\in VMOA$, that is,\n\t\t\\[\n\t\t\\displaystyle \\lim_{|w| \\to 1^-}\\| g\\circ\\varphi _w\\|_{H^2}=0.\n\t\t\\]\n\t\tThe membership of $g$ to $BMOA$ or to $VMOA$ can be expressed using Carleson windows and the measure $d\\nu_g(z) \\colonequals |g'(z)|^2 (1-|z|^2)dm(z)$. Namely, it is proved in \\cite{AS1} that\n\t\t\\[\n\t\tg \\in BMOA\\quad \\iff \\frac{\\nu_g(W(I_{n,k}))}{ |I_{n,k} |}=O(1)\n\t\t\\]\n\t\tand \n\t\t\\[\n\t\tg \\in VMOA \\quad \\iff \\displaystyle \\lim_{n\\to \\infty} \\sup _{1\\leqslant k \\leqslant 2^n}\\frac{\\nu_g(W(I_{n,k}))}{| I_{n,k}|}=0.\n\t\t\\]\n\t\t\n\t\n\t\n\t\t\\noindent\tFor $g \\in VMOA$, denote by \n\t\t{\\[\n\t\t\t(\\tau _n(\\nu_g))_{n\\geqslant 1}\\colonequals\\left (\\frac{\\nu_g(W(I_{n}))}{{| I_n| }} \\right )_{n\\geqslant 1}\n\t\t\t\\]\n\t\t\tthe nonincreasing rearrangement of the sequence $\\left (\\frac{\\nu_g(W(I_{n,k}))}{{| I_{n,k}| }} \\right )_{n,k}$.}\n\t\n\t\t\\\\\n\t\t\n\t\tThe main result in the case of the Hardy space is the following theorem.\t\n\t\t\n\t\n\t\t\n\t\t\\begin{theo}\\label{Hardy}\n\t\t\tLet $g\\in VMOA$. Then\n\t\t\t\n\t\t\t\\begin{enumerate}[label=$(\\roman*)$,leftmargin=* ,parsep=0cm,itemsep=0cm,topsep=0cm]\n\t\t\t\t\\item\\label{i} There exists an absolute integer $p\\geqslant 1$ such that \n\t\t\t\t\\[\n\t\t\t\ts_{p n}(I_g) \\lesssim \\sqrt{\\tau_n (\\nu_g)}, \\quad n\\geqslant 1.\n\t\t\t\t\\]\n\t\t\t\t\\item \\label{ii} There exists an absolute constant $B>0$ such that\n\t\t\t\t\\[\n\t\t\t\t\\frac{1}{B} \\displaystyle \\sum _{j=1}^n \n\t\t\t\t{\\tau_j(\\nu_g) } \\leqslant \\displaystyle \\sum _{j=1}^n s_j^2(I_g),\\quad n\\geqslant 1.\n\t\t\t\t\\]\n\t\t\t\\end{enumerate}\n\t\t\\end{theo}\n\t\t\n\t\tUnder some regularity conditions, we obtain a description of the asymptotic behavior of the singular values of $I_g$.\n\t\n\t\t\\\\ \\\\\n\t\t\\noindent Let $\\gamma >0$, a positive nonincreasing sequence $(u_n)_n$ is said to belong to ${\\mathcal{R}}_\\gamma$ if there exists a nonincreasing sequence $(x_n)_n$ such that $n^\\gamma x_n$ increases to infinity and $u_n \\asymp x_n$. If, in addition, there exists $\\alpha \\in (0,\\gamma)$ such that $n^\\alpha x_n$ decreases, $(u_n)_n$ is said to belong to ${\\mathcal{R}} _{\\gamma, \\alpha}$.\n\t\n\t\t\n\t\t\\begin{theo}\\label{B}\n\t\t\tLet $g\\in VMOA$. The following statements hold. \\\\\n\t\t\t\\begin{enumerate}[label=$(\\roman*)$,leftmargin=* ,parsep=0cm,itemsep=0cm,topsep=0cm]\n\t\t\t\t\\item \\label{2i} Let $\\gamma \\in (0,1\/2)$. If {$(s_n(I_g))_n \\in {\\mathcal{R}}_\\gamma$ or $(\\tau _n(\\nu_g))_n \\in {\\mathcal{R}}_{2\\gamma}$}. Then\n\t\t\t\t\\[\n\t\t\t\ts_n(I_g) \\asymp\\sqrt{ \\tau _n(\\nu_g)}.\n\t\t\t\t\\]\n\t\t\t\t\\item\\label{3i} Let $\\gamma \\in (0,1)$ and let $\\alpha \\in (0,\\gamma)$. If {$(s_n(I_g))_n \\in {\\mathcal{R}}_{\\gamma,\\alpha}$ or $(\\tau _n(\\nu_g))_n \\in {\\mathcal{R}}_{2\\gamma,2\\alpha}$}. Then\n\t\t\t\t\\[\n\t\t\t\ts_n(I_g) \\asymp\\sqrt{ \\tau _n(\\nu_g)}.\n\t\t\t\t\\]\n\t\t\t\\end{enumerate}\n\t\t\\end{theo}\n\t\t\\noindent The following corollary is a direct consequence of Theorem \\ref{B}.\n\t\t\\begin{cor} Let $g \\in VMOA$. Let $\\alpha \\in [0,1[$ and $\\beta \\geqslant 0$ such that $ \\alpha >0$ or $\\beta >0$ . We have\n\t\t\t\\[\n\t\t\ts_n(I_g) \\asymp \\frac{1}{n^\\alpha \\log^\\beta (n+1)} \\quad \\iff \\tau_n(\\nu_g) \\asymp \\frac{1}{n^{2\\alpha} \\log^{2\\beta} (n+1)}.\n\t\t\t\\]\n\t\t\\end{cor}\n\t\t\n\t\n\t\t\n\t\n\t\t\n\t\n\t\tThe paper is organized as follows. In Section \\ref{sec2}, we prove Theorem \\ref{Bergman}. Section \\ref{sec3} is devoted to the proofs of the main results in the Hardy space.\n\t\n\t\t\\section{Bergman spaces}\n\t\t\\label{sec2}\n\t\tWe start this section by recalling the definitions of sampling and interpolating sets for Bergman spaces which play an important role in this paper.\\\\\n\t\t\n\t\tLet $Z = (z_n)_{n\\geqslant 1}$ be a sequence of points in $\\mathbb{D}$.\\\\\n\t\t\\begin{enumerate}[label=$\\bullet$]\n\t\t\t\\item $Z$ is said to be separated if $\\displaystyle \\inf_{i\\neq j} \\rho(z_i,z_j) > 0$, where $\\rho(z,w) \\colonequals \\left|\\frac{z-w}{1-\\bar{z}w}\\right|$\n\t\t\tis the pseudohyperbolic metric on $\\mathbb{D}$.\n\t\t\t\n\t\t\t\\item $Z$ is called a sampling set for ${\\mathcal{A}}^2_\\alpha$ if \n\t\t\t\\[\n\t\t\t\\|f\\|^2_{{\\mathcal{A}}^2_\\alpha} \\asymp \\sum_{n\\geqslant 1} |f(z_n)|^2(1-|z_n|^2)^{2+\\alpha},\\quad f \\in {\\mathcal{A}}^2_\\alpha.\n\t\t\t\\]\n\t\t\t\\item $Z$ is called an interpolating set for ${\\mathcal{A}}^2_\\alpha$ if for every sequence $(\\lambda_n)_{n \\geqslant 1}$ satisfying \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum_{n\\geqslant 1} |\\lambda_n|^2(1-|z_n|^2)^{2+\\alpha}<\\infty,\n\t\t\t\\]\n\t\t\tthere exists $f$ in ${\\mathcal{A}}^2_\\alpha$ such that $f(z_n)=\\lambda_n$ for all $n\\geqslant 1.$ \n\t\t\\end{enumerate}\n\t\tThe closed graph theorem shows that if $Z$ is an\n\t\tinterpolating set for ${\\mathcal{A}}^2_\\alpha$, then there exists a constant $C$ such that the interpolation problem can be\n\t\tsolved by a function satisfying\n\t\t\\[\n\t\t\\|f\\|_{{\\mathcal{A}}^2_\\alpha} \\leqslant C \\left(\\sum_{n\\geqslant 1} |\\lambda_n|^2(1-|z_n|^2)^{2+\\alpha}\\right)^{1\/2}.\n\t\t\\]\n\t\tThe smallest such constant $C$ is called the constant of interpolation and we denoted it by $M_\\alpha(Z)$. We draw attention to the fact that every interpolating set for ${\\mathcal{A}}^2_\\alpha$ is separated. Interpolating and sampling sets for Bergman spaces have been studied by many authors, we refer the reader to \\cite{Seip1,Seip2} for more details.\\\\\n\t\t\n\t\n\t\tLet $\\mu$ be a positive Borel measure on $\\mathbb{D}$. Define the operator $T_\\mu$ by\n\t\t\\[ \n\t\tT_\\mu f(z)=\\int_\\mathbb{D} f(w)K^\\alpha(z,w)(1-|w|^2)^\\alpha d\\mu(w),\\ f\\in {\\mathcal{A}}^2_\\alpha,\n\t\t\\]\n\t\twhere $K^\\alpha(z,w)=(1-z\\bar{w})^{-2-\\alpha}$ is the reproducing kernel of ${\\mathcal{A}}^2_\\alpha.$ A simple computation yields\n\t\t\\[\n\t\t\\langle T_\\mu f , f\\rangle_{{\\mathcal{A}}^2_\\alpha} = \\displaystyle \\int _\\mathbb{D}|f(z)|^2 (1-|z|^2)^\\alpha d\\mu (z).\n\t\t\\]\n\t\t\n\t\t\\noindent These operators are called Toeplitz operators on Bergman spaces. Their boundedness was considered by Hasting \\cite{Has}. Later, Luecking obtained necessary and sufficient condition on $\\mu$ for which $T_\\mu$ belongs to a Schatten class $\\mathcal{S}_p({\\mathcal{A}}^2_\\alpha)$, with $p>0$ (see \\cite{Lue1}). Recently, the asymptotic behavior of the singular values of such operators has been studied on a large class of weighted Bergman spaces (see \\cite{APP, EE}).\\\\ \\\\\n\t\n\t\tIn the specific case of discrete measures, one can obtain explicit lower and upper bounds for the singular values of the associated Toeplitz operators. We have the following two lemmas.\n\t\t\\begin{lem}\\label{separation}\n\t\t\tLet $Z=(z_n)_{n\\geqslant 1}$ be a separated sequence and let $(c_n)_{n\\geqslant 1}$ be a sequence of positive numbers such that $\\left(c_n(1-|z_n|^2)^{-2}\\right)_{n\\geqslant 1}$ decreases to zero. For $\\mu= \\displaystyle \\sum_{n\\geqslant 1} c_n\\delta_{z_n}$, the operator $T_\\mu$ is compact on ${\\mathcal{A}}^2_\\alpha$ and \n\t\t\t\\[\n\t\t\ts_n(T_\\mu) \\lesssim \\dfrac{c_n}{(1-|z_n|^2)^2}.\n\t\t\t\\]\n\t\t\\end{lem}\n\t\t\n\t\t\\begin{proof}\n\t\t\n\t\t\tFirst, recall \\cite{GGK} that for a compact operator $T$ on a Hilbert space $H$,\n\t\t\t\\[\n\t\t\ts_n(T) = \\inf\\left\\{\\|T-R\\| : R \\colon H\\to H\\ \\text{is of rank}\\ 0$ such that the discs ${\\mathcal{D}}_\\delta(z_j)=\\{z\\in \\mathbb{D} : |z-z_j|<\\delta(1-|z_j|)\\}$ are pairwise disjoint. \n\t\t\tBy subharmonicity, we get\n\t\t\t\\[\n\t\t\t|f(z_j)|^2(1-|z_j|^2)^{2+\\alpha} \\lesssim \\int_{{\\mathcal{D}}_\\delta(z_j)} |f(z)|^2(1-|z|^2)^\\alpha\\ dm(z).\n\t\t\t\\]\n\t\t\tTherefore\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\langle \\left(T_\\mu-T_{\\mu_n}\\right)f,f\\rangle_{{\\mathcal{A}}^2_\\alpha} & \\lesssim & \\frac{c_n}{(1-|z_n|^2)^2} \\sum_{j\\geqslant n} \\displaystyle\\int_{{\\mathcal{D}}_\\delta(z_j)} |f(z)|^2(1-|z|^2)^\\alpha\\ dm(z)\\\\\n\t\t\t\t& \\lesssim & \\frac{c_n}{(1-|z_n|^2)^2}\\int_{\\mathbb{D}} |f(z)|^2(1-|z|^2)^\\alpha\\ dm(z)\\\\\n\t\t\t\t& = & \\frac{c_n}{(1-|z_n|^2)^2}\\|f\\| ^2_\\alpha.\n\t\t\t\\end{eqnarray*}\n\t\t\tSince the operator $T_\\mu-T_{\\mu_n}$ is positive, we obtain \n\t\t\t\\[\n\t\t\t\\| T_\\mu -T_{\\mu _n}\\| \\lesssim \\frac{c_n}{(1-|z_n|^2)^2}.\n\t\t\t\\]\n\t\t\n\t\t\\end{proof}\n\t\t\n\t\t\\begin{lem}\\label{interpolation}\n\t\t\tLet $Z=(z_n)_{n\\geqslant 1}$ be an interpolating set for ${\\mathcal{A}}^2_\\alpha$ and let $(c_n)_{n\\geqslant 1}$ be a sequence of positive numbers such that $\\left(c_n(1-|z_n|^2)^{-2}\\right)_{n\\geqslant 1}$ decreases to zero. For $\\mu= \\displaystyle \\sum_{n\\geqslant 1} c_n\\delta_{z_n}$, the operator $T_\\mu$ is compact on ${\\mathcal{A}}^2_\\alpha$ and \n\t\t\t\\[\n\t\t\ts_n(T_\\mu) \\asymp \\frac{c_n}{(1-|z_n|^2)^2}.\n\t\t\t\\]\n\t\t\\end{lem}\n\t\t\n\t\t\\begin{proof}\n\t\t\tThe upper bound result follows from Lemma \\ref{separation}. To obtain the lower bound, we use an alternative characterization {\\cite[p. 49]{GGK}} of the singular values:\n\t\t\n\t\t\t\\[\n\t\t\ts_n(T_\\mu) = \\underset{E,\\dim E=n}{\\sup}\\ \\left(\\underset{f\\in E\\setminus\\{0\\}}{\\inf}\\ \\frac{\\|T_\\mu f\\|_{{\\mathcal{A}}^2_\\alpha}}{\\|f\\|_{{\\mathcal{A}}^2_\\alpha}} \\right).\n\t\t\t\\] \n\t\t\tSet\n\t\t\t${\\bf \\mathcal{K}}_Z = span\\{k_{z_j}^\\alpha : j\\geqslant 1\\}$, where $k_{z_j}^\\alpha$ denotes the normalized reproducing kernel of ${\\mathcal{A}}^2_\\alpha$. We choose as subspace $E$ the orthogonal complement in ${\\bf \\mathcal{K}}_Z$ of the linear span of $\\{k_{z_j}^\\alpha : j > n\\}$. Since $Z$ is an interpolating set for ${\\mathcal{A}}^2_\\alpha$, the family $\\{k_{z_j}^\\alpha : j\\geqslant 1\\}$ is a Riesz basis of ${\\bf \\mathcal{K}}_Z$ and then the vector space $E$ is of dimension $n$. \n\t\t\n\t\t\tLet $f\\in E $, we have\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\|f\\|_{{\\mathcal{A}}^2_\\alpha}^2\\ \\leqslant\\ [M_\\alpha(Z)]^{2}\\sum_{j\\geqslant 1} |\\langle f,k_{z_j}^\\alpha\\rangle_{{\\mathcal{A}}^2_\\alpha}|^2\\ =\\ [M_\\alpha(Z)]^{2}\\sum_{j=1}^n |\\langle f,k_{z_j}^\\alpha\\rangle_{{\\mathcal{A}}^2_\\alpha} |^2.\n\t\t\t\n\t\t\t\\end{eqnarray*}\n\t\t\tFurthermore\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\langle T_\\mu f,f \\rangle_{{\\mathcal{A}}^2_\\alpha} & = & \\sum_{j\\geqslant 1} c_j (1-|z_j|^2)^\\alpha \\left| f(z_j) \\right|^2 \\\\\n\t\t\t\t& = & \\sum_{j=1}^n c_j (1-|z_j|^2)^\\alpha \\left| f(z_j) \\right|^2 \\\\\n\t\t\t\t& \\geqslant & \\frac{c_n}{(1-|z_n|^2)^2} \\sum_{j=1}^n |\\langle f,{k_{z_j}^\\alpha}\\rangle_{{\\mathcal{A}}^2_\\alpha}|^2,\n\t\t\t\\end{eqnarray*}\n\t\t\twhich finally leads to\n\t\t\t\\[\n\t\t\ts_n(T_\\mu)\\gtrsim \\left[M_\\alpha(Z)\\right]^{-2} \\dfrac{c_n}{(1-|z_n|^2)^2}.\n\t\t\t\\]\n\t\t\\end{proof} \n\t\n\t\n\t\n\t\n\t\t\n\t\n\t\t\\begin{proof}[\\textbf{Proof of Theorem \\ref{Bergman}}]\n\t\t\tWe start with the the upper bound of the singular values of $I_g$. Let $Z=(z_n)_{n\\geqslant 1}$ be a sampling set for ${\\mathcal{A}}^2_{\\alpha+2}$ such that each $R(I_n)$ contains $N$ points of $Z$, where $N$ is a large integer (see \\cite{Lue2}). We denote these points by $(z_{n,i})_{n\\geqslant 1}$, where $i=1,\\ldots,N$. \n\t\t\n\t\t\tUsing the fact that $Z$ is a sampling set for ${\\mathcal{A}}^2_{\\alpha +2}$ with the subharmoncity of $|g'|^2$, we get\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\int_{\\mathbb{D}}|f(z)g'(z)|^2(1-|z|^2)^{2+\\alpha}dm(z)& \\asymp & \\sum_{n\\geqslant 1}|f(z_{n})g'(z_{n})|^2(1-|z _{n}|^2)^{4+\\alpha}\\\\\n\t\t\t\t& \\lesssim & \\sum_n |f(z_{n})|^2(1-|z _{n}|^2)^\\alpha \\int _{R(I_n)}|g'(z)|^2(1-|z|^2)^2dm(z)\\\\\n\t\t\t\t& = & \\sum_{n,i} |f(z_{n,i})|^2(1-|z _{n,i}|^2)^\\alpha \\int _{R(I_n)}|g'(z)|^2(1-|z|^2)^2dm(z)\t\\\\\n\t\t\t\t& = & \\langle \\displaystyle \\sum_{i=1}^N T_{\\mu_i} f,f\\rangle_{{\\mathcal{A}}^2_\\alpha},\n\t\t\t\\end{eqnarray*}\n\t\t\twhere $\\mu_i = \\displaystyle \\sum _{n\\geqslant 1}m_g({R(I_n)})\\delta _{z_{n,i}}.$ This implies\n\t\t\t\\[\n\t\t\tI_g^*I_g \\lesssim \\displaystyle \\sum_{i=1}^N T_{\\mu_i}.\n\t\t\t\\] \n\t\t\n\t\t\t{We use a known inequality of the singular values from \\cite[p. 30]{GKG}. If $A$ and $B$ are two compact operators, then\n\t\t\t\t\\begin{equation}\\label{eq(2.1)}\n\t\t\t\ts_{m+n-1}(A+B)\\leqslant s_m(A)+s_n(B), \\quad m,n=1,2,\\ldots.\n\t\t\t\t\\end{equation}\n\t\t\t\tThus we obtain\t\n\t\t\t}\t \n\t\t\t\\begin{eqnarray*}\n\t\t\t\ts_{Nn}(I_g^*I_g) \\lesssim \\displaystyle \\sum_{i=1}^N s_n(T_{\\mu_i}) \\leqslant N \\max_{1\\leqslant i\\leqslant N} s_n(T_{\\mu_i}).\n\t\t\t\\end{eqnarray*}\n\t\t\tThis, together with Lemma \\ref{separation}, gives\n\t\t\t\\[\n\t\t\ts_{Nn}^2(I_g)\\lesssim N \\max_{1\\leqslant i\\leqslant N} s_n(T_{\\mu_i}) \\lesssim \\frac{m_g({R(I_n)})}{(1-|z_n|^2)^2}.\n\t\t\t\\]\n\t\t\n\t\t\n\t\t\n\t\t\tNow, we turn to prove the lower bound result. Choose a point $\\xi_n$ in $R(I_n)$ so that $|g'(z)| \\leqslant 2|g'(\\xi_n)|$ for all $z \\in R(I_n)$. Then\n\t\t\t\\begin{eqnarray}\\label{2}\n\t\t\tm_g(R(I_n))= \\int _{R(I_n)}|g'(z)|^2 (1-|z|^2)^2dm(z) \\leqslant |g'(\\xi_n)|^2 (1-|\\xi_n|^2)^4.\n\t\t\t\\end{eqnarray}\n\t\t\tNote that $(\\xi_n)_{n\\geqslant 1}$ can be written as a finite union of separated sets. Since every separated set itself can be expressed as a finite union of interpolating sets (see \\cite{Seip2}), there exist $\\Lambda_1, \\Lambda_2,\\cdots ,\\Lambda_M$ such that $\\{\\xi_n, \\ n\\in \\Lambda _i\\}$ is an interpolating sequence for ${\\mathcal{A}}^2_{\\alpha}$. Therefore \n\t\t\t\n\t\t\n\t\t\t\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\int_\\mathbb{D} |f(z)g'(z)|^2(1-|z|^2)^{\\alpha+2}dm(z)\n\t\t\t\t& \\gtrsim & \\sum_{i=1}^M \\sum_{n\\in \\Lambda _i} |f(\\xi_{n})|^2|g'(\\xi_{n})|^2(1-|\\xi_n|^2)^{\\alpha+4}\\\\\n\t\t\t\t& \\geqslant & \\sum_{i=1}^M \\sum_{n\\in \\Lambda _i}|f(\\xi_{n})|^2(1-|\\xi_{n}|^2)^\\alpha m_g(R(I_n))\\\\\n\t\t\t\t& = & \\langle \\sum_{i=1}^M T_{\\mu_i} f,f\\rangle_{{\\mathcal{A}}^2_{\\alpha}},\n\t\t\t\\end{eqnarray*}\t\n\t\t\twhere $\\mu_i=\\displaystyle \\sum_{n\\in \\Lambda _i} m_g({R(I_n)})\\delta_{\\xi_{n}}$.\n\t\t\tThe first inequality follows from the fact that $(\\xi_n)_{n\\geqslant 1}$ is a finite union of separated sets. Therefore\n\t\t\t\\[\n\t\t\tI_g^*I_g\\gtrsim \\sum_{i=1}^M T_{\\mu_i} \\geqslant T_{\\mu_k},\\quad k\\in \\{1,..,M\\}.\n\t\t\n\t\t\t\\] \n\t\t\tThis implies that \n\t\t\t\\[\n\t\t\ts_n(T_{\\mu_k}) \\lesssim s^2_n(I_g),\\quad k\\in \\{1,..,M\\}.\n\t\t\t\\]\n\t\t\tFinally, using the fact that each {set $\\{\\xi_n, \\ n\\in \\Lambda _i\\}$}, for $i=1,\\ldots,M$, is an interpolating set for ${{\\mathcal{A}}^2_{\\alpha}}$ along with Lemma \\ref{interpolation}, we get\n\t\t\t\\[\n\t\t\ts_n(T_{\\mu_k}) \\gtrsim a_n(\\mu_k),\n\t\t\t\\]\n\t\t\twhere $a_n(\\mu _k)$ is the nonincreasing rearrangement of the sequence of the sequence $(\\frac{\\mu _k({R(I_n)})}{m({R(I_n)})})_n$.\n\t\t\tConsequently, there exists an integer $p$ such that $s_{[\\frac{n}{p}]}^2(I_g)\\gtrsim \\frac{m_g({R(I_n)})}{(1-|z_n|^2)^2}.$ \n\t\t\\end{proof}\n\t\n\t\n\t\n\t\t\\section{The Hardy space $H^2$}\\label {hardy}\n\t\t\\label{sec3}\n\t\n\t\tThis section is devoted to prove Theorems \\ref{Hardy} and \\ref{B}. We start by recalling the connection between integration operators acting on the Hardy space $H^2$ and some standard operators.\\\\\\\\\n\t\tLet $\\mu$ be a positive Borel measure on $\\mathbb{D}$, and let $J_\\mu$ denote the embedding operator from $H^2$ into $L^2(\\mu)$. Carleson proved in \\cite{Car} that $J_\\mu$ is bounded if and only if $\\mu(W(I))\/|I |$ is uniformly bounded for all arcs $I\\subset\\mathbb{T}$. This result can be expressed in terms of the family of the dyadic arcs. \n\t\tIndeed, one can see that $J_\\mu$ is bounded if and only if there exists $C>0$ such that \n\t\t\\[\n\t\t\\mu ( W(I_{n,k}) ) \\leqslant C | I_{n,k}|, \\quad \\quad n\\geqslant 1\\ \\mbox{and}\\ 1\\leqslant k \\leqslant 2^n.\n\t\t\\]\n\t\tNote also that $J_\\mu$ is compact if and only if \n\t\t\\begin{eqnarray}\\label{compact}\n\t\t\\displaystyle \\lim _{n\\to \\infty} \\displaystyle \\sup _{1\\leqslant k\\leqslant 2^n} \\frac{\\mu ( W(I_{n,k}) )}{| I_{n,k} |}= 0.\n\t\t\\end{eqnarray}\n\t\tWe introduce the operator \n\t\t\\[\n\t\tS_{\\mu}f (z) = \\displaystyle \\int _\\mathbb{D}f(w)K(z,w)d\\mu(w),\\quad f\\in H^2,\n\t\t\\]\n\t\twhere $K(z,w)= (1-z\\bar{w})^{-1}$ is the reproducing kernel of $H^2$. \n\t\n\t\tA straightforward computation shows that\n\t\t\\[\n\t\t\\langle S_\\mu f , f\\rangle_{H^2} = \\displaystyle \\int _\\mathbb{D}|f(z)|^2d\\mu (z)= \\| J_\\mu f\\|^2_{L^2(\\mu)}.\n\t\t\\]\n\t\tThis implies that $S_\\mu$ is bounded, compact, or belongs to a Schatten class ${\\mathcal{S}}_{p}(H^2)$ if and only if $J_\\mu$ is. In \\cite{Lue1}, Luecking proved that $S_\\mu \\in {\\mathcal{S}}_p(H^2)$, for $p>0$, if and only if \n\t\t\\[\n\t\t\\displaystyle \\sum _{n,k} \\left (\\frac{\\mu (R(I_{n,k}))}{| I_{n,k} |}\\right )^{p}<\\infty.\n\t\t\\]\n\t\tIt is worth noting that Luecking's characterization can also be expressed in terms of Carleson windows $W(I_{n,k})$ (see \\cite{LLQRJFA}). However, this is no longer the case for Carleson boundedness criterion (see \\cite{PR}).\\\\\n\t\t\\\\\n\t\tNow, we fix and recall some notations. Let $\\mu$ be a positive Borel measure on $\\mathbb{D}$ which satisfies condition (\\ref {compact}), and let $(I_n(\\mu))_{n\\geqslant 1}$ be an enumeration of $(I_{n,k})_{n,k}$ such that the sequence $\\left(\\dfrac{\\mu(W(I_n(\\mu)))}{| I_n(\\mu)|}\\right)_{n\\geqslant 1}$ is nonincreasing. According to the notation used in the introduction, we define\n\t\t\\[\n\t\t\\tau _n(\\mu) \\colonequals \\frac{\\mu(W(I_n(\\mu))}{| I_n(\\mu)|}.\n\t\t\\]\n\t\n\t\t\n\t\t\n\t\t\\noindent The following lemma is the key to prove the first assertion of Theorem \\ref{Hardy}.\n\t\t\\begin{lem}\\label{KL}\n\t\t\tLet $\\mu$ be a positive Borel measure on $\\mathbb{D}$ such that $S_\\mu$ is compact on $H^2$. Let $I_n = I_n(\\mu)$, and let $(z_n)_{n\\geqslant 1} $ be a sequence such that $z _n \\in R(I_n)$. Fix an index $n\\geqslant 1$, and set $\\mu_n =\\displaystyle \\sum _{k\\geqslant n}\\mu (R(I_k))\\delta _{z _k}$. We have \n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item\\label{3} \n\t\t\t\t$\n\t\t\t\t\\displaystyle \\sup _{j\\geqslant 1}\\frac{\\mu_n(W(I_j))}{| I_j|} \\leqslant \\tau_n(\\mu).\n\t\t\t\t$\\\\\n\t\t\t\t\\item\\label{4} $s_n(S_{\\mu _1})\\ \\lesssim \\ \\tau_n(\\mu).$\\\\\n\t\t\t\\end{enumerate}\n\t\t\\end{lem}\t\n\t\t\n\t\t\\begin{proof}\n\t\t\tWe begin with part \\eqref{3}. First, note that if $n=1$, then $\\tau _j(\\mu_1)= \\tau _j(\\mu)$ and the result is obvious. Now, suppose that $n\\geqslant 2$. \\\\ \\\\\n\t\t\t{For $j\\geqslant 1$, let $\\Gamma_j=\\lbrace k\\geqslant n \\colon R(I_k) \\subset W(I_j) \\rbrace$. Then\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\t\\mu_n(W(I_j)) &=& \\sum_{k \\in \\Gamma_j} \\mu_n(R(I_k)) = \\sum_{k \\in \\Gamma_j} \\mu(R(I_k)) \\leqslant \\mu(W(I_j)).\n\t\t\t\t\\end{eqnarray*}\t\n\t\t\t\tThus for all $j\\geqslant n$, we have\n\t\t\t}\n\t\t\t\\[\n\t\t\t\\dfrac{\\mu_n(W(I_j))}{| I_j |} \\leqslant \\dfrac{\\mu(W(I_j))}{| I_j |} \\leqslant \\dfrac{\\mu(W(I_n))}{| I_n |} .\n\t\t\t\\]\n\t\t\tLet $ j\\in \\lbrace 1,\\ldots,n-1\\rbrace$. If we assume that\n\t\t\t\\begin{eqnarray}\\label{EL1}\n\t\t\t\\dfrac{\\mu_n(W(I_j))}{| I_j |} > \\dfrac{\\mu(W(I_n))}{| I_n |},\n\t\t\t\\end{eqnarray}\n\t\t\twe will end up with a contradiction. Indeed, we will prove that there exists $j_1\\in \\lbrace 1,\\ldots,n-1\\rbrace$ such that $W(I_{j_1})\\varsubsetneq W(I_j)$ and $\\dfrac{\\mu (W(I_n))}{| I_n |} < \\dfrac{\\mu_n(W(I_{j_1}))}{| I_{j_1}|}$, which is absurd since one can repeat this operation indefinitely.\\\\\n\t\t\t\n\t\t\tSuppose that assumption \\eqref{EL1} is satisfied, and write $W(I_j)= R(I_j)\\cup W(I_{j_1})\\cup W(I_{j'_1})$, where\n\t\t\t\\[\n\t\t\t| I_{j_1} |= | I_{j'_1}|= | I_{j}|\/2. \n\t\t\t\\]\n\t\t\tWithout loss of generality, we assume that $\\mu_n(W(I_{j_1}))\\geqslant \\mu_n(W(I_{j'_1}))$. Then, since $\\mu_n(R(I_j))=0$, we have\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\dfrac{\\mu_n(W(I_j))}{| I_j|} &=& \\dfrac{\\mu_n(W(I_{j_1}))+\\mu_n(W(I_{j'_1}))}{| I_{j_1}|+| I_{j'_1}|}\\leqslant \\dfrac{\\mu_n(W(I_{j_1}))}{| I_{j_1}|}.\n\t\t\t\\end{eqnarray*}\n\t\t\tThis implies $\\dfrac{\\mu (W(I_n))}{| I_n|} < \\dfrac{\\mu_n(W(I_{j_1}))}{| I_{j_1}|}$ and then $j_1\\in \\lbrace 1,\\ldots,n-1\\rbrace$. The proof of \\eqref{3} is complete.\\\\\n\t\t\t\n\t\t\tTo prove \\eqref{4}, recall that $s_n(S_{\\mu_1}) = \\inf\\left\\{\\left\\| S_{\\mu_1}- R\\right\\|\\ :\\ \\text{rank}\\ R0$. Then we have \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j=1}^n \\left|\\langle Tu_j, v_j\\rangle_{K} \\right|\\leqslant C^2 \\displaystyle \\sum _{j=1}^n s_j(T).\n\t\t\t\\]\n\t\t\\end{lem}\n\t\t\n\t\n\t\t\\begin{proof}\n\t\t\tLet $(e_n)_n$ and $(f_n)_n$ be two orthonormal bases of $H$ and $K$, respectively. Let $A$ and $B$ be the operators given by\n\t\t\t\\[\n\t\t\tAf=\\displaystyle\\sum_n\\langle u_n,f\\rangle_H e_n \\quad \\text{and}\\quad Bg=\\displaystyle\\sum_n\\langle v_n,g\\rangle_K f_n.\n\t\t\t\\]\n\t\t\tBy assumptions, $A$ and $B$ are bounded and $\\|A\\|, \\|B\\| \\leqslant C$. Note that \n\t\t\tthe adjoint operators $A^*$ and $B^*$ satisfy $A^*e_n=u_n$ and $B^*f_n=v_n$. By Theorem 3.5 of \\cite{GGK}, we have \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum_{j=1}^{n}\\left|\\langle BTA^* e_j,f_j\\rangle \\right| \\leqslant \\displaystyle \\sum_{j=1}^{n} s_j(BTA^*).\n\t\t\t\\]\n\t\t\tThus\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\displaystyle \\sum_{j=1}^{n}\\left|\\langle Tu_j,v_j \\rangle \\right| &=& \\displaystyle \\sum_{j=1}^{n}\\left|\\langle TA^* e_j,B^*f_j\\rangle \\right |\\\\\n\t\t\t\n\t\t\t\t& = & \\displaystyle \\sum_{j=1}^{n}\\left|\\langle BTA^* e_j,f_j\\rangle \\right|\\\\\n\t\t\t\n\t\t\t\t&\\leqslant& \\displaystyle \\sum_{j=1}^{n} s_j(BTA^*)\\\\\n\t\t\t\t&\\leqslant& \\|B\\|\\|A^*\\|\\displaystyle \\sum_{j=1}^{n}s_j(T)\\leqslant C^2 \\displaystyle \\sum_{j=1}^{n}s_j(T).\n\t\t\t\\end{eqnarray*} \n\t\t\t\n\t\t\\end{proof}\n\t\t\n\t\t\\noindent The classical Littlewood-Paley identity \\cite{Gar} states that\n\t\n\t\t\\[\n\t\t\\| f -f(0)\\|^2 _{H^2}= \\displaystyle \\int_{\\mathbb{D}}|f'(z)|^2\\log (1\/|z|^2)dm(z).\n\t\t\\]\n\t\n\t\tThis gives rise to the following formula which relates the operators $I_g$ and $S_{\\nu _g}$. We have\n\t\t\\begin{eqnarray}\\label{H-L}\n\t\t\\langle I^*_gI_g f, f \\rangle_{H^2} = \\displaystyle \\int _{\\mathbb{D}}|f(z)|^2|g'(z)|^2\\log (1\/|z|^2)dm(z)\n\t\t\\asymp \\langle S_{\\nu_g}f,f \\rangle_{H^2}.\n\t\t\\end{eqnarray}\n\t\n\t\t\n\t\n\t\n\t\t\\begin{proof}[\\textbf{Proof of Theorem \\ref{Hardy}}]\n\t\t\tIt is immediate, from formula (\\ref{H-L}), that if $f\\in H^2$ then $fg'\\in {\\mathcal{A}}^2_1$. We have seen in the proof of the upper bound result from Theorem \\ref{Bergman} that\n\t\t\tthere exist a sampling sequence $Z=(z_n)_{n\\geqslant 1}$ for ${\\mathcal{A}}^2_1$ and an integer $N$ such that each $R(I_n)$ contains $N$ points of $Z$ denoted by $(z_{n,i})_{n\\geqslant 1}$, where $i=1,\\ldots,N$. Then\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\displaystyle \\int _{\\mathbb{D}}|f(z)g'(z)|^2(1-|z|^2)dm(z)& \\asymp& \\displaystyle \\sum _{n\\geqslant 1} |f(z_{n})g'(z _{n})|^2(1-|z _{n}|^2)^3\\\\\n\t\t\t\t& \\lesssim & \\displaystyle \\sum _{n\\geqslant 1} |f(z_{n})|^2 \\displaystyle \\int _{R(I_n)}|g'(z)|^2(1-|z|^2)dm(z)\\\\\n\t\t\t\t& = & \\displaystyle \\sum _{n\\geqslant 1} \\displaystyle \\sum _{i=1}^N|f(z_{n,i})|^2 \\displaystyle \\int _{R(I_n)}|g'(z)|^2(1-|z|^2)dm(z)\\\\\n\t\t\t\t& = & \\langle \\displaystyle \\sum _{i=1}^N S_{\\nu_i} f,f\\rangle_{H^2},\n\t\t\t\\end{eqnarray*}\n\t\t\twhere $\\nu_i = \\displaystyle \\sum _{n\\geqslant 1}\\nu_g(R(I_n))\\delta _{z_{n,i}}.$ {Using inequality \\eqref{eq(2.1)}, we obtain}\n\t\t\t\\[\n\t\t\ts_{Nn}^2(I_g)\\asymp s_{Nn}(S_{\\nu_g}) \\lesssim \\max_{1\\leqslant i \\leqslant N} s_n(S_{\\nu_i}).\n\t\t\t\\]\n\t\t\t{By} Lemma \\ref{KL}, we deduce $s_{Nn}^2(I_g) \\lesssim \\tau _n(\\nu_g)$ as desired and the proof of assertion \\ref{i} is complete.\\\\\n\t\t\t\n\t\t\n\t\t\n\t\t\n\t\t\t\\noindent To prove assertion \\ref{ii}, we use Lemma \\ref{0}. Let $\\xi_n$ be the center of $R(I_n)$ and set\n\t\t\t\\[\n\t\t\tu_n(z)=v_n(z)=\\frac{(1-|\\xi_n|^2)^{3\/2}}{(1-\\bar{\\xi}_n z)^2}.\n\t\t\t\\]\n\t\t\tFor $f \\in H^2$, we have\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t|\\langle u _n, f\\rangle _{H^2}|^2 & \\lesssim & | u _n (0)f(0)|^2+\\left| \\displaystyle \\int _{\\mathbb{D}}\\frac{(1-|\\xi_n|^2)^{3\/2}}{(1-\\bar{\\xi}_n z)^3} \\overline{f'(z)} (1-|z|^2)dm (z)\\right|^2\\\\\n\t\t\t\t& = & (1-|\\xi_n|^2)^3 |f(0)|^2+|\\langle k_{\\xi_n}^1,f' \\rangle_{\\mathcal{A}^2_1} |^2.\n\t\t\t\\end{eqnarray*}\n\t\t\tThus \n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\displaystyle \\sum _n |\\langle u _n, f\\rangle _{H^2}|^2 & \\lesssim & \\displaystyle \\sum _n (1-|\\xi_n|^2)^3 |f(0)|^2+ \\displaystyle \\sum _n |\\langle k_{\\xi_n}^1,f' \\rangle_{\\mathcal{A}^2_1} |^2\\\\\n\t\t\t\t& \\lesssim & \\displaystyle \\sum _n (1-|\\xi_n|^2)^3 |f(0)|^2+\\|f'\\|^2_{{\\mathcal{A}}^2_1}\\\\\n\t\t\t\t& \\lesssim & \\|f\\|^2_{H^2}.\n\t\t\t\\end{eqnarray*}\n\t\t\tThe second inequality follows from the fact that $(\\xi _n)_n$ is a separated sequence.\n\t\t\n\t\t\tThen, by Lemma \\ref{0} applied to $S_{\\nu _g}$, there exists an absolute constant $c>0$ such that\n\t\t\t\\[\n\t\t\t\\displaystyle c \\sum _{j=1}^n |\\langle S_{\\nu_g} u_j, u_j\\rangle _{H^2}| \\leqslant \\displaystyle \\sum _{j=1}^n s_j(S_{\\nu_g}).\n\t\t\t\\]\n\t\t\tFurthermore\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\langle S_{\\nu_g} u_n,u_n\\rangle_{H^2} & = & \\displaystyle \\int_\\mathbb{D} \\dfrac{(1-|a_n|^2)^3}{|1-\\bar{a}_n z|^4}|g'(z)|^2(1-|z|^2) dm(z) \\\\\n\t\t\t\t& \\geqslant & \\displaystyle \\int_{W(I_n)}\\dfrac{(1-|a_n|^2)^3}{|1-\\bar{a}_n z|^4}|g'(z)|^2(1-|z|^2) dm(z)\\\\\n\t\t\t\t& \\gtrsim & \\dfrac{1}{|I_n|} \\displaystyle \\int_{W(I_n)} |g'(z)|^2(1-|z|^2) dm(z).\n\t\t\t\\end{eqnarray*}\n\t\t\tCombining these two facts with formula (\\ref{H-L}), we get \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j=1}^n \\frac{\\nu_g(W(I_j))}{|I_j|} \\lesssim \\displaystyle \\sum _{j=1}^n s_j(S_{\\nu_g}) \\lesssim \\displaystyle \\sum _{j=1}^n s^2_j({I_{g}}).\n\t\t\t\\]\n\t\t\\end{proof}\n\t\t\n\t\n\t\t\n\t\n\t\n\t\tWe turn now to prove Theorem \\ref{B}, we start with the proof of the first statement which is completely based on Theorem \\ref{Hardy}.\n\t\t\n\t\t\\begin{proof}[\\textbf{Proof of assertion \\ref{2i} of Theorem \\ref{B}} ]\n\t\t\tSuppose that $(s_n(I_g))_n \\in {\\mathcal{R}} _\\gamma$ for some $\\gamma \\in (0,1\/2)$. Then there exists $(x_n)_n$ a nonincreasing sequence of positive numbers such that ${n^{\\gamma} x_n}$ increases to infinity and $s_n(I_g) \\asymp x_n$. {For an integer $p\\geqslant 1$, we have \n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\tp^{-\\gamma}x_n \\leqslant x_{pn} \\leqslant x_n. \n\t\t\t\t\\end{eqnarray*}\n\t\t\t}\n\t\t\tBy assertion \\ref{i} of Theorem \\ref{Hardy}, we have \n\t\t\t\\begin{eqnarray}\\label{UB}\n\t\t\t{x^2_n \\asymp s_{np}^2(I_g)} \\lesssim \\tau_n(\\nu_g).\n\t\t\t\\end{eqnarray}\t\n\t\t\tMoreover, Theorem \\ref{Hardy} implies\n\t\t\t\\begin{eqnarray*}\\label{LB}\n\t\t\t\tn\\tau_n(\\nu_g) \\leqslant \\displaystyle \\sum_{j=1}^n \\tau_j(\\nu_g) & \\lesssim & \\displaystyle \\sum_{j=1}^n s_j^2(I_g)\\\\\n\t\t\t\t&\\lesssim & {\\displaystyle \\sum_{j=1}^n x^2_j} \\\\\n\t\t\t\t& \\leqslant & n^{2\\gamma} x^2_n \\displaystyle \\sum_{j=1}^n \\dfrac{1}{j^{2\\gamma}}. \n\t\t\t\n\t\t\t\n\t\t\t\\end{eqnarray*}\t\n\t\t\t{The last inequality follows from the fact that $(j^{2\\gamma}x_j)_j $ is increasing. Since $2\\gamma <1$, we have\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\t\\sum_{j=1}^n \\dfrac{1}{j^{2\\gamma}} &\\asymp& n^{1-2\\gamma}.\t\n\t\t\t\t\\end{eqnarray*}\t\n\t\t\t\tThus $n\\tau_n(\\nu_g)\t\\lesssim n x^2_n.$}\n\t\t\tThis along with inequality (\\ref{UB}) leads to {$ x_n \\asymp \\tau_n(\\nu_g)$.}\\\\ \\\\\n\t\t\n\t\t\tNow, suppose that $(\\tau_n(\\nu_g))_n \\in {\\mathcal{R}} _{2\\gamma}$ and let $(x_n)_n$ be a nonincreasing sequence of positive numbers such that $n^{2\\gamma} x_n$ increases to infinity and $\\tau_n(\\nu_g) \\asymp x_n$. It follows at once from assertion \\ref{i} of Theorem \\ref{Hardy} that \n\t\t\t{\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\ts_n^2(I_g) \\lesssim x_{[\\frac{n}{p}]} \\asymp x_n. \n\t\t\t\t\\end{eqnarray*}\n\t\t\t}\n\t\t\n\t\t\tLet $\\kappa$ be a large constant. Assertion \\ref{ii} of Theorem \\ref{Hardy} yields\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\displaystyle \\sum_{j=1}^{\\kappa n} x_j & \\lesssim & \\displaystyle \\sum_{j=1}^n s_j^2(I_g) + \\displaystyle \\sum_{j=n+1}^{\\kappa n} s_j^2(I_g) \\\\\n\t\t\t\t& \\lesssim & \\displaystyle \\sum_{j=1}^n x_j + \\kappa n s_n^2(I_g).\n\t\t\t\\end{eqnarray*}\n\t\t\t{Using the same line of reasoning as in the previous part, we obtain $\\displaystyle\\sum_{j=1}^n x_j \\lesssim n x_n.$ Therefore\t\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\t\\displaystyle \\sum_{j=1}^{\\kappa n} x_j &\\lesssim & n x_n + \\kappa n s_n^2(I_g).\t\n\t\t\t\t\\end{eqnarray*}\n\t\t\t\tIn contrast\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\t\t\\displaystyle \\sum_{j=1}^{\\kappa n} x_j & \\geqslant & \\kappa n x_{\\kappa n} \\\\ \n\t\t\t\t\t&\\geqslant& \\kappa^{1-2\\gamma} n x_n.\n\t\t\t\t\\end{eqnarray*}\n\t\t\t\tThe first inequality follows from the fact $(x_j)_j $ is nonincreasing and the second one uses the fact that $(j^{2\\gamma}x_j)_j $ is increasing. \n\t\t\t\tThus\n\t\t\t\n\t\t\t\t\\[\n\t\t\t\t\\kappa^{1-2\\gamma} x_n \\lesssim x_n + \\kappa s_n^2(I_g).\n\t\t\t\t\\]\n\t\t\t}\n\t\t\tFor $\\kappa$ large enough, we obtain the result.\n\t\t\\end{proof}\n\t\t\n\t\tThe proof of the second statement of Theorem \\ref{B} requires some preparatory results. The following technical lemma is a generalization of Proposition 3.3 in \\cite{LLQRJFA}.\n\t\t\\begin{lem}\\label{lemLLQP}\n\t\t\tLet $h: [0,\\infty[\\to [0,\\infty[$ be an increasing function such that $h(0)=0$. Suppose that there exist $\\varepsilon \\in (0,1)$ and $p\\geqslant 1$ for which $t\\to h(t^\\varepsilon)$ is concave and $t\\to h(t^p)$ is convex. Let $\\mu $ be a positive Borel measure on $\\mathbb{D}$. There exists $B>0$, which depends on $\\varepsilon$ and $p$, such that \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h\\left (\\frac{\\mu(W{(I_j)})}{| I_j |}\\right ) \\leqslant \\displaystyle \\sum _{j}h\\left (B\\frac{\\mu(R{(I_j)})}{| I_j|}\\right ).\n\t\t\t\\]\n\t\t\\end{lem}\n\t\t\\begin{proof}\n\t\t\tWe use the argument given in \\cite{LLQRJFA}. Let $\\alpha = 1\/\\varepsilon$ and let $\\beta =1\/(1-\\varepsilon)$ be the conjugate of $\\alpha$. Write\n\t\t\t\\[\n\t\t\t{W(I_{n,j})}=\\bigcup_{l\\geqslant n}\\bigcup_{k\\in H_{l,n,j}} {R(I_{l,k})},\\quad n\\geqslant 1\\ \\mbox{and}\\ 1\\leqslant j\\leqslant 2^n,\n\t\t\t\\]\n\t\t\twhere $H_{l,n,j}=\\left\\lbrace k\\in\\lbrace 1,\\ldots,2^l \\rbrace; \\dfrac{j}{2^n}\\leqslant\\dfrac{k}{2^l}<\\dfrac{j+1}{2^n}\\right\\rbrace.$ Consider $a \\in (1,2)$ such that $2< a^\\beta$. It is proved in \\cite{LLQRJFA} that \n\t\t\t\\[\n\t\t\t\\mu({W(I_{n,j})})^\\alpha \\lesssim \\displaystyle \\sum _{l\\geqslant n; \\ k\\in H_{l,n,k}}a^{(l-n)\\alpha }\\mu({R(I_{l,k})})^\\alpha.\n\t\t\t\\]\n\t\t\tRecall that the function $h_{1\/\\alpha}(t) \\colonequals h(t^{1\/\\alpha})$ is increasing and concave. Consequently,\n\t\t\t\\begin{eqnarray*}\n\t\t\t\th(2^n\\mu({W(I_{n,j})}) )& =& h_{1\/\\alpha} \\left ( (2^n\\mu({W(I_{n,j})}))^\\alpha \\right )\\\\\n\t\t\t\t& \\lesssim & \\displaystyle \\sum _{l\\geqslant n; \\ k\\in H_{l,n,k}} h_{1\/\\alpha} \\left (a^{(l-n)\\alpha}2^{n\\alpha}\\mu({R(I_{l,k})})^\\alpha \\right )\\\\\n\t\t\t\t& = & \\displaystyle \\sum _{l\\geqslant n; \\ k\\in H_{l,n,k}} h\\left (\\left (\\frac{a}{2}\\right )^{l-n}2^l\\mu({R(I_{l,k})}) \\right )\\\\\n\t\t\t\t& \\leqslant & \\displaystyle \\sum _{l\\geqslant n; \\ k\\in H_{l,n,k}} \\left (\\frac{a}{2}\\right ) ^{(l-n)\/p}h(2^l\\mu({R(I_{l,k})}) ).\\\\\n\t\t\t\\end{eqnarray*}\n\t\t\tThe last inequality follows from the fact $h(t^p)$ is convex. Then we obtain \n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\sum_{n= 1}^\\infty\\sum_{j=0}^{2^n-1}h\\left(2^n\\mu({W(I_{n,j})})\\right)&\\lesssim &\\sum_{l=1}^\\infty\\sum_{k=0}^{2^l-1}\\left(\\sum_{(n,j):l\\geqslant n,k\\in H_{l,n,j}}\\left (\\frac{a}{2}\\right ) ^{(l-n)\/p}\\right)h\\left(2^l\\mu({R(I_{l,k})})\\right)\\\\\n\t\t\t\t&\\leqslant&C{(p,\\varepsilon)}\\sum_{l=1}^\\infty\\sum_{k=0}^{2^l-1}h\\left(2^l\\mu({R(I_{l,k})})\\right).\n\t\t\t\\end{eqnarray*}\n\t\t\\end{proof}\n\t\t\n\t\n\t\t\\begin{theo}\\label {trace}\n\t\t\tLet $\\mu$ be a positive Borel measure on $\\mathbb{D}$ such that $J_\\mu$ is compact. Let $h: [0,\\infty[\\to [0,\\infty[$ be a convex increasing function with $h(0)=0$ and such that $t\\to h(t^\\varepsilon)$ is concave for some $\\varepsilon \\in (0,1)$. Then there exists $B>0$ which depends only on $\\varepsilon $ such that \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h\\left (\\frac{1}{B}\\sqrt{\\tau_j(\\mu)}\\right ) \\leqslant \\displaystyle \\sum _{j}h(s_j(J_\\mu)).\n\t\t\t\\]\n\t\t\\end{theo}\n\t\t\n\t\t\\begin{proof} The proof is based on Lemma \\ref{0}. Let $\\xi_n$ be the center of ${R(I_n)}$, and set\n\t\t\t\\[\n\t\t\tu_n(z)=\\frac{(1-|\\xi_n|^2)^{3\/2}}{(1-\\bar{\\xi}_n z)^2}\\ \\mbox{and}\\ v_n= c_n \\theta_n \\chi_{{R(I_n)}},\n\t\t\t\\]\n\t\t\twhere $\\theta _n = \\frac{u_n}{|u_n|}$ and $c_n= \\frac{1}{\\sqrt{\\mu ({R(I_n)})}}$ if $\\mu ({R(I_n)}) \\neq 0$.\\\\\n\t\t\tWe proved before \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _n |\\langle u _n, f\\rangle _{H^2}|^2 \\lesssim \\|f\\|^2_{H^2}, \\quad f \\in H^2.\n\t\t\t\\]\n\t\t\tBy Cauchy-Schwarz inequality, we have \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _n |\\langle v_n, v\\rangle _{L^2(\\mu)}|^2 \\leqslant \\displaystyle \\sum _n c^2_n\\mu ({R(I_n)})\\displaystyle \\int_{{R(I_n)}}|v|^2d\\mu = \\| v \\| ^2_{L^2(\\mu)},\\quad v \\in L^2(\\mu).\n\t\t\t\\]\n\t\t\tNote also that \n\t\t\t\\[\n\t\t\t\\langle J_\\mu u_n, v_n\\rangle _{L^2(\\mu)} = c_n \\displaystyle \\int _{{R(I_n)}}|u_n|d\\mu \\asymp \\sqrt{\\frac{\\mu({R(I_n)})}{| I_n |}}.\n\t\t\t\\]\n\t\t\tApplying Lemma \\ref{0}, we get\n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j=1}^n \\frac{1}{C}\\sqrt{\\frac{\\mu(R(I_j))}{| I_j |}} \\lesssim \\displaystyle \\sum _{j=1}^n s_j(J_\\mu),\n\t\t\t\\]\n\t\t\twhere $C$ is a positive constant. \n\t\t\tThen by Corollary 3.3 of \\cite{GGK}, we obtain\n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h\\left (\\frac{1}{C}\\sqrt{\\frac{\\mu(R{(I_j)})}{| I_j|}}\\right ) \\leqslant \\displaystyle \\sum _{j}h(s_j(J_\\mu)).\n\t\t\t\\]\n\t\t\tThe desired result follows from Lemma \\ref{lemLLQP}.\n\t\t\\end{proof}\n\t\t\\begin{cor}\\label {Ctrace}\n\t\t\tLet $g\\in VMOA$. Let $h: [0,\\infty[\\to [0,\\infty[$ be a convex increasing function with $h(0)=0$ and such that $t\\to h(t^\\varepsilon)$ is concave for some $\\varepsilon \\in (0,1)$. Then there exists $B>0$ which depends only on $\\varepsilon $ such that \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h\\left (\\frac{1}{B}\\sqrt{\\tau_j(\\nu_g)}\\right ) \\leqslant \\displaystyle \\sum _{j}h(s_j(I_g))).\n\t\t\t\\]\n\t\t\\end{cor}\n\t\t\\begin{proof}\n\t\t\tIt suffices to apply Theorem \\ref{trace} and to remark that $s_n(I_g)= s_n(J_{\\nu_g})$.\n\t\t\\end{proof}\n\t\t\n\t\tThe following lemma is proved in \\cite{EE}.\n\t\t\\begin{lem}\\label{lemEE}\n\t\t\tLet $(a_n)_n$ and $(b_n)_n$ be two positive {nonincreasing} sequences. Suppose that there exist $\\beta _1 >1$ and $\\beta _2>1 $ such that $(n^{\\beta _1} b_n)_n$ is {nonincreasing} and $(n^{\\beta _2}b_n)_n$ is increasing. Let $\\beta >\\beta _2$ and let $B>0$. If for each positive increasing concave function $h$ such that $h(t^\\beta)$ is convex we have\n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{n}h (\\frac{1}{B}a_n) \\leqslant \\displaystyle \\sum _{n}h (b_n) \\leqslant \\displaystyle \\sum _{n}h(B a_n),\n\t\t\t\\]\n\t\t\tthen $a_n \\asymp b_n$.\n\t\t\\end{lem}\n\t\t\\begin{proof}[\\textbf{Proof of assertion \\ref{3i} of Theorem \\ref{B}}] \n\t\t\tSuppose that $(s_n(I_g))_n \\in {\\mathcal{R}}_{\\gamma, \\alpha}$. Then there exists a {nonincreasing} sequence $(x_n)_n$ such that $(n^\\gamma x_n)_n$ is increasing, $(n^\\alpha x_n)_n$ is {nonincreasing}, and $x_n \\asymp s_n(I_g)$. We have to prove that $x_n \\asymp \\sqrt{\\tau _n(\\nu_g)}$. \n\t\t\tLet $\\beta>0$ be such that $\\beta \\alpha >1$. Put $\\beta _1= \\beta \\alpha$ and $\\beta _2= \\beta \\gamma$. Let $h$ be a positive increasing concave function such that $h(0)= 0$ and {$h_\\beta \\colonequals h(t^\\beta) $} is convex. The function $h_\\beta$ satisfies the assumptions of Corollary \\ref{Ctrace}, with $\\varepsilon = 1\/\\beta$. Then there exists $B_1$ such that\n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h_\\beta \\left (\\frac{1}{B_1}\\sqrt{\\tau _j(\\nu_g)}\\right ) \\leqslant \\displaystyle \\sum _{j}h_\\beta (x_j).\n\t\t\t\\]\n\t\t\tBy the first assertion of Theorem \\ref{Hardy}, we have\n\t\t\t\\[\n\t\t\tx_n \\asymp s_{p n}(I_g) \\lesssim \\sqrt{\\tau_n (\\nu_g)}, \\quad n\\geqslant 1.\n\t\t\t\\]\n\t\t\tThen there exists $B>0$ such that \n\t\t\t\\[\n\t\t\t\\displaystyle \\sum _{j}h_\\beta\\left (\\frac{1}{B}\\sqrt{\\tau _j(\\nu_g)}\\right ) \\leqslant \\displaystyle \\sum _{j}h_\\beta (x_j)\\leqslant \\displaystyle \\sum _{j}h_\\beta\\left (B\\sqrt{\\tau _j(\\nu_g)}\\right ).\n\t\t\t\\]\n\t\t\tLemma \\ref{lemEE} yields $x_n^{\\beta} \\asymp \\tau ^{\\beta\/2}_n (\\nu_g)$, that is, $x_n \\asymp \\sqrt{\\tau _n(\\nu_g)}$.\\\\\n\t\t\tThe result in the case $(\\tau _n (\\nu_g))_n \\in {\\mathcal{R}}_{2\\gamma, 2\\alpha}$ can be proved in the same way.\n\t\t\\end{proof}\n\t\t\n\t\tUsing Theorems \\ref{Hardy} and \\ref{Ctrace} with the same line of reasoning as in the proofs of Theorem 1.2 and Proposition 5.2 of \\cite{BEMN}, one can obtain the following result. We omit the proof here.\n\t\n\t\t\\begin{cor}\n\t\t\tLet $g\\in VMOA$. We have\\\\\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item $s_n(I_g)= o\\left ( \\frac{1}{n}\\right ) \\quad \\Longrightarrow \\quad I_g =0.$\n\t\t\t\t\\item $s_n(I_g)= O\\left ( \\frac{1}{n}\\right ) \\quad \\iff \\quad g' \\in H^1.$\n\t\t\t\\end{enumerate}\n\t\t\\end{cor}\n\t\n\t\n\t\t\n\t\n\t\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\bibliographystyle{plain}\n\t\n\t\t\\bigskip\n\t\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\n\nAn aspect of great importance in nuclear structure calculations at any\nexcitation energy concerns with the role of the effective interaction.\nAt low energies this problem has generated a considerable body of work\nin the last twenty years. On the contrary, this is a question not \nstudied yet in deep in the literature for higher energies.\n\nGiant resonances show an intricate mixture of multipolarities and the\nstudy of how the interaction affects it is a difficult\ntask. In the quasi--elastic peak region, the problem of the\nlongitudinal and transverse separation has occupied most of the\ninvestigations carried out till now and the discussion of the effects\ndue to changes in the effective interaction have not been considered\nin detail.\n\nAs an example, we mention the considerable number of Random--Phase \nApproximation (RPA) type calculations performed in this energy region,\nmuch of them using residual interactions which include basically a \nzero--range term plus meson--exchange potentials corresponding to\n$\\pi$, $\\rho$ and, eventually, other mesons~\\cite{al84}--\\cite{gi97}. \nAn important point concerning the interaction refers to the values\nchosen for the parameters entering in the zero--range piece.\nHowever, and to the best of our knowledge, only in\nRef.~\\cite{gi97} a certain discussion relative to the effects of\nvarying these parameters can be found. In fact, the common\npractice is to pick an interaction from the literature, which\nusually corresponds to a parameterization fixed for low energy\ncalculations, and afterwards use it to evaluate quasi--free\nobservables sometimes without taking care of the effective theory in \nwhich the interaction was adjusted. It is obvious that, to a certain\nlevel, doubtful results are possible because of the known link\nbetween effective theory and interaction.\n\nIn this work we want to address this question and investigate if \ndifferent parametrizations of the interaction can produce noticeable \ndifferences in the results and the extent to which the use of an \ninteraction fixed for a given effective theory affects the results \nobtained within a different one. In Sec.~II we give the details about \nthe effective theories and interactions used to perform the\ncalculations. In Sec.~III we show and discuss the results we have\nobtained. Finally, we present our conclusions in Sec.~IV.\n\n\\section{EFFECTIVE THEORIES AND INTERACTIONS}\n\nThe first interaction we consider in this work is the so--called \nJ\\\"ulich--Stony Brook interaction~\\cite{sp80} which is an effective \nforce widely used for calculations in the quasi--elastic peak. It is \ngiven as follows: \n\\begin{equation}\n\\label{int-I}\n\\displaystyle\nV^{\\rm I}_{\\rm res} \\, = \\, V_{\\rm LM} \\, + \\, V_\\pi \\, + \\,\n\\tilde{V}_\\rho \\, . \n\\end{equation}\nHere $V_{\\rm LM}$ is a zero--range force of Landau--Migdal type,\nwhich takes care of the short--range piece of the NN\ninteraction: %\n\\begin{equation}\nV_{\\rm LM} \\, = \\,C_0 \\,[ g_0\\, \n{\\mbox{\\boldmath $\\sigma$}}^1\\cdot {\\mbox{\\boldmath $\\sigma$}}^2 \n+\\, g_0^\\prime\\,\n{\\mbox{\\boldmath $\\sigma$}}^1\\cdot {\\mbox{\\boldmath $\\sigma$}}^2\n{\\mbox{\\boldmath $\\tau$}}^1 \\cdot {\\mbox{\\boldmath $\\tau$}}^2]\n\\, .\n\\end{equation}\nOn the other hand, a finite--range component generated by the\n($\\pi$+ $\\rho$)--meson exchange potentials is also included. The\ntilde in $\\tilde{V}_\\rho$ means that the bare $\\rho$--exchange\npotential is slightly modified in order to take into account the\neffect of the exchange of more massive mesons. In particular, a \nfactor $r=0.4$ is multiplying the finite--range non--tensor piece\nof the potential (see Ref.~\\cite{sp80} for details). This force\nwas fitted to reproduce low energy magnetic properties in the\nlead region (specifically, magnetic resonances in $^{208}$Pb and\nmagnetic moments and transition probabilities in the\nneighboring nuclei). The calculations were performed in the\nframework of the RPA and Woods--Saxon single--particle wave\nfunctions were used in the configuration space. The values \n$g_0=0.57$ and $g_0^\\prime=0.717$ (with $C_0=386.04$~MeV~fm$^3$) \nwere found to be adequate to describe the properties \nstudied.\n\nAs previously stated, this interaction has been considered in \ndifferent calculations in the quasi--elastic peak (see\ne.g.~\\cite{gi97}). The problem arise because some of them have been\ndone within the Fermi gas (FG) formalism, with local density\napproximation to describe finite nuclei, in the ring approximation \n(RA), where the exchange terms are not taken into account, and with \nthe full unmodified $\\rho-$exchange potential. Under these \ncircumstances, the possible effects in the nuclear \nresponses due to the inconsistency between the model and the effective\ninteraction could be non--negligible. This is precisely the first aspect\nwe want to investigate. To do that we compare the responses obtained\nwith the J\\\"ulich--Stony Brook interaction with those calculated with\na second effective force of the form: \n\\begin{equation}\n\\label{int-II}\n\\displaystyle\nV^{\\rm II}_{\\rm res} \\, = \\, V_{\\rm LM} \\, + \\, V_\\pi \\, + \\,\nV_\\rho \\, ,\n\\end{equation}\nby considering the same values for the zero--range parameters in both\ncases. The force in Eq.~(\\ref{int-II}) only differs from \n$V^{\\rm I}_{\\rm res}$ in the $\\rho$-potential which, in this case,\ndoes not include any reduction factor. Both RPA and RA effective\ntheories are used to analyze the results.\n\nA second question of interest to us is to determine how the change of\nthe zero--range parameters affects the responses calculated within a\ngiven theory. This will inform us about the necessity of considering\nor not in detail the role of these parameters. This aspect is analyzed\nby considering $V^{\\rm II}_{\\rm res}$ with parameters $g_0$ and \n$g_0^\\prime$ fixed, as in the case of the J\\\"ulich--Stony Brook \ninteraction, to reproduce some low energy properties in the lead\nregion (see details in the next section). It is worth to point out\nthat $V^{\\rm II}_{\\rm res}$ is precisely the interaction used in \npractice in much of the calculations mentioned above and that is why we \nwant to use it for this analysis.\n\nOur analysis focuses on the transverse response functions in the\nquasi--elastic peak. We will not consider the longitudinal ones\nbecause they are strongly influenced by the spin independent pieces of\nthe interaction (in particular, the $f_0$ and $f_0^\\prime$ channels)\nand these are difficult to fix at low energy because of the role\nplayed by the scalar mesons not usually taken into account. \n\n\\section{RESULTS OF THE CALCULATIONS}\n\nThe investigation of the various questions we are interested in has been \ncarried out by comparing different calculations of the transverse (e,e')\nresponses in $^{40}$Ca for three different momentum transfer ($q=300$,\n410 and 550~MeV\/$c$).\n\n\\newpage\n\n\\vspace*{1cm}\n\n\\begin{center}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(100,180)(35,0)\n\\put(0,0){\\ewxy{figu1.ps}{180mm}}\n\\end{picture} \n\\end{center} \n\n\\vspace*{-1.2cm}\n\\noindent\n{\\small {\\sc Fig.~1}. Transverse nuclear responses for $^{40}$Ca, \ncalculated for the three momentum transfers we have considered in this\nwork. Dotted lines correspond to an RPA calculation with \n$V^{\\rm I}_{\\rm res}$, while solid curves represent the RA results for\n$V^{\\rm II}_{\\rm res}$. In both cases the values $g_0=0.57$ and \n$g_0^\\prime=0.717$ (with $C_0=386.04$~MeV~fm$^3$) have been used. \nDashed curves give the free FG responses. In all the calculations a \nvalue of $k_{\\rm F}=235$~MeV\/$c$ has been used.}\n\\vspace{1cm}\n\n\\newpage\n\nFirst we have study the effects produced when an effective\ninteraction, which has been determined in a given effective theory \n(e.g. RPA), is used to calculate (e,e') transverse responses in a \ndifferent framework (e.g. RA).\n\nBy considering the parameterization of Ref.~\\cite{sp80} (that is \n$g_0=0.57$, $g_0^\\prime=0.717$ and $C_0=386.04$~MeV~fm$^3$), we have \ncarried out two different calculations, the results of which are shown\nFig.~1. Therein, solid curves correspond to the calculations performed\nin the FG approach within the RA and with the interaction \n$V^{\\rm II}_{\\rm res}$ in Eq.~(\\ref{int-II}). On the other hand, \ndotted lines have been obtained within the RPA, also for the FG. The \nmodel used in this case is the one developed in Ref.~\\cite{ba96}, \nwhich, contrary to what happens for the RA approach, includes \nexplicitly the exchange terms in the RPA expansion. In this case we \nhave used the interaction $V^{\\rm I}_{\\rm res}$ in Eq.~(\\ref{int-I}) \nand we have adopted the factor $r=0.4$, which is consistent with the \nparameterization used. Also in Fig.~1, we have plotted the free FG \nresponses for comparison (dashed curves). \n\nThe first comment one can draw from these results is that the use of \nthe interaction, as it was fixed at low energy, leads to transverse \nresponses which are quite different from those obtained in the RA\n(with $V^{\\rm II}_{\\rm res}$) calculation, though the differences \nreduce with increasing momentum transfer. As we can see, the results \nobtained in the RPA are peaked at lower energies and this is a clear \nevidence of a more attractive residual interaction. It is \nstraightforward to check this point because the central piece of the \n$V^{\\rm I}_{\\rm res}$ is attractive, while the contrary happens for\n$V^{\\rm II}_{\\rm res}$, at least for $q\\leq 2k_{\\rm F}$. On the other \nhand it is interesting to note how the RA results are more similar to \nthe free response as long as $q$ increases, while the same does not \noccurs for the RPA responses. \n\nObviously, the reason for the discrepancies between both calculations \ncan be ascribed to the two basic ingredients of the effective theories\nused in each case: the exchange terms, which are included in the RPA \ncalculations but not in the RA ones, and the reduction factor $r$ \nmodifying the $\\rho-$exchange potential.\n\nBefore going deeper in this question, it is worth to comment on the\nnuclear wave functions used in the calculations discussed above. As in\nany FG type calculation, plane--waves have been considered here to \ndescribe the single--particle states. The fact that the interaction was \nfixed in a framework which considered microscopic RPA wave functions,\nbased on Woods--Saxon single--particle states, is an obvious \ninconsistency. Despite that, it has been shown~\\cite{am94,am96}\nthat, in this energy region, the details concerning the nuclear wave \nfunctions are not extremely important and, at least to some extent, the \nshell--model response can be reasonably described with the FG model, \nprovided an adequate value of the Fermi momentum, $k_{\\rm F}$, is\nused. In the present work, where we study the response in $^{40}$Ca,\nwe have taken $k_{\\rm F}=235$~MeV\/$c$ which gives a good agreement \nbetween FG and finite nuclei calculations~\\cite{am94}.\n\nWe come back to investigate the reasons for the large discrepancy\nbetween the RPA and RA calculations presented above. To do that we have \ndone two new calculations: RA with $V^{\\rm I}_{\\rm res}$ and RPA with \n$V^{\\rm II}_{\\rm res}$. These calculations have been compared with the\ntwo previous ones by means of the two following quantities:\n\\begin{equation}\n\\label{relexc}\n\\displaystyle\n\\gamma_{\\rm exc}^r(q,\\omega) \\, = \\,\n\\displaystyle\n\\frac{R_T^{{\\rm RPA}(r)}(q,\\omega) \\, - \\, \n R_T^{{\\rm RA}(r)}(q,\\omega) } \n{R_T^{{\\rm RA}(r)}(q,\\omega)}\n\\end{equation}\nand\n\\begin{equation}\n\\label{relfac}\n\\displaystyle\n\\gamma_r^{\\rm mod}(q,\\omega) \\, = \\,\n\\displaystyle\n\\frac{R_T^{{\\rm mod}(r=1.0)}(q,\\omega) \\, - \\, \n R_T^{{\\rm mod}(r=0.4)}(q,\\omega) } \n{R_T^{{\\rm mod}(r=0.4)}(q,\\omega) } \\, . \n\\end{equation}\n\nThe first one gives us information about the effect of the consideration\nof the exchange terms in the calculation. The corresponding results have\nbeen plotted in Fig.~2 (left panels). The first aspect to be noted is \nthat the exchange terms produce effects considerably larger for \n$V^{\\rm I}_{\\rm res}$ (solid lines) than for $V^{\\rm II}_{\\rm res}$ \n(dashed curves). These effects reduce with increasing momentum\ntransfer and they are rather small for $V^{\\rm II}_{\\rm res}$ above\n$q=410$~MeV\/$c$. \n\nOn the other hand, the effect of the reduction factor $r$ in the \n$\\rho$--exchange potential is measured with the parameter $\\gamma_r$. \nThe values of this parameter for the two effective models considered, \nthese are RPA and RA, are shown in Fig.~2 (right panels), with solid\nand dashed curves respectively. It is apparent that the effects of \nconsidering the $r$ factor are much larger than those due to the \nexchange. In general they are more important for the RA calculations \nthan for the RPA ones, and reduce the higher $q$ is. \n\n\\newpage\n\n\\vspace*{1cm}\n\n\\begin{center}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(100,180)(30,0)\n\\put(0,0){\\ewxy{figu2.ps}{180mm}}\n\\end{picture} \n\\end{center} \n\n\\vspace*{-1.2cm}\n\\noindent\n{\\small {\\sc Fig.~2}. Left panels: $\\gamma_{\\rm exc}$, in percentage, \nas defined in Eq.~(\\ref{relexc}). Dashed (solid) curves give the results \nobtained for $r=1.0 (0.4)$. Right panels: $\\gamma_r$, in \\%, calculated \nas in Eq.~(\\ref{relfac}), for RPA (solid curves) and RA (dashed curves).}\n\\vspace{1cm}\n\n\\newpage\n\nThe first conclusion to be noted is that when using a given\ninteraction is mandatory to take care of the effective theory where \nits parameterization was fixed. The change of the framework produces \nresults which could not be under control.\n\nThe open question in this respect is how different becomes the \nresponses calculated within different effective theories but with an\ninteraction fixed consistently with the theory. This is the second\naspect we investigate. To do that we have considered the \n$V^{\\rm II}_{\\rm res}$ and have determined the parameters $g_0$ and \n$g_0^\\prime$ of the Landau--Migdal piece in such a way that the \nenergies and B-values of the two $1^+$ states in $^{208}$Pb at 5.85 \nand 7.30~MeV are reproduced. This has been done both in RPA and RA. \nThe reason for choosing these two states lies in their respective \nisoscalar and isovector character, what makes them particularly \nadequate to permit the determination of both parameters almost \nindependently. The values obtained in this procedure are shown in \nTable~I. It is remarkable the small value of $g_0$ needed for the RA\ncalculation. A similar result is found when a pure zero--range \nLandau--Migdal interaction is adjusted, with the same criterion, in \nRPA type calculations (see Refs.~\\cite{co90,hi92}). This points out\nthe importance of the exchange, at least at low energy.\n\n\\vspace{.5cm}\n\\noindent\n{\\small {\\sc TABLE I.} Values of the Landau--Migdal parameters $g_0$ and \n$g_0^\\prime$ obtained in the procedure of fixing the effective interaction \n$V^{\\rm II}_{\\rm res}$ (see text). The values quoted ``RPA'' (``RA'') \ncorrespond to calculations performed with (without) the consideration\nof the exchange terms.}\n\\begin{center}\n\\begin{tabular}{cccc}\n\\hline\\hline\nEffective theory && $g_0$ & $g_0^\\prime$ \\\\ \\hline\nRPA && $~0.470$ & $~0.760$ \\\\\n RA && $~0.038$ & $~0.717$ \\\\\\hline\\hline\n\\end{tabular}\n\\end{center}\n\n\\vspace{1cm}\n\nWith the interaction fixed in this way we have evaluated the\ntransverse responses for the three momentum transfer we are\nconsidering throughout this work. The results are shown in Fig.~3\nwhere dotted (solid) curves correspond to the RPA (RA) calculations.\nDashed lines represent the free FG responses. As we can see, the\ndifferences between the results obtained with the two effective\ntheories are now much smaller than in Fig.~1.\n\n\\newpage\n\n\\vspace*{1cm}\n\n\\begin{center}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(100,180)(35,0)\n\\put(0,0){\\ewxy{figu3.ps}{180mm}}\n\\end{picture} \n\\end{center} \n\n\\vspace*{-1.2cm}\n\\noindent\n{\\small {\\sc Fig.~3}. Transverse nuclear responses for $^{40}$Ca,\ncalculated with the $V^{\\rm II}_{\\rm res}$ interaction. Dotted lines \ncorrespond to an RPA calculation while solid curves represent the RA \nresults. The values of $g_0$ and $g_0^\\prime$ in Table~I have been \nused. Dashed curves give the free FG responses. In all the \ncalculations a value of $k_{\\rm F}=235$~MeV\/$c$ has been used.}\n\n\\vspace{1cm}\n\n\\newpage\n\nTwo points deserve a comment. First, it is clear that the large \ndifferences observed between the RPA calculation here discussed and\nthat shown in Fig.~1 are mainly due to the presence of the reduction \nfactor $r=0.4$ in the $V^{\\rm I}_{\\rm res}$ interaction. Second,\nthe similitude of the results obtained with the two calculations done\nnow with $V^{\\rm II}_{\\rm res}$, shows up the relevance of the link \nbetween effective theories and interactions.\n\nThe last aspect we want to analyze is how the responses calculated in a\ngiven approach change when the zero--range parameters are modified. In\nother words, we want to determine what is the role of these\nparameters. How $g_0^\\prime$ affects the responses is a point which\nhas been investigated with a certain detail in different previous\nworks (see e.g., Ref.~\\cite{ba96}) and then we focus here in $g_0$.\nIts influence can be seen in Fig.~4, where we compare the responses\nplotted in Fig.~3 (solid curves), with those obtained by changing the \n$g_0$ parameter in order to use values considered by different\nauthors. Dashed--dotted curves correspond to $g_0=0$. Dashed lines \nrepresent the responses obtained with $g_0=0.70$ (0.57) for the RPA \n(RA) calculation. The values of $g_0^\\prime$ have not been changed. \nThe first point to be noted is the insensibility of the RA responses \nto the changes in $g_0$. As we can see, strong changes in $g_0$ \nproduce almost no effect on the RA result. This can be easily \nunderstood because in the ring series the $g_0$ contribution is \nweighted with the magnetic moment $\\mu_s^2$ while the $g_0^\\prime$ \npiece appears with $\\mu_v^2$. That means, the $g_0$ contribution is\n$\\mu_s^2\/\\mu_v^2 \\, \\approx \\, 1\/28$ of the $g_0^\\prime$\ncontribution. The situation is different in the RPA case, where the \n$g_0$ contribution is as important as the $g_0^\\prime$ one because \nof the presence of the exchange terms (see Ref.~\\cite{ba96}). This \nmakes that some of the RA calculations performed by other authors can \nbe considered as ``consistent'' in practice, of course despite the \nfact that these parametrizations are unable to reproduce low energy \nproperties. For example, in Ref.~\\cite{gi97}, the parameterization of \nthe J\\\"ulich--Stony Brook interaction was considered and this \ncoincides with one of those used here ($g_0=0.57$ and \n$g_0^\\prime=0.717$).\n\nThe results obtained in this work open a series of questions which we\nconsider worth for nuclear calculations in this energy region. In the\nfollowing we enumerate and comment them:\n\n\\newpage\n\n\\vspace*{.5cm}\n\\begin{center}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(100,170)(24,0)\n\\put(0,0){\\ewxy{figu4.ps}{170mm}}\n\\end{picture} \n\\end{center} \n\n\\vspace*{-1.5cm}\n\\noindent\n{\\small {\\sc Fig.~4}. $R_T$ responses calculated in the RPA (left\npanels) and RA (right panels). Solid curves correspond to\nthe parametrizations of Table~I. Dashed--dotted curves have been \nobtained with $g_0=0$, while the dashed ones correspond to \n$g_0=0.70$ (0.57) for the RPA (RA) calculation,\nwith the same values of $g_0^\\prime$ as for the solid curves.}\n\n\\newpage\n\n\n\\begin{enumerate}\n\n\\item It has been shown that the strength of the tensor piece of \n$V^{\\rm I}_{\\rm res}$ is too strong to describe low energy properties \n(see, e.g., Ref.~\\cite{co90}) and different mechanisms have been \nproposed to cure this problem (core--polarization effects~\\cite{na85},\ntwo--particles two--holes excitations~\\cite{dr86}, {\\it in--medium} \nscaling law~\\cite{br91}, etc.) The role of the tensor part of the \ninteraction in the quasi-elastic peak should be investigated in order \nto establish the effective force to be used.\n\n\\item The presence of the exchange terms increase the sensitivity of \nthe responses to the details of the interaction. How important can be\nthe interference between these terms and other physical mechanisms\nbasic in this energy region (such as, e.g., meson--exchange currents,\nfinal state interactions, short--range correlations, etc.) is a matter\nof relevance in order to fully understand the nuclear response. The\nanalysis of the possible differences between RA and RPA with respect \nto these effects is of special interest in view of the fact that RA \ncalculations are the most usual in the quasi--elastic peak.\n\n\\item The procedure of fixing the interaction is basic in order to deal\nwith the possibility of having an unique framework to calculate the \nnuclear response at any momentum transfer and excitation energy. The \nproblem of developing such ``unified'' model is still unsolved, but\nthe cross analysis of low energy nuclear properties and quasi--elastic\npeak responses could give valuable hints.\n\n\\end{enumerate} \n\n\\section{CONCLUSIONS}\n\nIn this work we have analyzed the role of the effective interaction in\nthe quasi--elastic peak region by comparing the results obtained with\ndifferent effective theories and forces previously fixed in order to \ngive a reasonable description of several low energy nuclear properties.\n\nSome conclusions can be drawn after our analysis. First, it has been \nfound that the interaction plays a role that, similarly to what\nhappens at low excitation energy, cannot be neglected. The particular \npoint to be noted is the necessity of using effective interactions \nwhich have been fixed within an effective theory.\n\nSecond, the procedure we have followed to perform the calculations, \nthat is to determine the interaction at low energy before calculating\nat the quasi--elastic peak, seems to be adequate to look for an\n``unique'' framework to calculate the nuclear response in different\nenergy and momentum regimes. \n\nThe role of the tensor piece of the interaction must be\ninvestigated. At low energy is a basic ingredient of the nuclear\nstructure calculations. Thus it is important to disentangle its\ncontribution in other excitation energy regions. Additionally, it \nseems encouraging to analyze the problem by including other physical \nmechanisms (meson--exchange currents, short--range correlations, \nfinal state interactions, etc.) which are known to be important in \nthe description of the nuclear response and which depend on the \ninteraction. \n\n\\acknowledgements\n\nDiscussions with G. Co' are kindly acknowledged.\nThis work has been supported in part by the DGES (Spain) under \ncontract PB95-1204 and by the Junta de Andaluc\\'{\\i}a (Spain). \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe local mass density of massive black holes residing in the nuclei\nof galaxies (Magorrian et al 1998; Richstone et al 1998) is in good\nagreement with that expected from the intensity of the X-ray\nbackground at 30~keV (Fabian \\& Iwasawa 1999; Salucci et al 1999). The\nX-ray background is assumed to be due to radiatively-efficient\naccretion onto black holes at redshifts $z\\sim 1-2$, with accretion\naccounting for the bulk of their mass. An important requirement is the\npresence of large column densities of absorbing matter, $N_{\\rm H}\\sim\n10^{24}\\hbox{$\\cm^{-2}\\,$}$, around many of the X-ray sources. Intrinsically they\nare assumed to radiate the broad-band spectrum of a typical quasar\n(Elvis et al 1994), with a large UV bump, and a power-law X-ray\ncontinuum of photon index, $\\Gamma\\approx2$, up to a few $100{\\rm\\thinspace keV}$.\nPhotoelectric absorption in this matter then hardens the observed\nspectrum considerably below 30~keV so that the cumulative spectrum\nfrom a population of sources with a range of column densities and\nredshifts resembles that of the observed X-ray Background (Setti \\&\nWoltjer 1989; Madau et al 1994; Celotti et al 1995; Matt \\& Fabian 1994;\nComastri et al 1995; Wilman \\& Fabian 1999). Provided that the sources\nare not too Thomson-thick, it is only above about 30~keV that the\nunabsorbed radiation from growing black holes is observed.\n\nTwo major issues for the model are discussed here; a) the typical mass\nand luminosity of the obscured objects and b) the source of the\nobscuring matter. The first a) is an issue because few luminous\nobscured quasars $L(2-10{\\rm\\thinspace keV})>10^{44}\\hbox{$\\erg\\s^{-1}\\,$}$ have yet been found by\nX-ray (or other waveband) surveys or studies (Boyle et al 1998;\nBrandt et al 1997; Halpern et al 1999), so it might seem that highly\nobscured objects are only of low luminosity. This could be a serious\nproblem for the growth of black holes of mass $10^9\\hbox{$\\rm\\thinspace M_{\\odot}$}$ or more,\nwhich might have luminosities considerably more than that. The second\nb) requires large column densities to obscure most of the Sky as seen\nfrom the sources, in order that most accretion power in the Universe\nis absorbed (Fabian \\& Iwasawa 1999 estimate that about 85 per cent is\nabsorbed). We have previously argued that a circumnuclear starburst is\nresponsible for the obscuration in nearby objects (Fabian et al 1998),\nbut in its simple form could be difficult to sustain if the central\nblack hole takes a Gyr or more to grow to its present mass.\n\nA model is presented here in which a black hole grows with its\nsurrounding stellar spheroid by the accretion of hot gas. The gas is\nalso cooling and forming a distributed gaseous cold component within\nwhich stars slowly form. In the central regions the cold component\nprovides the required column density for absorption of the quasar\nradiation from the accreting black hole. The quasar is assumed to make\na slow wind, which eventually becomes powerful enough to blow away the\nabsorbing gas, probably due to a wind, and an unobscured quasar is\nseen (see Silk \\& Rees 1998 and Blandford 1999 for a similar wind\nmodel). Because the fuel supply for the black hole has also been\nejected however, the quasar dies when its accretion disc is exhausted.\nThe high obscuration phase occurs only during the growing phase and\nthus at redshifts greater than unity which have not been explored yet\nin the hard X-ray band. Reprocessing of the radiation by the absorbing\ngas into far infrared emission may make these objects detectable in\nthe sub-millimetre band.\n\n\\section{Growth of the black hole}\n\nMagorrian et al (1998) find from the demographics of nearby massive\nblack holes that the mass of the hole $M_{\\rm BH}$ is proportional to\nthe mass of the surrounding spheroid $M_{\\rm sph}$ or bulge; $M_{\\rm\nBH}\\approx 0.005M_{\\rm sph}$. There is considerable scatter about this\nrelation of order $\\pm1$~dex. This can be combined with the mass\nfunction of galactic bulges, where most of the mass resides. Salucci et\nal (1999) use the Schechter function luminosity function of bulges to\nobtain the local mass function of black holes. Most of the mass lies\nin black holes of individual mass around the break in the function,\ni.e. $3\\times 10^8\\hbox{$\\rm\\thinspace M_{\\odot}$}$.\n\nTaking this as a typical mass, and assuming growth by accretion with a\nradiative efficiency of $\\eta=0.1 \\eta_{-1}$, the bolometric\nluminosity of the final object $L_{\\rm B}=3\\times 10^{46}-3\\times\n10^{44}\\hbox{$\\erg\\s^{-1}\\,$}$ if it is at one and 0.01 times the Eddington limit\n$L_{\\rm Edd}$, respectively. The Salpeter time for the growth of the\nblack hole, i.e. the mass doubling time, $t_{\\rm s}=3\\times 10^7\n(L_{\\rm Edd}\/L_{\\rm B})\\eta_{0.1}{\\rm\\thinspace yr}.$ If we now assume that the\ninitial mass of all massive black holes is less than that of the\ncentral black hole in our Galaxy, say $\\sim 10^6\\hbox{$\\rm\\thinspace M_{\\odot}$}$, and has grown\nto $3\\times 10^6\\hbox{$\\rm\\thinspace M_{\\odot}$}$ within 3~Gyr, which corresponds to $z\\sim2$,\nthen $8t_{\\rm s}<3\\times 10^9{\\rm\\thinspace yr}$ and the black hole must have grown\nat a rate of at about 10 per cent of the Eddington rate or more. Thus\nover the final $3\\times 10^8{\\rm\\thinspace yr},$ $L_{\\rm B}>3\\times 10^{45}\\hbox{$\\erg\\s^{-1}\\,$}$.\nFrom the work of Elvis et al (1994) the observed 2--10~keV luminosity\nis about 3 per cent of the bolometric one for a quasar so we can\nconclude that $L(2-10{\\rm\\thinspace keV})>10^{44}\\hbox{$\\erg\\s^{-1}\\,$}$ during the growth of a\ntypical black hole and we are dealing with powerful, obscured quasars.\n\nIt is clear that the objects making up the X-ray background, and the\nradiative growth phase of most of the mass in nearby black holes are\nnot represented by any object observed so far. \n\n\\section{The obscuration}\n\nThe spectrum of the X-ray Background requires that most accretion\npower is absorbed. This means that some absorption (say $N_{\\rm\nH}>10^{22}\\hbox{$\\cm^{-2}\\,$}$) occurs in 90 per cent of objects and heavy\nabsorption ($N_{\\rm H}>10^{24}\\hbox{$\\cm^{-2}\\,$}$) in 30--50 per cent of them. 10\nper cent are unabsorbed and are the quasars identified in blue optical\nsurveys or are bright in the soft X-ray band of ROSAT. So far these\ncould be distinct different populations of quasars or different phases\nin the growth of all quasars.\n\nSuch high absorbed fractions mean that the covering fraction of the\nsky by high column density material seen from a growing black hole\nmust approach $4\\pi {\\rm\\thinspace sr}$. This cannot be provided by any thin disc or\nby the standard torus of unified models. The absorbing matter is\nprobably concentrated within the innermost few 100~kpc, or its total\nmass becomes high. It must presumably be cold and fairly neutral, or\nit will not absorb the X-rays. At first sight this is at variance with\nit being space covering, especially if it must be so for a Gyr or\nmore. Such matter should collide and dissipate into a disclike\nstructure.\n\nFollowing our earlier work on a circumnuclear starburst (Fabian et al\n1998), it is plausible that collisions do take place leading to\ndissipation but that some massive star formation occurs in the shocked\nand cooled dense gas. The winds and supernovae from those stars then\nsupply the energy to keep the rest of the cold matter in a chaotic and\nspace covering state.\n\nA more detailed model can be developed by assuming that the black hole\nis growing at the same time as the galaxy does. The stellar spheroid\ncontinues to grow by cooling of the gas heated by gravitational\ncollapse of the protogalaxy. It is likely that the hot phase density\nwhile the galaxy continues to grow is the maximum possible, which\nmeans that the radiative cooling time of the gas equals the\ngravitational infall time. This condition has been studied in the\ncontext of quasar fuelling and growth by Nulsen \\& Fabian (1999). The\ngas accretes in a Bondi flow, probably forming a disc well within the\nBondi radius. The accretion rate is such that it can typically be 10\nper cent of the Eddington value.\n\nThe situation as envisaged here is essentially a maximal cooling flow\nand we can use the properties of observed cooling flows (Fabian 1994\nand references therein) to indicate how the cooled gas is distributed\nand how it may lead to absorption. X-ray observations of cluster\ncooling flows show that the mass deposited by cooling is distributed\nwith $\\dot M(L^{\\rm\nw}_{\\rm Edd}.\\eqno(10)$$ If $L_{\\rm w}=a L_{\\rm Edd}$ then the limit occurs\nwhen $$a {{4 \\pi G M m_{\\rm p} c}\\over \\sigma_{\\rm T}}= {{v^4 f v_{\\rm\nw}}\\over{2G}},\\eqno(11)$$ or at the critical mass $$M_{\\rm c}={v^4\\over{2\\pi\nG^2 m_{\\rm p}N_{\\rm T}}}{v_{\\rm w}\\over c}{f\\over a}={M_{\\rm T}\\over\n2a}{v_{\\rm w}\\over c}.\\eqno(12)$$ Thus if $v_{\\rm w}\\sim 0.1 c$, and $a\\sim\n0.1$ then $M_{\\rm c}\\sim M_{\\rm T}\/2.$ Assuming that most of the mass\nwithin $r_{\\rm T}$ lies in the central black hole again means that\n$M_{\\rm BH}\\sim M_{\\rm T}\\propto M_{\\rm sph}.$ Most of the power\nduring the main growth phase of a massive black hole is then radiated\ninto gas with a Thomson depth of about unity, as required for\nmodelling the X-ray Background spectrum.\n\nClearly $f$ cannot equal unity if a galaxy is to be formed. In\npractice it will be a function of time. The important issue here is\nthat it cannot be small, i.e. it probably lies in the range of 0.1 --\n0.5.\n\nThe black hole mass is therefore proportional to the spheroid mass, as\nobserved (Magorrian et al 1998). The quasar is now unobscured and is\nobservable as an ordinary blue excess object for as long as it has\nfuel. A reasonable estimate would be about a million yr for the disc to\nempty. The ejection phase is tentatively identified with broad\nabsorption line quasars (BAL quasars; see e.g. Weymann 1997).\n\nThe normalization for the relation between $M_{\\rm BH}$ and $M_{\\rm\nsph}$ is obtained by using the Faber-Jackson relation given by Binney\n\\& Tremaine (1992), where $v=220(L\/L_\\star)^{0.25}\\hbox{$\\km\\s^{-1}\\,$}$ and\n$L_\\star=4\\times 10^{10}\\hbox{$\\rm\\thinspace L_{\\odot}$}$ in the V-band, and a mass-to-light\nratio in that band of 6 (both for a Hubble constant of $50\\hbox{$\\kmps\\Mpc^{-1}$}$).\nThe result is $${M_{\\rm T}\\over M_\\star} =0.01\\eqno(13) $$ so\n$${M_{\\rm BH}\\over M_{\\rm sph}}\\approx 0.005,\\eqno(14)$$ for the\nvalues of $a$ and $v_{\\rm w}$ above. This is in good agreement with\nthe relation found by Magorrian et al (1998). In detail, the weak\nvariations in $M\/L$ such as summarized by Salucci \\& Persic (1997)\ncould shift the observed relation from being strictly linear. \n\nNote that the ordinary quasar phase markes the end of the main growth\nphase of both the black hole and its host spheroid. The central black\nhole in galaxies therefore has a profound effect on the whole galaxy,\nin providing a limit to the stellar component. In this context, Silk\n\\& Rees (1998) have shown that a powerful quasar wind could end the\nformation of a galaxy and Blandford (1999) has noted that it could\nprevent the formation of a galactic disc.\n\n\\section{Discussion}\n\nIt has been shown here that a black hole, growing by accretion in the\ncentre of a young, isothermal, spheroidal galaxy which has a large\nmass component of cold gas, will appear highly obscured ($\\tau_{\\rm\nT}>1$). Provided that a significant fraction of the accretion power,\nassumed to be close to the Eddington limit for the black hole, is\nreleased as a sub-relativistic wind, then, as suggested by Silk \\&\nRees (1998) the cold, and accompanying hot, gas components are ejected\nfrom the galaxy when the Thomson depth to the outside $\\tau_{\\rm T}$\ndrops to about unity. The central accretion source then appears as an\nunobscured quasar which lasts until its disc empties. The growth of\nboth the central black hole and the stellar body of the galaxy then\nterminate, unless fresh gas and stars are brought in from outside, for\nexample by a merger. Assuming that most of the mass within the region\nwhere $\\tau_{\\rm T}\\sim 1$ has been accreted into the black hole, it\nis found that its mass $M_{\\rm BH}\\propto M_{\\rm sph}$, the mass of\nthe spheroid of the galaxy.\n\nThe rough explanation for the proportionality is that a more massive\ngalaxy has more cold gas and so requires a more powerful wind to eject\nthe gas. A stronger wind requires a more massive black hole. The\nnormalization results by relating the wind power to the Eddington\nlimit.\n\nThe model accounts for the bulk of the X-ray Background which requires\nthat most accretion power, resulting from the growth of massive black\nholes, is obscured. Also, such obscured accretion leads to agreement\nwith the local mass density of black holes (Fabian \\& Iwasawa 1999).\nMost of the absorbed radiation will be reradiated in the far\ninfrared\/sub-mm bands and contribute to the source counts and\nbackgrounds there. It can plausibly account for some of the sub-mm\nsources recently discovered by SCUBA (Barger et al 1998; Hughes et al\n1998, Blain et al 1999). Indeed it predicts a population of distant\nUltra-Luminous Infrared Galaxies (ULIRGs; Sanders \\& Mirabel 1996)\nassociated with the main growth phase of massive black holes and\ndistinct from the nearby population, which is probably due to mergers\nbriefly fuelling central starbursts and fully grown black holes.\n(Whether radiation from the black hole can then dominate the ULIRG\nemission depends on the Eddington limit, i.e. the mass of the black\nhole, relative to the strength of the starburst; see e.g. Wilman et al\n1999.) The bolometric power of a typical growing black hole must be\nhigh, $L_{\\rm Bol}>10^{46}\\hbox{$\\erg\\s^{-1}\\,$}$; which means an observed 2--10~keV\nluminosity when $N_{\\rm H}\\sim 2\\times 10^{24}\\hbox{$\\cm^{-2}\\,$}$ of about\n$3\\times 10^{-15}\\hbox{$\\erg\\cm^{-2}\\s^{-1}\\,$}$. Current ASCA (e.g. Ueda et al 1998) and\nBeppoSAX (Fiore et al 1999) hard X-ray surveys have not probed deep\nenough to reveal these objects, although they should be obvious in the\ndeeper surveys planned for Chandra and XMM. These surveys will enable\nthe major growth phase of black holes and, with optical\/infrared\nidentifications, their redshift distribution to be studied.\n\nNote that the obscured growth phase of a massive black hole represents\na distinctly different phase from the briefer unobscured phase\npredicted after the gas is ejected and the quasar dies or any later\nphase, obscured or unobscured, when the quasar is revived by a merger\nor other transient fuelling event. These last phases are the ones\nwhich have been observed so far; the major growth phase has not. It is\nunlikely therefore that the properties of the major growth phase are\nobtainable by any simple extrapolation from observations of any\ntransient recent phases, i.e. from studies of the properties of active\ngalaxies at low redshift ($z<0.5$).\n\nIt is important that the cold gas in the young galaxy, which provides\nthe X-ray (and other waveband) obscuration, be metal enriched, and\nprobably dusty. This is likely to be a consequence of continued star\nformation, particularly of massive stars, throughout the galaxy. The\nenergy of stellar winds and supernovae help to keep the cold gas space\ncovering. Note that the injected metals and stellar mass loss will be\ndistributed both as assumed for the stars and the cold gas. Indeed the\nmetallicity of the obscuring gas closest to the black hole may be\nhigher than the solar values, which leads to better fits to the X-ray\nBackground spectrum (Wilman \\& Fabian 1999). This also accounts for\nthe high metallicity inferred in the broad-line region gas for many\nquasars (Hamann \\& Ferland 1999).\n\nThe column density distribution of the gas obscuring the radiation\nfrom the growing black hole will be such that most power is emitted\njust before the gas is ejected, which happens around $\\tau_{\\rm T}\\sim\n1$. The fraction of the power radiated at optical depths greater than\n$\\tau_{\\rm T}$ scales roughly as $\\tau_{\\rm T}^{-1}$. Angular momentum\nand other factors may cause the gas in the young spheroidal galaxy,\nand the wind, to not be completely spherical. Thus when the wind\nejects the gas it may do so most along one axis and only later eject\ngas in other directions (if at all). This means that the column\ndensity distribution for $\\tau_{\\rm T}<1$ is complicated to predict\nand is best found from the shape of the X-ray Background spectrum.\n\nAn important consequence for the stellar radiation in young growing\ngalaxies is that most of it too is obscured. This agrees with recent\nevidence for the star formation history of the Universe from sub-mm and\nother observations. \n\nThe ejection of the metal-rich cold gas in the young galaxy should\nterminate its stellar growth, as well as growth of the black hole. The\nfinal appearance of a galaxy is thus significantly affected by its\ncentral black hole. How far the gas is ejected depends on how long the\nunobscured quasar phase lasts and what the surrounding gas mass and\ndensity is; whether for example the galaxy is in a group or cluster.\nThe most massive black holes will be in the most massive galaxies and\nmay last longest in the unobscured quasar phase. They might also be\nsurrounded by a hot intragroup medium which could prevent much of the\nhotter space-filling phase from being ejected. If a surrounding hot\nphase is a necessary ingredient for a radio source then such objects\nmight be more likely to be radio galaxies.\n\nDuring the ejection phase the quasar might be classed as a BAL and\nlater it might be seen to be surrounded by extended metal-rich\nfilaments, depending on the velocity of ejection of the cold gas. The\nmetal-rich gas, if mixed with surrounding hot intracluster gas, will\nenhance the local metallicity, providing one source for the extensive\nmetallicity gradients found by X-ray spectroscopy around many cD\ngalaxies in clusters (Fukazawa et al 1994). \n\nFinally, it is noted that the model requires a significant power\noutput in the form of a wind associated with the growth of black\nholes. This wind power is dissipated as heat in the surrounding\nmedium. It may have a marked effect on surrounding intracluster gas\n(Ensslin et al 1998; Wu, Fabian \\& Nulsen 1999), possibly contributing\nto the heating required to change the X-ray luminosity--temperature\nrelation, $L_{\\rm x}\\propto T_{\\rm x}^\\alpha$, from the predicted one\nwith $\\alpha\\sim 2$ to the observed one with $\\alpha\\sim 3.$ The\nestimates of Wu et al (1999) indicate that it will also heat the\ngeneral intergalactic medium to a temperature of $\\sim 10^7{\\rm\\thinspace K}$ at\n$z\\sim 1-2$.\n\nIn summary, the growth of both massive black holes and galactic bulges\nis a highly obscured, and related, process, best observed directly in\nthe hard X-ray band and indirectly, through radiation of the absorbed\nenergy, in the sub-mm band.\n\n\n\n\n\\section{Acknowledgements}\n\nI thank the referee for comments and The Royal Society for support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSince its inception in 1959 by Erd\\\"os and R\\'enyi \\cite{ER}, the theory of random graphs has developed into \na rapidly growing and widely applicable branch of discrete mathematics, bringing together ideas from graph theory, combinatorics, and probability theory. In one model, a random graph is a subgraph $\\Gamma$ of a complete graph on $n$ vertices such that every edge of the complete graph \nis included in $\\Gamma$ with probability $p$, independently of the other edges. One is interested in probabilistic features of $\\Gamma$ and their dependence on $p$ when $n$ is large. Here $03\/n$, the homology group $H_2(Y;{\\mathbf Z})$ is nontrivial with probability tending to $1$; see also \\cite{Ko}. Thus, \nfor $p>3\/n$, the random 2-complex $Y$ is homotopically two-dimensional a.a.s. \n\nOur main result is as follows: \n\n\n\\begin{theorem}\\label{main} \n (a) If for some $k\\ge 1$ the probability parameter $p$ satisfies\\footnote{Recall that the symbol $a_n\\ll b_n$ means that $a_n>0$ and $a_n\/b_n \\to 0$ as $n \\to \\infty$.}\n\\[\n\\label{ass} p\\ll n^{-1-\\frac{2}{k+1}},\n\\] then a random 2-complex \n$Y\\in G(\\Delta_n^{(2)},p)$ is collapsible to a graph in at most $k$ steps, asymptotically almost surely (a.a.s). \n(b) If for some $k\\ge 1$ the probability parameter $p$ satisfies \n\\[\np\\gg n^{-1-\\frac{1}{3\\cdot 2^{k-1}-1}},\n\\]\nthen $Y$ is not collapsible to a graph in $k$ or fewer steps, a.a.s.\n\\end{theorem}\n\nLoosely speaking, Theorem \\ref{main} combines with previously known results to suggest that {\\it a random 2-complex with vanishing 2-dimensional homology is homotopically one-dimensional. }\n\nTheorem \\ref{main} implies:\n\n\n\\begin{corollary}\nIf for some $k\\ge 1$ the probability parameter $p$ satisfies $$p\\ll n^{-1-\\frac{2}{k+1}}$$\nthen the fundamental group $\\pi_1(Y)$ of a random 2-complex \n$Y\\in G(\\Delta_n^{(2)},p)$ is free and $H_2(Y;{\\mathbf Z})=0$, a.a.s.\n\\end{corollary}\n\nThe proof of Theorem \\ref{main} is given at the very end of the paper. A key role is played by Theorem \\ref{thm1}, \nwhich states that there exists a finite list of forbidden 2-complexes $\\mathcal L_{k,r}$ with $k\\ge 0$ and $r\\ge 2$, \nsuch that an arbitrary 2-complex of degree at most $r$ (see below) is collapsible to a graph in $k$ steps if and only if it does not contain any of the 2-complexes from $\\mathcal L_{k,r}$. This allows us to reduce the collapsiblity problem to the containment problem for random complexes which was studied in \\cite{CFK}. \n\n\n\\subsection*{Acknowledgments}\nThis research was implemented during visits of M. Farber to FIM ETH Z\\\"urich and Louisiana State University, and a visit of D. Cohen to the University of Z\\\"urich. Portions of this work were carried out during the Spring of 2010, when the first two authors participated in the Mathematisches Forschungsinstitut Oberwolfach Research in Pairs program. \n We thank the FIM ETH, LSU, the University of Z\\\"urich, and the MFO for their support and hospitality, and for providing productive mathematical environments.\n\n\\section{Collapsibility of a 2-complex to a graph} \n\n\\subsection{Basic definitions}\n\nLet $Y$ be a finite 2-dimensional simplicial complex. An edge of $Y$ is called {\\it free} if it is included in exactly one 2-simplex.\n\nThe {\\it boundary} $\\partial Y$ is defined as the union of free edges. We say that a 2-complex $Y$ is {\\it closed} if $\\partial Y=\\emptyset$. \n\nA $2$-complex \n$Y$ is called {\\it pure} if every maximal simplex is 2-dimensional. By the {\\it pure part} of a 2-complex we mean the maximal pure subcomplex, i.e. the union of all 2-simplexes. \n\n\n\nLet $Y$ be a simplicial 2-complex and let $\\sigma$ and $\\tau$ be two 2-simplexes of $Y$. \nWe say that $\\sigma$ and $\\tau$ are adjacent if they intersect in an edge.\nThe {\\it distance} between $\\sigma$ and $\\tau$, $d_Y(\\sigma, \\tau)$, is the minimal integer $k$ such that there exists a sequence of 2-simplexes $\\sigma=\\sigma_0, \\sigma_1, \\dots, \\sigma_k=\\tau$ with the property that $\\sigma_i$ is adjacent to $\\sigma_{i+1}$ for every $0\\le i0$, an {\\it $r$-pseudo-surface} is a finite, pure, strongly connected 2-dimensional simplicial complex of degree at most $r$.\n\n \n\n\\subsection{Simplicial collapse} Let $Y$ be a 2-complex. A 2-simplex of $Y$ is called {\\it free} if at least one of its edges is free. \nLet $\\sigma_1, \\dots, \\sigma_k$ be all free 2-simplexes in $Y$, and let \n$e_1, \\dots, e_k$ be free edges with $e_i\\subset \\sigma_i$. We say that the complex \n$$Y'={Y-\\cup_{i=1}^k {\\rm {int}}(\\sigma_i) - \\cup_{i=1}^k {\\rm {int}}(e_i)}$$ \nis obtained from $Y$ by collapsing all free 2-simplexes. Clearly $Y'\\subset Y$ is a deformation retract. The operation $Y\\searrow Y'$ is called a {\\it simplicial collapse}. \nNote that $Y'$ is not uniquely determined if one of the free simplexes of $Y$ has two free edges; however the pure part of $Y'$ (i.e. the union of 2-simplexes of $Y'$) is uniquely determined. \n\nThis process can be iterated $Y'\\searrow Y''$, $Y''\\searrow Y'''$, etc. We denote $Y=Y^{(0)}$, $Y'=Y^{(1)}$, $Y''=Y^{(2)}$ etc. \nThe sequence of subcomplexes $Y^{(0)}\\supset Y^{(1)}\\supset Y^{(2)}\\supset \\dots$ is decreasing and there are two possibilities: either (a) for some $k$, the complex $Y^{(k)}$ is one-dimensional (a graph), or (b) for some $k$, the complex $Y^{(k)}$ is 2-dimensional and closed, i.e., $\\partial Y^{(k)}=\\emptyset$. \n\n\\begin{definition} We say that $Y$ is collapsible to a graph in at most $k$ steps if $Y^{(k)}$ is a graph. \nWe say that $Y$ is collapsible to a graph in $k$ steps if $Y^{(k)}$ is a graph and $\\dim Y^{(k-1)}=2$. \n\\end{definition}\n\n\nObserve that {\\it if $Y$ is collapsible to a graph in at most $k$ steps then any simplicial subcomplex $S\\subset Y$ is also collapsible to a graph in at most $k$ steps}. \nAt each step one removes the free triangles in $Y^{(i)}$ which belong to $S$. \n\nLet $Y$ be a 2-complex, and consider the sequence of collapses\n\\[\nY^{(0)}\\searrow Y^{(1)}\\searrow Y^{(2)}\\searrow\\dots \\searrow Y^{(k)} \\searrow \\dots.\n\\] \nFor a 2-simplex $\\sigma\\in Y$ define $$ D_Y(\\sigma) = \\sup \\{i; \\, \\sigma\\subset Y^{(i)}\\}\\, \\, \\in \\{0, 1, \\dots, \\infty\\}.$$ \nA 2-simplex $\\sigma$ is free if and only if $D_Y(\\sigma)=0$. \n\nA 2-complex $Y$ is collapsible to a graph in at most $k+1$ steps if and only if $D_Y(\\sigma)\\le k$ for any $2$-simplex $\\sigma$. \nIf after performing several collapses $Y^{(0)}\\searrow Y^{(1)}\\searrow Y^{(2)}\\searrow\\dots $ \nwe obtain a subcomplex $Y^{(r)}\\subset Y$ with empty boundary $\\partial Y^{(r)}=\\emptyset$, \nthen $Y^{(r)}=Y^{(r+1)}=Y^{(r+2)}=\\dots$ and $D_Y(\\sigma)=\\infty$ for any simplex $\\sigma$ in $Y^{(r)}$.\n\n\\begin{lemma} Let $\\sigma$ be a 2-simplex with $D_Y(\\sigma)=k$ where $00$. If $Y\\searrow Y'$ is the first collapse, then \n$\\sigma\\subset Y'$ and clearl\n$$D_{Y'}(\\sigma)=k-1$$\nand $Y'$ has degree at most $r$. \nBy the inductive hypothesis, there exists $S'\\in \\mathcal L_{k-1,r}$ and a simplicial embedding $S'\\to Y'$, mapping the central simplex of $S'$ onto $\\sigma$. \n\nFor each edge \n$e$ lying in $A_{S'}(\\sigma)$ choose a 2-simplex $\\sigma_e\\subset Y$ as follows. If $e\\subset \\partial Y'$, let $\\sigma_e$ be any free triangle in $Y$ containing $e$. \nIf $e\\not\\subset \\partial Y'$, let $\\sigma_e$ be any triangle in $Y'$ containing $e$ which is not in $S'$; such $\\sigma_e$ exists since $e\\not\\subset \\partial Y'$. \n\nNext we define a subcomplex $S\\subset Y$ as the union\n$$S=S'\\cup \\bigcup_{e}\n\\sigma_e \\, \\subset Y,$$\nwhere $e$ runs over the edges in $A_{S'}(\\sigma)$. Note that $S$ is finite, pure, and strongly connected since $S'$ is an $r$-pseudo-surface. \nMoreover, the degree of $S$ is at most $r$ since it is a subcomplex of $Y$. One has \n$D_S(\\sigma)\\ge k$\nby Corollary \\ref{corgood}. More precisely, we obtain that $D_S(\\sigma)=k$\nby Lemma \\ref{ge}. Finally we observe that obviously $d_S(\\sigma, \\sigma')\\le k$ for any 2-simplex $\\sigma'$ of $S$.\nThus, $S\\in \\mathcal L_{k,r}$. \\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm1}] Consider the sequence of successive collapses $Y\\searrow Y^{(1)}\\searrow Y^{(2)} \\searrow Y^{(3)}\\searrow \\dots$. \nWe assume that $Y$ is not collapsible to a graph in $k$ steps, which implies that there are two possibilities: either (a) $Y^{(i)}\\not= Y^{(i+1)}$ for any $ik$, then consider the subcomplex \n$Z\\subset Y^{(i)}$ defined as the union of all $\\sigma$ with $d_{Y^{(i)}}(\\sigma_\\ast, \\sigma)\\le k$. \n\nClearly $Z$ is not collapsible to a graph in $k$ steps. Therefore, in the sequence of collapses $Z\\searrow Z^{(1)}\\searrow Z^{(2)} \\searrow Z^{(3)}\\searrow \\dots$, we again \nhave either case (a) or (b) as above. In case (a), we apply Lemma \\ref{lm10}; and in case (b), we obtain a subcomplex $S\\subset Z$ with $\\partial S=\\emptyset$ such that\n$d(\\sigma_\\ast, \\sigma)\\le k$ for any $\\sigma\\subset S$. We have $S\\in \\mathcal L_{k,r}$ in either case, completing the proof. \n\\end{proof}\n\n\\section{Collapsibility of a random 2-complex}\n\n\\subsection{The degree sequence}\nRecall that the degree of an edge $e$ in a 2-complex is defined as the number of 2-simplexes which contain $e$. The degree of an edge in a random 2-complex \n$Y\\in G(\\Delta_n^{(2)},p)$ is an integer in the set $\\{0, 1, \\dots, n-2\\}$.\n\n\nLet $X_k: G(\\Delta_n^{(2)},p)\\to {\\mathbf Z}$ be the random variable counting the number of edges of degree $k$ in a random $2$-complex, where $k=0,1, 2, \\dots, n-2$.\nA straightforward calculation reveals that \n\\[\n{\\mathbb E}(X_k) = {n \\choose 2} {{n-2}\\choose k}p^k(1-p)^{n-2-k}.\\]\nThe expectation of the number of edges of degree at least $r$ in a random $2$-complex is \n\\begin{equation} \\label{edegree}\n\\sum_{k=r}^{n-2} {\\mathbb E}(X_k) \\le n^2\\sum_{k=r}^{n-2}(pn)^k \\le \\frac{n^2(pn)^r}{1-pn}.\n\\end{equation}\n\n\n\n\n\\begin{corollary}\\label{cor12} The probability that a random 2-complex $Y\\in G(\\Delta_n^{(2)}, p)$ has an edge of degree at least $r$ is less than or equal to \n$$\\frac{n^{2+r}p^r}{1-pn}.$$ \nThus, if $$p\\ll n^{-1-\\frac{2}{r}},$$\nthen a random 2-complex $Y\\in G(\\Delta_n^{(2)}, p)$ has no edges of degree $r$ or greater, a.a.s.\n\\end{corollary} \n\\begin{proof} This follows from inequality \\eqref{edegree} by applying the first moment method, see, for instance, \\cite{JLR}. \n\\end{proof}\n\n\n\n\\subsection{The invariant $\\tilde \\mu(S)$.} \nFollowing \\cite{BHK} and \\cite{CFK}, for a $2$-complex $S$ with $v=v(S)$ vertices and $f=f(S)>0$ faces one defines\n\\[\n\\mu(S) = \\frac{v}{f} \\in \\mathbb Q,\n\\]\nand \n\\[\n\\tilde\\mu(S)=\\min_{S'\\subset S} \\mu(S'),\n\\]\nwhere $S'$ runs over all subcomplexes of $S$ or, equivalently, over all pure subcomplexes $S'\\subset S$. Note the following {\\it monotonicity property} of $\\tilde \\mu$:\n\\begin{eqnarray}\\label{mon}\n\\mbox{if} \\quad S\\subset T, \\quad \\mbox{then}\\quad \\tilde \\mu(S) \\ge \\tilde \\mu(T). \n\\end{eqnarray}\n\nThe invariant $\\tilde \\mu$ controls embeddability of finite 2-complexes into random 2-complexes as illustrated by the following result.\n\n\\begin{theorem}[\\cite{CFK}]\\label{tilda} Let $S$ be a finite simplicial complex. \n\\begin{enumerate}\n \\item[(a)] If $p\\ll n^{-\\tilde \\mu (S)}$, the probability that $S$ admits a simplicial embedding into a random 2-complex $Y\\subset G(\\Delta_n^{(2)}, p)$ tends to zero as $n\\to \\infty$;\n \\item[(b)] If $p\\gg n^{-\\tilde \\mu (S)}$, the probability that $S$ admits a simplicial embedding into a random 2-complex $Y\\subset G(\\Delta_n^{(2)}, p)$ tends to one as $n\\to \\infty$. \n\\end{enumerate}\n\\end{theorem}\n\n\n\\begin{definition} \nA 2-complex $S$ is called balanced if $\\tilde \\mu(S) = \\mu(S)$, or, equivalently, $\\mu(S')\\ge \\mu(S)$ for any subcomplex $S'\\subset S$. \n\\end{definition}\n\nAny triangulated surface is balanced, see \\cite{CFK}.\n\n\\begin{example}{\\rm \nSuppose that a 2-complex $S$ has a free triangle with two free edges, and that the result $S'$ of removing this triangle satisfies $\\mu(S')<1$. Then $\\mu(S)>\\mu(S')$ and $S$ is unbalanced. \nIndeed, if $\\mu(S')=v\/f$, where $v=v(S')$ and $f=f(S')$, then $vv\/f$. \nIn this way one produces many unbalance 2-complexes, including 2-disks. }\n\\end{example}\n\nNext, we examine the $\\tilde \\mu$ invariants of 2-complexes $S\\in \\mathcal L_{k, r}$.\n\n\\begin{lemma}\\label{closed} Let $S$ be a closed 2-complex, i.e., $\\partial S=\\emptyset$. Then $\\tilde \\mu(S)\\le 1$. \n\\end{lemma}\n\\begin{proof} Without loss of generality, we may assume that $S$ is connected, since otherwise we can apply the following arguments to a connected component of $S$ and use the monotonicity property (\\ref{mon}). Moreover, we may assume that $S$ is pure, since otherwise we may deal with the maximal pure subcomplex of $S$ instead of $S$. \n\nSuppose first that $H_2(S;{\\mathbf Z}_2)=0$. Then by the Euler--Poincar\\'e theorem, $\\chi(S)\\le 1$, and we have\n$$v-e+f=\\chi(S)\\le 1,\\quad \\mbox{and} \\quad 3f\\ge 2e,$$\nwhere $v, e, f$ denote the numbers of vertices, edges and faces in $S$. In the latter inequality we used the assumptions that $S$ is pure and closed. These inequalities imply\n$$v-f\/2 \\le \\chi(S) \\le 1, \\quad \\mbox{and}\\quad \\mu(S) \\le 1\/2 + 1\/f.$$\nSince $f\\ge 4$ we obtain that $\\tilde \\mu(S) \\le \\mu(S)\\le 3\/4 <1.$\n\nAssume now that $H_2(S;{\\mathbf Z}_2)\\not=0$. We will show that there is a subcomplex $S'\\subset S$ which is also closed, $\\partial S'=\\emptyset$, and satisfies $H_2(S';{\\mathbf Z}_2)={\\mathbf Z}_2$. \nIndeed, consider a nonzero two-dimensional cycle $c=\\sum_{i\\in I} \\sigma_i$ with ${\\mathbf Z}_2$ coefficients, where the $\\sigma_i$ are distinct 2-simplexes of $S$. Let $I' \\subseteq I$ be the minimal subset of the indexing set $I$ for which $c'=\\sum_{i\\in I'} \\sigma_i$ is still a cycle, and let $S'=\\bigcup_{i\\in I'} \\sigma_i$ be the corresponding subcomplex of $S$. Then clearly $H_2(S';{\\mathbf Z}_2)={\\mathbf Z}_2$ and $S'$ is closed and pure. \n\nBy the Euler--Poincar\\'e theorem, $\\chi(S')\\le 2$, and we have \n$$v'-e'+f'=\\chi(S') \\le 2, \\quad \\mbox{and}\\quad 3f'\\ge 2e',$$\nwhere $v',e ', f'$ denote the numbers of vertices, edges and faces in $S'$. This gives\n$$v'-f'\/2\\le \\chi(S') \\le 2,$$\nand\n\\begin{eqnarray}\\label{two}\n\\mu(S') \\le \\frac{1}{2} + \\frac{2}{f'}. \n\\end{eqnarray}\nSince $f'\\ge 4$, the last inequality gives $\\mu(S')\\le 1$. Finally, we have $\\tilde \\mu(S) \\le \\mu(S') \\le 1$. \n\\end{proof}\n\n\\begin{lemma}\\label{nonclosed} If $S\\in \\mathcal L_{k,r}$ for some $k\\ge 0$, \\, $r\\ge 2$ then one has \n\\begin{eqnarray}\\label{finally}\n\\tilde \\mu(S) \\le 1+\\frac{2}{k+1}.\\end{eqnarray}\n\\end{lemma}\n\\begin{proof} If $S$ is closed the result follows from Lemma \\ref{closed}. \nAssume now that $\\partial S\\not=\\emptyset$. \nLet $\\sigma_\\ast$ be the central simplex of $S$ and let $\\sigma_0, \\sigma_1, \\dots, \\sigma_k=\\sigma_\\ast$ be a collapsing path leading to $\\sigma_\\ast$. \nHere $D_S(\\sigma_i) = i$ and $\\sigma_i\\cap \\sigma_{i+1}$ is an edge, see Definition \\ref{defaccess}. \nThen the union \n$S'=\\cup_{i=0}^k\\sigma_i$ is a subcomplex having exactly $k+1$ faces and at most $k+3$ vertices.\nThus, \n$$\\mu(S') \\le \\frac{k+3}{k+1}=1+\\frac{2}{k+1},$$ \nestablishing (\\ref{finally}). \n\\end{proof}\n\n\n\n\n\\subsection{The threshold for $k$-collapsibility.}\n\\begin{definition}\nLet $\\tilde\\mu_{k,r}$ denote the largest possible value of the invariant $\\tilde\\mu(S)$ for $S$ a forbidden $r$-pseudo-surface, \n\\[\n\\tilde \\mu_{k,r} \\, = \\, \\max_{S\\in \\mathcal L_{k,r}}\\tilde \\mu(S) \\, \\in \\mathbb Q.\n\\]\n\\end{definition}\n\nFor instance, examining the surfaces shown in Figure \\ref{lone} reveals that $\\tilde \\mu_{1,2} =3\/2$.\n\n\\begin{theorem}\\label{thm21} Consider a random 2-complex $Y\\in G(\\Delta_n^{(2)},p)$. \n\\begin{enumerate}\n\\item[(a)] If for some $r\\ge 2$ and $k\\ge 1$, one has $$p\\ll n^{-1-\\frac{2}{r+1}} \\quad \\mbox{and}\\quad p\\ll n^{-\\tilde \\mu_{k,r}},$$ then $Y$ is collapsible to a graph in at most $k$ steps, a.a.s.\n\\item[(b)] If for some $r\\ge 2$ and $k\\ge 1$, one has $p\\gg n^{-\\tilde \\mu_{k,r}}$, then $Y$ is not collapsible to a graph in $k$ or fewer steps, a.a.s.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof} By Corollary \\ref{cor12}, if $p\\ll n^{-1-\\frac{2}{r+1}}$, then a random 2-complex $Y\\in G(\\Delta_n^{(2)},p)$ has degree at most $r$, a.a.s. \nNext, we apply Theorem \\ref{thm1} and examine the embeddability of complexes $S\\in \\mathcal L_{k,r}$ into $Y$. \nBy Theorem \\ref{tilda} (a), if $p\\ll n^{-\\tilde \\mu(S)}$, then \n$S$ does not embed \ninto $Y$, a.a.s. Since $\\tilde \\mu_{k,r}\\ge \\tilde \\mu(S)$, we see that the assumption $p\\ll n^{-\\tilde \\mu_{k,r}}$ implies that no $S\\in \\mathcal L_{k,r}$ can be embedded into $Y$, a.a.s.\nThus, by Theorem \\ref{thm1}, we see that $Y$ is collapsible to a graph in $ k$ or fewer steps. This proves part (a). \n\nTo prove part (b), we apply Theorem \\ref{tilda} (b) to conclude that if $p\\gg n^{-\\tilde \\mu_{k,r}}$, then there exists $S\\in \\mathcal L_{k,r}$ which is embeddable into $Y$, a.a.s. \nThis implies that \n$Y$ is not collapsible to a graph in at most $k$ steps, a.a.s.\n\\end{proof}\n\n\n\n\\begin{example} {\\rm Consider the surface $S_k\\in \\mathcal L_{k,2}$ introduced in Example \\ref{ex1}. Note that $S_k\\in \\mathcal L_{k,r}$ for any $r\\ge 2$. The numbers of vertices $v_k$ and faces $f_k$ of $S_k$ satisfy\nthe recurrence relations\n\\begin{eqnarray}\\label{musk}v_k=2\\cdot v_{k-1}\\quad \\mbox{and}\\quad f_k = v_{k-1}+f_{k-1}.\\end{eqnarray}\nIndeed, viewing $S_{k-1}$ as a subcomplex of $S_k$, we see that all vertices of $S_{k-1}$ lie on the boundary, and each edge of the boundary of $S_{k-1}$ adds a vertex to $S_k$. This explains the first equation. For the second, note that the number of new triangles\nin $S_k$ is equal to the number of edges on $\\partial S_{k-1}$. \n\nSince $v_0=3$ and $f_0=1$, solving the recurrence relations \\eqref{musk} yields \n$$v_k = 3\\cdot 2^{k} \\quad \\mbox{and}\\quad f_k = 3\\cdot 2^{k}-2. $$\nConsequently, \n$$\\mu(S_k)= 1+ \\frac{1}{3\\cdot 2^{k-1} -1}.$$\n}\n\\end{example}\n\n\\begin{lemma}\\label{star}\nThe surface $S_k$ is balanced, and hence $$\\tilde \\mu(S_k) = \\mu(S_k)= 1+ \\frac{1}{3\\cdot 2^{k-1} -1}.$$\n\\end{lemma} \n\\begin{proof} \nLet $S$ be a pure subcomplex of $S_k$ with $v=v(S)$ vertices and $f=f(S)$ faces. Write $v=v_k-m$ and $f=f_k-n$, where \n$v_k$ and $f_k$ are as above and\n$m$ and $n$ are the number of vertices and faces which are in $S_k$, but not in $S$. We claim that $m=v_k-v \\le f_k-f =n$. This assertion is established by induction.\n\nThe case $k=0$ is trivial. So assume inductively that for any $i