Content-Type: multipart/mixed; boundary="-------------0606011335584" This is a multi-part message in MIME format. ---------------0606011335584 Content-Type: text/plain; name="06-171.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-171.comments" NO E-MAIL ADDRESS ---------------0606011335584 Content-Type: text/plain; name="06-171.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-171.keywords" singular spectrum, localization transition, Anderson model ---------------0606011335584 Content-Type: application/x-tex; name="DISORDER.TEX" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="DISORDER.TEX" \documentstyle[11pt]{article} \textwidth 17truecm \textheight 22.5truecm \setcounter {section} {0} \font\newten=msbm10 \font\newseight=eurb7 \font\newsix=eurb5 %\font\newseight=msbm7 %\font\newsix=msbm5 \newfam\newfont \textfont\newfont=\newten \scriptfont\newfont=\newseight \scriptscriptfont\newfont=\newsix \def\Iii#1{{\newten\fam\newfont#1}} \renewcommand{\theequation}{\thesection.\arabic{equation}} \newcommand{\bn}{\begin{equation}} \newcommand{\en}{\end{equation}} \newtheorem{theorem}{Theorem} \newtheorem{proposition}{Proposition} \newtheorem{definition}{Definition} \newtheorem{lemma}{Lemma}[section] \newtheorem{state}{Statement} \def\N{\Iii N} \def\R{\Iii R} \def\Z{\Iii Z} \def\L{\Iii L} \def\C{\Iii C} \def\P{\Iii P} \def\1{\bf 1} \def\0{\bf 0} \def\lplus{\ell_\delta^2} \def\lminus{\ell_{-\delta}^2} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{Anderson Model\\ and Absence\\ of Pure Singular Spectrum \thanks{personal research of 1999} } \author{V.(-"D.") Grinshpun\thanks{no e-mail address, {\bf self alone}, non-slavonic, non-committee's, surname adopted, no relatives,\newline slavonic non-speaking, medical certificate of 1996, never requested any third parties neither to receive, nor to entrust any of my correspondence, nor to communicate on my behalf} } \date{} %\date{March 7, 2000} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} Absence of singular continuous component, with probability one, in the spectra of random perturbations of multidimensional finite-difference Hamiltonians, is for the first time rigorously established under certain conditions ensuring either absence of point component, or absence of absolutely continuous component in the corresponding regions of spectra. The main technical tool applied is the theory of rank-one perturbations of singular spectra (\cite{AD,STW}). \noindent The respective new result (the non-mixing property) is applied to establish existence and bounds of the (non-empty) pure absolutely continuous component in the spectrum of the Anderson model with bounded random potential in dimension $d=2$ at low disorder. The same proof holds for $d>4$. \noindent The new (1999) result implies, via the trace-class perturbation analysis (\cite{SS}), the Anderson model with the unbounded random potential to have only pure point spectrum (complete system of localized wave-functions) with probability one in arbitrary dimension. \noindent The original approximation scheme, based on the resolvent reduction formula, and analogue of the Lippman-Schwinger equations for the generalized eigenfunctions, is applicable in order to establish absence of the singular spectral component (i.e. existence of the non-empty pure absolutely continuous spectral component) in the range of the conductivity spectrum of the arbitrary bounded perturbation in the exactly solvable model, in the surface model, and of some other multidimensional Hamiltonians on $\ell^2(\Z^d)$ and $\L^2(\R^d)$, random and non-random as well. \noindent The new original results imply non-zero value of conductivity in the energy regime corresponding to the high impurity concentration and zero temperature (at low disorder), providing rigorous proof for the Mott conjecture (on existence of localization transition, from the metal-type diffusion, corresponding to the non-empty pure absolutely continuous spectral component at low disorder, to the quantum jump diffusion corresponding to the pure point spectrum at high disorder). \end{abstract} \newpage \tableofcontents % INTRODUCTION \section {Introduction} \label{s:1} \setcounter {equation} {0} The purpose of the following paper is to announce the author's new (1999) rigorous proof of absence of pure singular continuous component in the spectra of random perturbations of multidimensional Hamiltonians, and to describe some important applications. The new results imply, in particular, presence of non-empty pure absolutely continuous component in the spectrum of the Anderson tight-binding model at low disorder. \noindent Position of the corresponding mobility edge separating the spectral regions where the eigenstates are localized (Anderson localization of impurity spectrum), and are extended (via delocalization effect of conductivity spectrum), is determined as $$ E_m\: \sim\: \inf\: (\sigma(H_A)\, \cap\, \sigma(-\Delta))\: <\: \infty, $$ depending on the degree of the impurity concentration. \noindent Thus existence of the so-called localization transition, from the metal-type diffusion (corresponding to the non-zero diffusion coefficient at low disorder), to the quantum jump diffusion (corresponding to the zero diffusion coefficient at high disorder), now is rigorously established. \noindent Existence of the corresponding finite critical energy $E_m$ at the high concentration (low disorder) was predicted previously for the Anderson tight-binding Hamiltonian by the physical theory of impurity conduction (\cite{MT,M,A}), and the corresponding hypothesis is known in the physical literature as the Mott conjecture. \noindent In the following paper the new results establish rigorously and for the first time: \noindent A) pure point spectrum in the Anderson model with the unbounded random potential (Appendix B); \noindent B) existence of non-empty pure absolutely continuous component in the spectrum of the Anderson model with random potential with single-site probability distribution having bounded density of compact support (Section \ref{s:2}), in dimension 2 at low disorder. \noindent Similar proofs hold at the same time for $d>4$. \noindent Some extensions and further generalizations of the new (1999) results are valid for the random and non-random Anderson-type Hamiltonians defined on \ $\ell^2(\Z^d)$ \ and \ $\L^2(\R^d)$ as well, and are supposed to be described by the author's possible forthcoming publication(s). The main technical tool is the approximation scheme based on the generalized eigenfunction's formalism (represented in part by the following paper), and rigorous study of the properties of C.M\" oller's operator. Theorem 3 (the new result on absolutely continuous spectrum) is valid, in particular for the disordered surface model (\cite{G6}), disordered exactly solvable model \cite{BG,G4}, also in many examples not described by the following paper. Existence of non-empty pure absolutely continuous component in the spectrum of the Cayley tree (\cite{ICMP}), and of (possibly non-pure) absolutely continuous component in the spectrum of the surface model (\cite{JMP}) had been established by the different methods prior to the results described by the following paper. The main assumption imposed in order to deduce a.s. absence of the singular continuous component in the spectrum of a random perturbation, is absence either of its point spectrum, or of its absolutely continuous spectrum established \' a priori. Specifically, the pure absolutely continuous spectrum is rigorously established to exist in the intervals free of the point spectrum of a random (ergodic) Hamiltonian. By the same way, the pure point spectrum is proved to exist in the intervals where the absolutely continuous spectrum is empty. \noindent The main technical tool applied to establish absence of spectra of mixed types is the theory of rank-one perturbations of singular spectra of random Hamiltonians developed by \cite{STW,SW} in order to study the point spectrum in the Anderson tight-binding model, applying some results \cite{AD} had established previously. In the Anderson tight-binding model, a non-empty pure absolutely continuous spectrum is rigorously established, via proving absence of the mixed point spectrum within its conductivity spectral component, for the bounded random potentials (with single-site probability distribution of compact support and bounded density), satisfying the following condition: if \bn\label{1.1} \sup\limits_{q\in {\rm supp}\, dP(q)} (|q|\: +\: |q|^{-1}) \: <\: \infty, \en then $$ \sigma_{pp}(H_A)\: \cap \sigma(H_0)\: =\: \emptyset, $$ where $H_A$, $H_0$ are defined by (\ref{2.1}), (\ref{2.2}). \noindent As it had been previously established, in the Anderson model, the continuous spectrum vanishes when the disorder parameter increases, or when the degree of the impurity concentration decreases, so when the disorder is sufficiently high (at the low concentration of impurities), the whole spectrum is pure point, with probability 1 (\cite{FS,DLS,STW,SW}). \noindent For the unbounded random (strongly unbounded non-random) potentials (cf. definition in Appendix B), it is rigorously proved (Theorems \ref{t:7}, \ref{t:8}) absence of the absolutely continuous spectrum in arbitrary dimension with probability 1: $$ \sigma_{ac}(H_U)\: =\: \emptyset, $$ which, according to the main new result (Theorems \ref{t:1}, \ref{t:2}), implies that the Anderson tight binding model with unbounded random potential exhibits only pure point spectrum, with probability 1 (Theorem \ref{t:4}). Consider the random Hamiltonian defined on $\ell^2(\Z^d)$, $d\geq 1$: \bn\label{1.2} \overline{H}_\lambda(\omega)\: =\: \overline{H}_0\: +\: \lambda Q_\omega, \en where the non-perturbed $\overline{H}_0$ is defined by the Laplace operator, and $Q_\omega$ is the random perturbation defined by \bn \label{1.3} Q(\omega)\psi(x)\: =\: q_\omega (x)\psi(x), \;\; \psi\in\ell^2(\Z^d), \;\; x\in\Z^d, \en $\{q_\omega(x)\}_{x\in\Z^d}$ are independent random variables with identical probability distributions of compact support and bounded density \bn\label{1.4} {\rm Prob}\{q(0)\in dq\}\: =\: g(q)dq, \;\; g_0^{-1}=\sup\limits_q\, g(q)<\infty. \en \noindent The corresponding probability space $$ (\Omega,\P)=\prod_{j\in\Z^d}(\Z_j, dP(q_j)). $$ \begin{theorem}[The non-mixing property (A)]\label{t:1} \noindent Suppose \bn\label{1.5} \sigma_{pp}(\overline{H}(\omega))\cap(a,b)\: =\: \emptyset, \en $(a,b)\subset\R$, with probability 1. \noindent Then $$ \sigma_{sc}(\overline{H}(\omega))\cap (a,b)\: =\: \emptyset,\;\; \hbox{and}\;\; \sigma(\overline{H}(\omega))\cap (a,b)\: \subset\: \sigma_{ac}(\overline{H}) $$ (i.e. the spectrum of $\overline{H}$ in $(a,b)$ is pure absolutely continuous), with probability 1. \end{theorem} \begin{theorem}[The non-mixing property (B)]\label{t:2} \noindent Suppose \bn\label{1.6} \sigma_{ac}(\overline{H}(\omega))\cap(a,b)\: =\: \emptyset, \en $(a,b)\subset\R$, with probability 1. \noindent Then $$ \sigma_{sc}(\overline{H}(\omega))\cap (a,b)\: =\: \emptyset, \;\;\hbox{and}\;\; \sigma(\overline{H}(\omega))\cap (a,b)\: \subset\: \sigma_{pp}(\overline{H}) $$ (i.e. the spectrum of $\overline{H}$ in $(a,b)$ is pure point), with probability 1. \end{theorem} \bigskip \noindent {\bf Remark 1}. Theorems \ref{t:1}, \ref{t:2} hold for arbitrary ergodic self-adjoint $\overline{H}_0=\overline{H}_0(\overline{\omega})$ on $\ell^2(\Z^d)$, $d\geq 1$. \bigskip \noindent The main applications of the new general result are described by the following theorems. \begin{theorem}[Absolutely continuous spectrum]\label{t:3} \noindent Consider the random operator $H(\omega)$ defined on \- $\ell^2(\Z^d)$ \- ($d>1$) by (\ref{2.1})\- -(\ref{2.2}) and (\ref{1.3})-(\ref{1.4}) (the Anderson model), or by (\ref{1.7})-(\ref{1.9}) (the disordered surface model), with single-site probability distribution having bounded density, satisfying condition (\ref{1.1}), with probability 1 (Theorem \ref{t:3} -(2)), or $\forall\omega\in\Omega$ (Theorem \ref{t:3} -(1)). Then: \begin{enumerate} \item[{\bf (1)}] there exists $\lambda_0>0$, such that if $0<\lambda<\lambda_0$, $$ \sigma_{ac}(H(\omega))\: \ne\: \emptyset $$ (i.e. there is non-empty absolutely continuous spectrum at low disorder); \item[{\bf (2)}] $$ \sigma(H(\omega))\cap \sigma(H_0)\: \subset\: \sigma_{ac}(H) $$ (i.e. the conductivity component of the spectrum is pure absolutely continuous) with probability 1. \end{enumerate} \end{theorem} \bigskip \noindent {\bf Remark 2}. The corresponding "mobility edges" $E_{m\pm}$, separating the (non-random) point and continuous components of the spectrum of $H(\omega)$, are determined by $$ \inf \{\sigma(H(\omega)\}\: \leq E_{m-}\: \leq \inf\{\sigma(H_0)\}\}\; <\; {\rm sup}\{\sigma(H_0)\}\: \leq\: E_{m+}\: \leq {\rm sup}\{\sigma(H(\omega)\}. $$ \bigskip \begin{theorem}[Pure point spectrum for unbounded random potential] \label{t:4} \noindent Consider Hamiltonian \newline $H(\omega)$\ defined on $\ell^2(\Z^d)$ ($d\geq 1$) by (\ref{2.1})\- -(\ref{2.2}) and (\ref{1.3})-(\ref{1.4}) (the Anderson model), with random potential (i.i.d.v.). Suppose the single-site probability distribution $dP(q)$ has unbounded support and bounded density, then $$ \sigma(H(\omega))\: =\: \sigma_{pp} $$ (i.e. the spectrum is pure point), with probability 1. \end{theorem} \bigskip \noindent {\bf Remark 3}. The condition (\ref{1.1}) is necessary to ensure a.s. existence of the pure absolutely continuous component in the spectrum of the Anderson model (Section \ref{s:2}) at low disorder, and cannot be weakened, as it is seen via the following examples. \bigskip \noindent {\bf Example 1}. Consider the Anderson Hamiltonian with random potential formed by the independent identically distributed random variables (i.i.d.v.) with the Gauss probability distribution: $$ g(q)\: =\: {1\over \sqrt{2\pi}\sigma}\: e^{-(q-m)^2\over 2\sigma^2}, $$ $g_0=\sqrt{2\pi}\sigma$. As it had been previously established, there exist $0<\overline{g_0} <\infty$, and $0\leq E_0(g_0)< \infty$, such that $E_0=0$, if $g_0>\overline{g_0}$, and $$ \sigma(H_G)\: \cap\: (\pm E_0,\pm\infty)\: \subset\: \sigma_{pp}(H_G) $$ (i.e. the impurity spectrum is pure point), with probability 1 (\cite{DLS,SW,STW}, 1986). At the other hand, it had been known that the spectrum of $H_G$ is pure point if $d=1$ (\cite{p1986}). \noindent The new Theorem \ref{t:4} provides interesting application (strong new result formulated via Example 1), proving that all the spectrum of $H_G$ is pure point for arbitrary value of the disorder parameter $g_0$, in any dimension $d\geq 1$. \bigskip \noindent {\bf Example 2 (The disordered surface model)}. \noindent Consider the disordered surface Hamiltonian $H_s$ with the random potential formed by the "subspace" lattice $\Z^\nu$ ($1\leq\nu\leq d$) of i.i.d.v.: \bn \label{1.7} H_s(\omega)\: =\: H_0\: +\: \lambda Q_\omega, \en where $H_0$ is finite-difference Laplace operator: \bn \label{1.8} H_0\Psi(X)\: =\: \sum\limits_{\|Y-X\|=1}\: \Psi(Y),\;\;\;\; X,Y\in\Z^d, \;\;\Psi\in\ell^2(\Z^d); \en $\sigma(H_0)=[-2d,2d]$, the potential $Q_\omega$ is random multiplication operator on the subspace $\ell^2(\Z^\nu)$, $1\leq\nu \lambda_0$, $$ \sigma(H_0)\: =\: [-2d,2d]\: \subset \: \sigma_{pp}(H_A) $$ (i.e. the spectrum is pure point at high disorder), with probabiliy 1 (\cite{DLS,SW,STW}). In Example 2, strength of the impurity disorder introduced by the random sources concentrated on a subspace of lower dimension $\nu < d$, is insufficient to ensure localization in the region of the conductivity spectral component. \bigskip \noindent {\bf Remark 4}. Theorem \ref{t:4} holds for arbitrary infinite-order $H_0$ on $\ell^2(\Z^d)$ satisfying (\ref{4.3a}), if the random potential $q_\omega$ satisfies (\ref{4.1}) (cf. Appendix B). \bigskip \noindent {\bf Remark 5}. The results are applicable to study the corresponding spectral properties of the $N$-particle Hamiltonians, random and non-random as well. \bigskip \noindent Theorems \ref{t:1}, \ref{t:2} are proved in Appendix A. Theorem \ref{t:3} is proved in Section \ref{s:2} via Theorems \ref{t:1}, \ref{t:5}, \ref{t:6}, except the Statement 1 for the Anderson model with bounded non-random potential, which proof is not presented by the following paper. Theorem \ref{t:4} is proved in Appendix B via Theorems \ref{t:2}, \ref{t:7}, \ref{t:8}. The rigorous results presented are in accordance with the physical theory of impurity conduction, and with the previous original research in physics (\cite{A,ET,Il,M,YO}). \noindent These results were new (i.e. were not available in the literature on the subject at the moment), as it had been recognized via private communications by author, and is seen by the corresponding surveys (\cite{S,p1999}). \noindent {\bf Remark 6}. The wrong statement in \cite{p1999} is nowhere proven. \bigskip \section {Anderson model and absence of singular spectrum} \label{s:2} \setcounter {equation} {0} The Anderson model was initially introduced by P.Anderson \cite{A} in 1958 to model the quantum processes of spin diffusion, impurity conduction, and localization. The corresponding Hamiltonian is defined by the finite-difference operator \bn \label{2.1} H_A(\omega)\: =\: H_0\: +\: \lambda Q(\omega), \en \noindent where $H_0$ is the Laplace operator \bn \label{2.2} H_0\psi(x)\: =\: \sum\limits_{\|x-y\|=1} (\psi(x)-\psi(y)), \;\; \psi\in\ell^2(\Z^d), \; x,y\in\Z^d, \en $\|x\|=\sum\limits_{1\leq j\leq d}|x_j|$, $Q_\omega$ is the random potential defined by (\ref{1.3})-(\ref{1.4}). \noindent Operator $H_A$ is ergodic self-adjoint operator, which spectrum $\sigma(H_A)$, as well as its corresponding point component $\sigma_{pp}(H_A)$, absolutely continuous component $\sigma_{ac}(H_A)$, and singular continuous component $\sigma_{sc}(H_A)$ are non-random subsets of $\R$ (\cite{KS}): $$ \sigma(H_A) = \sigma(H_0) \dot + {\rm supp}\, dP(\lambda q), $$ where $\dot +$ denotes the algebraic sum of subsets of $\R$, and $\sigma(H_0)$ denotes the spectrum of the non-perturbed Laplace operator, which is pure absolutely continuous:\ $\sigma(H_0)=[0,4d]=\sigma_{ac}$. The non-random parameters $g_0$ (supposing $\lambda$ is fixed), or $\lambda$ (supposing $g_0$ is fixed), are usually used to measure strength of the disorder produced by the impurity sources of random amplitudes, while the value $g_0^{-1}{\rm dist}(q_j,q_{j+1})$ is used to denote degree of the concentration of corresponding impurities. In the following there will be convenient to consider $g_0$ as the disorder parameter, so that "low disorder" means sufficiently small values of $g_0$, and consequently the "high concentration" (of impurities). The model considered had been intensively studied in the recent years, and it had been rigorously established via different approximation schemes ({\cite{FS,DLS,STW,SW,D,G2,G3,G4}) that the respective spectrum exhibits exponential localization (i.e. it is pure point, and the eigenfunctions decay exponentially at infinity) in the regions of impurity spectrum corresponding to the certain values of disorder parameter (the so-called "high disorder localization"). At the same time the structure of the conductivity spectral component (corresponding to the region of the spectrum of non-perturbed Laplace operator) until recently has not been described in the available literature. Rigorous study of the corresponding spectral properties was made possible via the new approximation scheme based on the following observation. \noindent The Anderson model may be understood as the limiting case $\nu=d$ of the disordered surface model studied previously by \cite{G3,G6}. This implies, in particular, that all the statements of Theorem \ref{t:3} are valid, except theorem on existence of non-empty absolutely continuous component in the spectrum of the operator with full-space non-random potential at low disorder, which proof could compose a part of possible separate publication by the author. \noindent Denote by $G(E;x,y)=(H_A-E)^{-1}(x,y)$, $G_0(E;x,y)=(H_0-E)^{-1}(x,y)$, $x,y\in\Z^d$, the resolvent kernels of operators $H_A(\omega)$, $H_0$ on $\ell^2(\Z^d)$. \noindent Also denote $\ell^2_{\pm\delta}(\Z^d)=\ell^2(\Z^d,d\mu_{\pm\delta})$, where $d\mu_{\pm\delta}(x)=(1+|x|)^{\pm 2\delta}dx$, $\delta>{d\over 2}$. \noindent The sequence $Q=\{q(\xi)\subset\, {\rm supp}\, dP(q)\}_{\xi\in\Z^d}$ is called an admissible potential. Denote by ${\cal A}_Q$ the set of all admissible potentials, and by $H_Q$ operator with the fixed potential $Q=Q(\omega_0)\in {\cal A}_Q$ (corresponding to some fixed $\omega=\omega_0\in\Omega$). \bigskip \begin{theorem}[General properties of spectrum]\label{t:5} \noindent Consider the Anderson model, defined by (\ref{2.1})-(\ref{2.2}), (\ref{1.3})-(\ref{1.4}). \begin{enumerate} \item[{\bf 1)}] {\rm The ergodic property.} $$ \sigma(H_A)\: =\: \bigcup_{Q\in {\cal A}_Q}\: \sigma(H_Q). $$ \bigskip \item[{\bf 2)}] {\rm The resolvent identity.} \smallskip \noindent {\bf A)} \noindent Suppose $E\not\in \sigma(H_A)\cup\sigma(H_0)$. Then \begin{eqnarray*} G(E;x,y)\: & = & \: G_0(E;x,y) \\ & + &\: \sum\limits_{\zeta,\mu\in\Z^d}\: G_0(E;x,\zeta)\: \Gamma_q^{-1}(E;\zeta,\mu)\: G_0(E;\mu,y), \end{eqnarray*} where $x,y\in\Z^d$, and operator $\Gamma_q(E):\ell^2(\Z^d)\to\ell^2(\Z^d)$ is defined as follows: \begin{eqnarray}\label{2.3} \Gamma_q(E)\: & = &\: -(\lambda q)^{-1}\: -\: h_0(E),\\ h_0(E;\zeta,\mu)\: & = &\: G_0(E; \zeta,\mu),\;\;\;\;\zeta,\mu\in\Z^d.\nonumber \end{eqnarray} \medskip \noindent {\bf B)} $$ \sigma(H_A)\setminus \sigma(H_0)\: =\: \{E\in\R|\; 0\in \sigma(\Gamma_q(E))\}, $$ where $\Gamma_q(E)$ is defined by (\ref{2.3}) \bigskip \item[{\bf 3)}] {\rm Eigenfunctions.} \smallskip \noindent Suppose $$ \sup\limits_{\xi\in\Z^d} |q(\xi)|^{-1}<\infty. $$ \noindent {\bf A)} A function $\Psi_E(x)$, $x\in\Z^d$ is the generalized eigenfunction of $H_A$, corresponding to the generalized eigenvalue $E\in\sigma(H_A)$, if and only if $$ \Psi_E(x)\: =\: \Phi_E(x)\: +\: \sum\limits_{\zeta\in\Z^d}\: \varphi_E(\zeta) G_0(E;x,\zeta)\: \in\: \ell^2_{-\delta}(\Z^d), $$ where $\varphi_E\in\ell^2_{-\delta}(\Z^d)$, $x\in\Z^d$, $\delta > {1\over 2}$, $$ \Gamma_q(E)\varphi_E(\zeta)\: =\: \Phi_E(\zeta), $$ where $\Gamma_q(E)$ is defined by (\ref{2.3}), $\Phi_E(x)$ is the distributional solution of the Laplace equation \newline $(H_0-E)\Phi_E\: =\: 0$. \medskip \noindent {\bf B)} $E\in \sigma(H_A)\setminus \sigma(H_0)$ is the eigenvalue, and $\Psi_E$ is the corresponding eigenfunction of $H_A$ if and only if $$ \Psi_E(x)\: =\: \sum\limits_{\zeta\in\Z^d}\: \varphi_E(\zeta) G_0(E;x,\zeta),\;\;\;x\in\Z^d, $$ where $\Psi_E\in\ell^2(\Z^d)$, and $\varphi_E\in\ell^2(\Z^d)$ is eigenfunction corresponding to the eigenvalue $0$ of $\Gamma_q(E)$: $$ \Gamma_q(E)\varphi_E\: =\: 0. $$ \medskip \noindent {\bf C)} Suppose $E\in\sigma(H_A)$, $$ \sup\limits_{\xi\in\Z^d}\, |q(\xi)|<\infty, $$ \bn\label{2.4} (H_A-E)\Psi_E\: =\: 0. \en Consider \bn\label{2.5} \varphi_E\: = \: -\lambda q \Psi_E. \en \noindent Then: \smallskip \noindent {\bf I}. $\Psi_E\in\ell^2(\Z^d)$ is a solution of (\ref{2.4}) (i.e. is the eigenfunction of $H_A$), if and only if \bn\label{2.6} \varphi_E\: =\: (H_0-E)\Psi_E\: \in\: \ell^2(\Z^d). \en \smallskip \noindent {\bf II}. $\Psi_E\in\ell^2(\Z^d)$ is a solution of (\ref{2.4}) (the eigenfunction of $H_A$), if and only if for some $z\not\in\sigma(H_A)\cup\sigma(H_0)$, \bn\label{2.7} \Psi_E\: =\: \Phi_z\: +\: (H_0-z)^{-1}\varphi_E, \en where \bn\label{2.8} \Phi_z\: =\: (E-z)(H_0-z)^{-1}\Psi_E\: \in\: \ell^2(\Z^d). \en \bigskip \item[{\bf 4)}] {\rm The point spectrum.} \smallskip \noindent {\bf A)} {\rm Geometrical structure.} $$ \sigma_{pp}(H_A)\cap (\sigma(H_A)\setminus\sigma(H_0))\: =\: \{E\in\R|\; 0\in\sigma_{pp}(\Gamma_q(E))\}, $$ where $\Gamma_q(E)$ is defined by (\ref{2.3}). \medskip \noindent {\bf B)} {\rm Absence of the mixed point spectrum.} \noindent Suppose \bn\label{2.10} \sup\limits_{x\in\Z^d}\: (|q(x)|+|q(x)|^{-1})\: <\: \infty. \en Then $$ \sigma_{pp}(H_q)\cap \sigma(H_0)\: =\: \emptyset. $$ \medskip \noindent {\bf C)} {\rm Localization at high disorder.} \noindent Consider the disordered Anderson model defined by the random Hamiltonian (\ref{2.1})\- -(\ref{2.2}), (\ref{1.3})\- -(\ref{1.4}). \noindent Given $\varepsilon>0$ there exist $\delta_0(\varepsilon)>0$, and $E_0=E_0(\delta)>2d$, $E_0(\delta_0)=2d+\varepsilon$, such that $$ \sigma(H_A)\: \cap\: (\pm E_0, \pm\infty)\: =\: \sigma_{pp} $$ (i.e. at high disorder impurity spectrum is pure point), and the corresponding eigenfunctions decay exponentially fast at infity, with probability 1. The point spectrum is non-empty (i.e. $E_0\in (2d,\sup\{\sigma(H_A)\}$), if $\delta>\delta_0(E_0)$. \end{enumerate} \end{theorem} \bigskip \begin{theorem}\label{t:6} {\bf (Pure absolutely continuous spectrum)} Suppose (\ref{2.10}) holds with probability 1. Then $$ \sigma(H_A)\cap \sigma(H_0)\: \subset\: \sigma_{ac}(H_A) $$ (i.e. the spectrum of $H_A$ in $[0,4d]$ is pure absolutely continuous), with probability 1. \end{theorem} \noindent Theorem \ref{t:6} is a consequence of Theorem \ref{t:5}-4b) and Theorem \ref{t:1}. \noindent {\it Proof of Theorem \ref{t:5} .} Statement 2a) (the resolvent identity) is proved in \cite{G3}. \noindent Proofs of 3a), b), 2b), 4a) are analogous to the proofs of Theorems 3a, 3b, and of Corollary 4 of \cite{BG,G4}. \noindent Theorem \ref{t:5}-4c) is proved in \cite{G3}, and could be seen as some strong version of the theorem on existence of the pure point spectrum in the Anderson model. It implies bounds for the mobility edges and admits certain extension to cover the weak disorder localization (\cite{G8}). \noindent Theorem \ref{t:5}- 1) is proved in \cite{G6}. \noindent {\it Proof of Theorem \ref{t:5}-3c)} {\bf I}. The statement (\ref{2.6}) is a trivial consequence of (\ref{2.4}) and (\ref{2.5}). {\bf II}. Relations (\ref{2.7}) and (\ref{2.5}) imply \begin{eqnarray}\label{2.11} (H_A-z)\Psi_E\: & = &\: (H_A-z)\Phi_z\: +\: (H_A-z)(H_0-z)^{-1}\varphi_E\nonumber\\ & = &\: (H_0-z)\Phi_z\: +\: \lambda q (\Phi_z+(H_0-z)^{-1}\varphi_E)\: +\: \varphi_E\nonumber\\ & = &\: (H_0-z)\Phi_z\: +\: \lambda q\Psi_E\: +\: \varphi_E\nonumber\\ & = &\: (H_0-z)\Phi_z. \end{eqnarray} \noindent Since (\ref{2.8}) implies \begin{eqnarray*} (H_0-z)\Phi_z\: & = &\: (E-z)\Psi_E\\ & = &\: (H_A-z)\Psi_E\: -\: (H_A-E)\Psi_E, \end{eqnarray*} (\ref{2.11}) holds if and only if (\ref{2.4}) is valid. Theorem \ref{t:5}-3c) is proved. \noindent {\bf Remark 7}. Theorem \ref{t:5}-4b) is proved in the following paper in full details for $d=2$ only (because of the reasons explained by Remark 6). The same approximation scheme (with precision to the values of the dimension-dependent coefficients) step-by-step could be repeated to verify the results for arbitrary $d\geq 5$. The results are valid also if $d\geq 5$. \bigskip \noindent {\it Proof of Theorem \ref{t:5}-4b), $d=2$.} \noindent Suppose the potential $Q$ satisfies (\ref{2.10}). Consider $\Psi_E\in \ell^2(\Z^2)$, the eigenfunction of $H_A$ corresponding to eigenvalue $E\in\sigma(H_0)$: $$ (H_A-E)\Psi_E\: = \: 0. $$ Theorem \ref{t:5}-3c) implies \begin{eqnarray*} \Psi_E(x)\: & = &\: \Phi_z(x)\: +\: \sum\limits_{\zeta\in\Z^d}G_z^0(x,\zeta)\varphi_E(\zeta) \nonumber\\ & = & \:\Phi_z(x)\: +\: (H_0-z)^{-1}\varphi_E(x)\: \in\: \ell^2(\Z^2), \;\; x\in\Z^2, \end{eqnarray*} where $$ \Phi_z(x)\: =\: (E-z)(H_0-z)^{-1}\: \Psi_E\: \in \ell^2(\Z^2), $$ and $$ \varphi_E\: =\: -\lambda q \Psi_E\:\in\:\ell^2(\Z^2), $$ $z\in\R\setminus (\sigma(H_A)\cup\sigma(H_0))$. \noindent Denote \begin{eqnarray*} J_{++}\: & = &\: \{(j_1,j_2)\in \Z^2|\: j_1\geq 0, j_2\geq 0\},\\ J_{+-}\: & = &\: \{(j_1,j_2)\in \Z^2|\: j_1\geq 1, j_2\leq -1\},\\ J_{-+}\: & = &\: \{(j_1,j_2)\in \Z^2|\: j_1\leq -1, j_2\geq 1\},\\ J_{--}\: & = &\: \{(j_1,j_2)\in \Z^2|\: j_1\leq 0, j_2\leq 0\}, \end{eqnarray*} $$ \varphi_{\pm\pm}(j)\: =\: \cases{\varphi(j),\; &if $j\in J_{\pm\pm}$, \cr 0,\; &otherwise}, $$ \begin{eqnarray*} \Delta_{++}\: =\: \{(p_1,p_2)\in\C^2|\; \Im\, p_1\leq 0,\: \Im\, p_2\leq 0\},\\ \Delta_{+-}\: =\: \{(p_1,p_2)\in\C^2|\; \Im\, p_1\leq 0,\: \Im\, p_2\geq 0\},\\ \Delta_{-+}\: =\: \{(p_1,p_2)\in\C^2|\; \Im\, p_1\geq 0,\: \Im\, p_2\leq 0\},\\ \Delta_{--}\: =\: \{(p_1,p_2)\in\C^2|\; \Im\, p_1\geq 0,\: \Im\, p_2\geq 0\}, \end{eqnarray*} $$ E(p,j)\: =\: e^{-i(\Re\, p_1j_1\, +\, \Re\, p_2j_2)} \: e^{\Im\, p_1j_1+\Im\, p_2j_2}. $$ By $\hat{\varphi}_{\pm\pm}(p)$, $p\in \R^2$, denote the Fourier transform of $\varphi_{\pm\pm}(j)\in\ell^2(\Z^2)$ ($\hat{\varphi}_{\pm\pm}(p)\in \L^2(\R^2)$ by the Riesz-Fischer theorem \cite{RF}). \noindent Denote by ${\cal H}(\Delta)$ the set of functions holomorfic in $\Delta\subset\C^2$, by $\widetilde{\Delta}_{\pm\pm}$ an open subset of $\overline{\Delta}_{\pm\pm}(\epsilon)\: =\: \{p\in \Delta_{\pm\pm}:\: |p|\leq \epsilon <\infty\}$. \noindent Since \begin{eqnarray}\label{2.12} \sum_{j\in J_{\pm\pm}}|E(p,j)\varphi_{\pm\pm}(j)|\: & \leq &\: (\sum\limits_{(j_1,j_2)\in J_{\pm\pm}}e^{-2(|\Im\, p_1j_1|\, +\, |\Im\, p_2j_2|)})^{1\over 2}\: \|\varphi_{\pm\pm}\|\nonumber\\ & < & \infty,\;\;\; p\in\Delta_{\pm\pm}\subset\C^2, \end{eqnarray} it follows that $\hat{\varphi}_{\pm\pm}(p)$ may be continued to ${\cal H}(\widetilde{\Delta}_{\pm\pm})$ by the Osgood lemma, since the series (\ref{2.12}) converges absolutely and uniformly in each compact $\overline{\Delta}_{\pm\pm}\subset\C^2$ to define continuous in $\widetilde{\Delta}_{\pm\pm}$ function which is holomorphic in each separate variable $p_j\in\widetilde{\Delta}^j_{\pm\pm}$, $j=1,2$. \noindent Denote $$ \Psi_{\pm\pm}(E;j)\: =\: \cases{\Psi_E(j),\;\; &if $j\in J_{\pm\pm}$, \cr 0,\;\; &otherwise.} $$ Since $\Psi_{\pm\pm}\in\ell^2(\Z^2)$ and $\hat{\Psi}_{\pm\pm}(p)\in {\cal H}(\widetilde{\Delta}_{\pm\pm})$, it follows by Theorem \ref{t:5}-3c) that if $q$ satisfies (\ref{2.10}) then \begin{eqnarray}\label{2.14} \varphi_{\pm\pm}(E;j)\: & = &\: -\lambda q\Psi_{\pm\pm}(E;j)\nonumber\\ & = &\: (H_0-E)\Psi(E;j)\nonumber\\ & = &\: (H_0-E)\Psi_{\pm\pm}(E,j)\nonumber\\ & + &\: \delta(j_1\mp 1)\Psi_{\pm\pm}(\pm 1,j_2) + \: \delta(j_2\mp 1)\Psi_{\pm\pm}(j_1,\pm 1)\nonumber\\ & + &\: \delta(j_1)\: (\Psi_{\pm\pm}(0,j_2+1) \: + \: \Psi_{\pm\pm}(0,j_2-1) \: - \: 3\Psi_{\pm\pm}(0,j_2))\nonumber\\ & + &\: \delta(j_2)\: (\Psi_{\pm\pm}(j_1+1,0) \: + \: \Psi_{\pm\pm}(j_1-1,0) \: - \: 3\Psi_{\pm\pm}(j_1,0)). \end{eqnarray} \noindent By passing to the Fourier transform in (\ref{2.14}) (\cite{Hd}), \begin{eqnarray}\label{2.15} 2\pi\hat{\varphi}_{\pm\pm}(p)\: & = &\: 2\pi(|p|^2-E)\: \hat{\Psi}_{\pm\pm}(p)\nonumber\\ & + &\: e^{-i(p_1(j_1\mp 1)+p_2j_2)}\: \Psi_{\pm\pm}(\pm 1,j_2)\nonumber\\ & + &\: e^{-i(p_1j_1+p_2(j_2\mp 1))}\: \Psi_{\pm\pm}(j_1,\pm 1)\nonumber\\ & + &\: e^{-ip_2(j_2+1)}\: \Psi_{\pm\pm}(0,j_2+1)\nonumber\\ & + &\: e^{-ip_2(j_2- 1)}\: \Psi_{\pm\pm}(0,j_2-1)\nonumber\\ & - &\: 3e^{-ip_2j_2}\: \Psi_{\pm\pm}(0,j_2)\nonumber\\ & + &\: e^{-ip_1(j_1+ 1)}\: \Psi_{\pm\pm}(j_1+1,0)\nonumber\\ & + &\: e^{-ip_1(j_1- 1)}\: \Psi_{\pm\pm}(j_1-1,0)\nonumber\\ & - &\: 3e^{-ip_1j_1}\: \Psi_{\pm\pm}(j_1,0), \;\; (j_1,j_2)\in J_{\pm\pm}. \end{eqnarray} It follows that \bn\label{2.16} \hat{\Psi}_{\pm\pm}(p)\: =\: {\Theta_{\pm\pm}(p)\over |p|^2-E}\: \in {\cal H}(\widetilde{\Delta}_{\pm\pm}), \en where $\Theta_{\pm\pm}(p)$ is defined by (\ref{2.15}). Choose $\widetilde{\Delta}_{\pm\pm}$: $E\in \widetilde{\Delta}_{\pm\pm}\subset \Delta_{\pm\pm}$. Since $\hat{\Psi}_{\pm\pm}(p)$ is bounded in $\widetilde{\Delta}_{\pm\pm}$, (\ref{2.16}) implies \bn\label{2.17} \Theta_{\pm\pm}(p)_{|p|^2-E=0;\; p\in\widetilde{\Delta}_{\pm\pm}}\: =\: 0. \en \noindent Since $\hat{\varphi}_{\pm\pm}(p)\in {\cal H}(\widetilde{\Delta}_{\pm\pm})$, and $e^{-ipj}\in {\cal H}(\widetilde{\Delta}_{\pm\pm})$, $j\in\Z^2$, it follows that $$ \Theta_{\pm\pm}(p)\: \in\: {\cal H}(\widetilde{\Delta}_{\pm\pm}). $$ Now since $$ \Delta_{\pm\pm}\: \backslash\: (\Delta_{\pm\pm}^0\: \stackrel{\rm def}{=}\: \{p\in\Delta_{\pm\pm}:\; |p|^2-E=0\}) $$ is the non-connected domain, it follows by the Riemann theorem (the unique continuation property for holomorphic functions on $C^n$, $n>1$), that $\Delta_{\pm\pm}^0$ could serve as the zero-surface for the holomorhic function $\Theta_{\pm\pm}(p)\in {\cal H}(\widetilde{\Delta}_{\pm\pm})$, $\Delta_{\pm\pm}^0\subset \widetilde{\Delta}_{\pm\pm}$,\- only if this function equals identically to\- $0$\- (\cite{GRo} Theorem 2, Part II/E): \bn\label{2.18} \Theta_{\pm\pm}(p)\: \equiv\: 0, \;\; p\in\widetilde{\Delta}_{\pm\pm}. \en Relations (\ref{2.16}) and (\ref{2.18}) imply $$ \hat{\Psi}_{\pm\pm}(p)\: \equiv\: 0, \;\; p\in\widetilde{\Delta}_{\pm\pm}, $$ which implies \bn\label{2.20} \Psi_{\pm\pm}(E)\: \equiv\: 0. \en Since (\ref{2.20}) is established for arbitrary combination of index $\pm\pm$, it follows $$ \Psi_E\: \equiv\: 0. $$ Theorem \ref{t:5}-3c) is proved, if $d=2$. $\Box$ \bigskip \section{Appendix A. Absence of pure singular continuous spectrum}\label{s:A} \setcounter {section} {3} \setcounter {equation} {0} {\it Proof of Theorems \ref{t:1}, \ref{t:2}}. Denote by $\psi$ the unit vector in $\Z^d$, by ${\cal L}(A)$ is the Lebesgue measure of ${\cal L}$- measurable set $A\subset \R$, $d\rho_0=d\rho(\overline{H}(\omega_0),\psi)$ the spectral measure of $\overline{H}(0)=\overline{H}(\omega_0)$, $\omega_0\in\Omega$, associated with $\psi$ (e.g. there is considered operator defined by (\ref{1.2}) with the fixed value of admissible potential $q(\omega_0)$), \begin{eqnarray}\label{3.1} {\cal F}_0(z)\: & =\: &\int\limits_{\R}\: {d\rho_0(\lambda)\over \lambda-z}\nonumber\\ \: & =\: &\langle \psi,(\overline{H}(0)-z)^{-1}\psi \rangle, \end{eqnarray} where $z\not\in\sigma(\overline{H})$ ($z=x+i\varepsilon$, $\varepsilon\ne 0$), $(\overline{H}(0)-z)^{-1}$ denotes the resolvent of $\overline{H}(0)$. ${\cal F}_0(z)$ is called the Stiltjes transform of the spectral measure $d\rho_0$: \bn\label{3.2} \Im\, {\cal F}_0(z)\: =\: \int\limits_{\R}\: {\varepsilon d\rho_0(\lambda)\over (\lambda-x)^2+\varepsilon^2}, \en \bn\label{3.3} \Re\, {\cal F}_0(z)\: =\: \int\limits_{\R}\: {(\lambda-x) d\rho_0(\lambda)\over (\lambda-x)^2+\varepsilon^2}. \en \noindent Denote by $d\rho_\gamma=d\rho(\overline{H}(\gamma),\psi)$ the spectral measure of $$ \overline{H}(\gamma)=\overline{H}(0)+\gamma\langle .,\psi\rangle\psi, $$ associated with $\psi$. If $\psi$ is cyclic for $\overline{H}(0)$ (i.e. the set of finite linear combinations of $\{\overline{H}(0)^n\}_{n=1}^\infty$ is dense in $\ell^2(\Z^d)$), then $\psi$ is also cyclic for $\overline{H}(\gamma)$, $\gamma\in {\rm supp}\, dP(q)$. \noindent It follows by (\ref{3.1}) and by the rank-one perturbation formula for the resolvent the Stiltjes transform of $d\rho_\gamma$ satisfies \bn\label{3.4} {\cal F}_\gamma(z)\: =\: { {\cal F}_0(z)\over 1+\gamma {\cal F}_0(z) }, \en $\Im\, z\ne 0$. \noindent Consider also \bn\label{3.5} B_0(x)\: =\: ( \int\limits_{\R}\: { d\rho_0(\lambda)\over (\lambda-x)^2 })^{-1}, \en then $0\leq B_0(x)<\infty$, $x\in\sigma(\overline{H}(0))$. \noindent Since $\gamma\langle .,\psi\rangle\psi$ is a rank-one perturbation, $$ {\rm supp}\, d\rho^{ac}(\gamma)\: =\: {\rm supp}\, d\rho^{ac}_0, $$ and $$ {\rm supp}\, d\rho^{ac}_0 \: \cap\: (a,b)\: \subseteq\: \{x\in (a,b)|\: \Im\, F_0(x)>0\}. $$ \noindent {\bf (A)} Choose $\omega_0$ such that $$ \sigma_{pp}(\overline{H}(\omega_0))\cap (a,b)\: =\: \emptyset. $$ Since $$ d\rho_0^{pp}(a,b)\: =\: 0, $$ and $$ {\cal L}\{{\rm supp}\, d\rho_0^{sc}\}\: =\: 0, $$ it follows \begin{eqnarray}\label{3.6} &\; &{\cal L}\: \{(a,b)\, \cap\, {\rm supp}\, d\rho_0\, \setminus\, S_{reg}\}\nonumber\\ & = &\: {\cal L}\{ x\in (a,b)\, \cap\, {\rm supp}\, d\rho_0|\: \Im\, {\cal F}_0(x+i0)\, =\, B_0(x+i0)\, =\, 0\}\nonumber\\ & \leq &\: {\cal L}\{\, {\rm supp}\, d\rho_0^{sc}\, \cap\, (a,b)\, \}\: +\: {\cal L}\{\, {\rm supp}\, d\rho_0^{pp}\, \cap\, (a,b)\, \}\nonumber\\ & = &\; 0, \end{eqnarray} where $S_{reg}$ is defined by \ref{3.14}. \noindent Proposition 1 (Appendix A) and (\ref{3.6}) imply \bn\label{3.7} d\rho_\gamma^{sc}(a,b)\: =\: 0 \en for ${\cal L}$- a.e. $\gamma\ne 0$. \noindent (\ref{3.7}) implies \begin{eqnarray}\label{3.8} &\; &\P\{\omega|\; d\rho^{sc}(\overline{H}(\omega))\{ (a,b) \}\: \ne\: 0\}\nonumber\\ & \leq &\: \int\limits_{\prod\limits_{j\ne j_0}\R_j}\; g_0^{-1}\: {\cal L}\{\gamma\in\R|\, d\rho_\gamma^{sc}\{ (a,b) \}\: \ne\: 0\}\: dP(q_j)\nonumber\\ & = &\: 0. \end{eqnarray} Since $\psi$ may be chosen arbitrary over the dense in $\ell^2(\Z^d)$ set of basis vectors $\{e(j)\}_{j\in\Z^d}$, (\ref{3.7}) and (\ref{3.8}) imply \bn\label{3.9} \sigma_{sc}(\overline{H}(\omega))\: \cap\: (a,b)\: =\: \emptyset \en with probability 1. (\ref{1.5}) and (\ref{3.9}) imply $$ \sigma(\overline{H}(\omega))\: \cap\: (a,b)\: \subset\: \sigma_{ac}(\overline{H}) $$ with probability 1. Theorem \ref{t:1} is proved. \noindent {\bf (B)} Choose $\omega_0$ such that $$ \sigma_{ac}(\overline{H}(\omega_0))\cap (a,b)\: =\: \emptyset. $$ Proposition 2 (Appendix A) implies \begin{eqnarray}\label{3.10} &\; &{\cal L}\: \{(a,b)\, \cap\, {\rm supp}\, d\rho_0\, \setminus\, {\cal B}\}\nonumber\\ & = &\: {\cal L}\{ x\in (a,b)\, \cap\, {\rm supp}\, d\rho_0|\: B_0(x+i0)\, =\, 0\}\nonumber\\ & \leq &\: {\cal L}\: \{ (a,b)\, \cap\, {\rm supp}\, d\rho_\gamma^{ac} \} \: + \: {\cal L}\: \{ (a,b)\, \cap\, {\rm supp}\, d\rho_\gamma^{sc} \}\nonumber\\ & = &\: 0, \end{eqnarray} where ${\cal B}$ is defined by \ref{3.13}, since $$ {\cal L}\{{\rm supp}\, d\rho_\gamma^{sc}\}\: =\: 0, $$ $$ d\rho_\gamma^{ac}(a,b)\: =\: 0, $$ $\gamma\ne 0$. \noindent Proposition 1 (Appendix A) and (\ref{3.10}) imply $$ d\rho_\gamma(a,b)\: =\: d\rho_\gamma^{pp}(a,b) $$ for ${\cal L}$- a.e. $\gamma\ne 0$, hence \bn\label{3.11} d\rho_\gamma^{sc}(a,b)\: =\: 0 \en for ${\cal L}$- a.e. $\gamma\ne 0$. Since $dP(q)$ has bounded density, (\ref{3.11}) implies \bn\label{3.12} \sigma_{sc}(\overline{H}(\omega))\: \cap\: (a,b)\: =\: \emptyset \en with probability 1. Relations (\ref{1.6}) and (\ref{3.12}) imply $$ \sigma(\overline{H}(\omega))\: \cap (a,b)\: \subset\: \sigma_{pp}(\overline{H}) $$ with probability 1. Theorem \ref{t:2} is proved. $\Box$ \bigskip \noindent The following propositions are the previously established results of \cite{STW,SW}. \bigskip \noindent {\bf Proposition 1}. \noindent {\bf A}. \noindent Denote \bn\label{3.13} {\cal B}\: =\: \{x\in (a,b)|\: B_0(x)>0\}\: =\: 0. \en Then $$ {\cal L}\{{\cal B}\}\: =\: 0 $$ if and only if $$ d\rho_\gamma^{\rm pp}(a,b)\: =\: 0 $$ for ${\cal L}$- a.e. $\gamma\ne 0$. \medskip \noindent {\bf B}. \noindent Denote \bn\label{3.14} S_{\rm reg}\: =\: \{x\in (a,b)|\: \Im\, {\cal F}_0(x+i0)+B_0(x)>0\}. \en Then $$ {\cal L}\{ {\rm supp}\, d\rho_0\cap (a,b)\setminus S_{\rm reg} \}\: =\: 0, $$ if and only if $$ d\rho_\gamma^{sc}(a,b)\: =\: 0 $$ for ${\cal L}$- a.e. $\gamma\ne 0$. \bigskip \noindent A measure $d\rho$ is said to be supported on $A\subset\R$, if $d\rho(\R\setminus A)=0$ (i.e. ${\rm supp}\, d\rho\subseteq A$). \bigskip \noindent {\bf Proposition 2.} (\cite{AD,SW}). \noindent Suppose $\gamma\ne 0$, then \medskip \noindent {\bf (A)} $$ {\rm supp}\, d\rho_\gamma^{ac}\:\cap (a,b)\: \subseteq \: {\cal A}\: \stackrel{\rm def}=\: \{x\in (a,b)|\: \Im\, {\cal F}_0(x+i0)>0\} $$ (i.e. $d\rho_\gamma^{ac}$ is supported on ${\cal A}$), \medskip \noindent {\bf (B)} $$ {\rm supp}\, d\rho_\gamma^{pp}\: \cap (a,b)\: \subseteq \: {\cal B} \: \stackrel{\rm def}=\: \{x\in (a,b)|\: B_0(x)>0\} $$ (i.e. $d\rho_\gamma^{pp}$ is supported on ${\cal B}$), \medskip \noindent {\bf (C)} $$ {\rm supp}\, d\rho_\gamma^{sc}\: \cap (a,b)\: \subseteq \: {\cal C}\: \stackrel{\rm def}=\: {\rm supp}\, d\rho_\gamma\cap (a,b)\setminus \{{\cal A}\cup {\cal B}\} $$ (i.e. $d\rho_\gamma^{sc}$ is supported on ${\cal C}$). \bigskip \noindent {\bf Proposition 3.} (\cite{SW}) \noindent Define $$ \eta(\Delta)\: =\: \int\limits_{\R} { \rho_\gamma(\Delta)\: d\gamma \over 1+\gamma^2 }, $$ where $\Delta$ is $\rho_\gamma$ - measurable subset of $\R$, \ $d\rho_0(\Delta)\: \ne\: 0.$ \noindent Then $d\eta$ is mutually equivalent to the Lebesgue measure. \bigskip Propositions 1 - 3 are proved in \cite{SW} (cf. also \cite{G6}). \bigskip \section{Appendix B. The pure point spectrum for unbounded random potential} \label{s:B} \setcounter {section} {4} \setcounter {equation} {0} While the pure point spectrum had been established to appear in particular in the Anderson model at the edges of its impurity spectral component (the so-called strong disorder localization phenomenon \cite{A,FS,DLS,SW,D,G2,G3,G4}, etc.), the pure absolutely continuous spectrum is established to exist within the conductivity component of the corresponding spectrum (the so-called low disorder delocalization effect), in particular in the Anderson model with the bounded random potential (having single-site probability distribution of bounded density and compact support, Theorem \ref{t:6}, Section \ref{s:2}). \noindent At the same time, it is well-known that self-adjoint operators (on $\ell^2(\Z^d)$, or $\L^2(\R^d)$) with growing potential have no absolutely continuous spectrum, for example, if $\lim\limits_{|x|\rightarrow\infty}|q(x)|=\infty$, then the spectrum is pure discrete (\cite{Gl}). \noindent In the following section there is proved the analogues theorem for the multidimensional finite-difference operators with random (ergodic) potential (Theorem \ref{t:4}). All the spectrum is established to be pure point with probability 1 assuming that the random potential has unbounded support. \noindent Consider the Anderson tight-binding Hamiltonian $H_U$ defined by (\ref{2.1}) - (\ref{2.2}) with the random potential having single-site probability distribution of unbounded support: \begin{eqnarray}\label{4.1} \sup\limits_q\, {\rm supp}\, dP(q)\: & = &\: +\infty\;\;\;\;\hbox{\rm or}\nonumber\\ \inf\limits_q\, {\rm supp}\, dP(q)\: & = &\: -\infty. \end{eqnarray} \noindent Theorem \ref{t:4} is a consequence of Theorem \ref{t:7} and of Theorem \ref{t:2}. \bigskip \begin{theorem} \label{t:7} \noindent{\bf Absence of absolutely continuous spectrum \- for unbounded random potential} $$ \sigma_{ac}(H_U)\: =\: \emptyset $$ with probability 1. \end{theorem} \bigskip \noindent The proof follows from the result for the strongly unbounded non-random potentials (Theorem \ref{t:8}). \noindent Consider the operator $H$ defined on $\ell^2(\Z^d)$ by (\ref{2.1})-(\ref{2.2}), and assume \bn\label{4.3} \limsup\limits_{\Lambda_{L_j}\nearrow\infty}\: \inf\limits_{x\in\partial\Lambda_{L_j}}\: |q(x)|=\: \infty, \en where $\Lambda_{L_j}=\{x\in\Z^d:\: \|x\|\leq L_j\}$, i.e. the potential $q(x)$ is unbounded over increasing to infinity sequence of concentric spheres $\Lambda_{L_j}\subset\Lambda_{L_{j+1}}$ of radius $L_j$, $j\in\N$. \bigskip \noindent {\bf Definition}. {\it The potential satisfying (\ref{4.3}) is referred in the following paper as the strongly unbounded.} \bigskip \begin{theorem}\label{t:8} \noindent{\bf Absence of absolutely continuous spectrum \- for the strongly \- unbounded \- potential} $$ \sigma_{ac}(H)\: =\: \emptyset. $$ \end{theorem} \bigskip \noindent{\bf Remark 8}. Theorem \ref{t:8} holds for arbitrary infinite-order operator $H=H_0+q$ on $\ell^2(\Z^d)$, with off-diagonal part satisfying \bn\label{4.3a} H_0(x,y)\: =\: H_0(x-y),\;\;\; |H_0(x)|\: \leq\: C|x|^{-(d+\varepsilon)}, \en $x,y\in\Z^d$, for some $\varepsilon >0$, if the non-random potential $q$ is such that that there exist sequences $\{l_n\}_{n\in\N}$, $l_n>0$, and $\{L_n\}_{n\in \N}$, $L_n>0$: \begin{eqnarray}\label{4.3b} &\: &\limsup\limits_{\Lambda_{L_n}\nearrow\infty}\: \inf\limits_{x\in\partial^*_{l_n}(\Lambda_{L_n})}\: |q(x)|=\: \infty,\\ &\: &\lim\limits_{n\rightarrow\infty}l_n\: =\: \infty,\nonumber\\ &\: &\lim\limits_{n\rightarrow\infty} L_n\: =\: \infty,\nonumber\\ &\: &\partial^*_l(\Lambda_L)\: =\: \{x\in\Z^d:\: {L-l\over 2}\leq |x|\leq {L\over 2}\}. \end{eqnarray} \bigskip \noindent {\bf Remark 9}. Theorem \ref{t:7} holds for arbitrary infinite-order operator $H$ on $\ell^2(\Z^d)$ with random potential satisfying (\ref{4.1}), (\ref{4.3a}). \bigskip \noindent Theorem \ref{t:8} (the main technical result involved to establish Theorem \ref{t:4}) is the extension for the case $d>1$ of the result on absence of the absolutely continuous spectrum of the one-dimensional Jacoby matrix with the unbounded diagonal potential (\cite{SS}, 1989), it is explicitly proved in \cite{G6}. The generalization of this result for the case of one-dimensional finite-difference operator of infinite order (Remark 8, $d=1$) could be found in \cite{G1}. \noindent {\it Proof of Theorem \ref{t:7}.} Denote as before by $(\Omega, {\cal S}, \P)$ the probability space of realizations of the random potential (\ref{1.3})-(\ref{1.4}), where ${\cal S}$ denotes the $\sigma$- algebra of $\P$- measurable subsets of $\Omega$. Consider the sequence $L_{n+1}>L_n>0$, $n\in\N$, and denote $$ \Omega(b,\Lambda_l)\: =\: \{q\in {\cal A}_Q|\: \inf_{x\in\partial\Lambda_l}\: |q(x)|\geq b\}\: \in\: {\cal S}, $$ $$ \Omega_n\: =\: \bigcup_{\Lambda_{L_n}\subset\Lambda_{L_{n+1}} }\: \Omega(b_n,\Lambda_{L_n})\: \in\: {\cal S}. $$ Then \bn\label{4.4} \P\{\Omega_n\}\: \geq \: {L_{n+1}\over L_n}\: \P\{\Omega(b_n,\Lambda_{L_n})\}. \en So choose $\{L_n\}_{n\in\N}$: $$ L_{n+1}\: \geq\:{1\over |n|dP\{(\pm b_n,\pm\infty)\}^{C_dL_n^{d-1}}}\: L_n. $$ Condition (\ref{4.1}) implies that it is possible to choose $\{b_n\}_{n\in\N}$: \begin{eqnarray}\label{4.5} \lim\limits_{n\rightarrow\infty}\: |b_n|\: =\: \infty,\nonumber\\ dP\{(\pm b_n,\pm\infty)\}\: \ne\: 0. \end{eqnarray} Then (\ref{4.4}), (\ref{4.5}) imply \bn\label{4.6} \sum\limits_{n\rightarrow\infty}\: \P\{\Omega_n\}\: =\: \infty. \en \noindent Now Borel-Kantelli lemma via (\ref{4.6}) implies $$ \P\{\overline{\Omega}=\bigcap_{n\geq 0}\bigcup_{k\geq n}\Omega_n\}\: =\: 1, $$ i.e. the potential $Q_\omega$ satisfying (\ref{4.1}), is strongly unbounded with probability 1. \noindent Theorem \ref{t:8} implies $$ \sigma_{ac}(H_U)\: =\: \emptyset $$ holds with probability 1. Theorem \ref{t:7} is proved. $\Box$ \newpage \section{Appendix C. Authorship's proofs} \label{s:C} \bigskip \noindent {\bf A}. \noindent From vbach@mathematik.uni-mainz.de Wed Mar 1 01:06:06 2000 Received: from uran.kharkiv.net (relay.kharkiv.net [194.44.156.30]) by burda.kharkiv.net (8.9.3/8.9.1/burda) with ESMTP id BAA20721 \noindent for ; Wed, 1 Mar 2000 01:04:56 +0200 (EET) Received: from dune.kharkiv.net (dune.kharkiv.net [194.44.156.50]) by uran.kharkiv.net (8.9.3/8.9.3/uran) with ESMTP id BAA31526 \noindent for ; Wed, 1 Mar 2000 01:00:20 +0200 (EET) \noindent (envelope-from vbach@mathematik.uni-mainz.de) Received: from lima.mathematik.uni-mainz.de (lima.Mathematik.Uni-Mainz.DE Received: from imamz108.mathematik.uni-mainz.de (really [134.93.142.208]) Received: from vbach by imamz108.mathematik.uni-mainz.de with local (Exim 2.05 1 (Debian)) \noindent From: Volker Bach \noindent Reply-To: vbach@mathematik.uni-mainz.de \noindent Organization: FB Mathematik, Uni Mainz, D-55099 Mainz CONFERENCE PROGRAM AS OF FEB 29, 2000 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \centerline{\bf International Conference on} \centerline {\bf Differential Geometry and Quantum Physics} \centerline {\bf Berlin, March 6--10, 2000.} {\bf Organized by Sonderforschungsbereich 288, \newline Volker Bach, Jochen Br\"uning, \newline Georg Lang, and Bianca Toltz} {\bf Address:} \newline FB Mathematik 8-5, TU Berlin, \newline Stra{\ss}e des 17.~Juni 136, \newline D-10623 Berlin, Germany; \newline {\bf email:} \newline sfb288@sfb288.math.tu-berlin.de, \newline vbach@mathematik.uni-mainz.de, \newline bruening@mathematik.hu-berlin.de; \newline {\bf Conference Homepage:} \newline http://www-sfb288.math.tu-berlin.de/conference/ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \centerline{\bf Scientific Program} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% **************************************************************** \bigskip % -------------------------------------------------------------- Grinshpun, Vadim Fri Dec 31 12:28:44 MET 1999 % \bigskip \noindent{Grinshpun, Vadim \ \ \ \ Mon, 14:00--14.25} \noindent{grinshpun@ilt.kharkov.ua} \bigskip \centerline{\bf Absolutely Continuous Spectrum in the Tight Binding Anderson Model} It is proved that the operator $H = H_0 + \lambda Q$, acting on $\ell^2(Z^d)$, $d \geq 2$, where $H_0$ is the finite-difference Laplace operator, and $Q$ is the multiplication operator (independent random variables with identical distribution $P$ of compact support), exhibits pure absolutely continuous spectrum at weak disorder: $\sigma_{ac}(H) \neq \emptyset$ with probablity $1$, if $0 < \lambda \leq \lambda_1(d,P)$. It is proved, by applying the analogue of the multiscale analysis scheme, that $\sigma(H) \setminus \sigma(H_0 \pm \varepsilon) \subset \sigma_{pp}(H)$ with probablity $1$ for $0 < \lambda \leq \lambda_2(d,P)$, if $P$ is H{\"o}lder continuous of finite order $\rho >0$. It was established (1984) that $\sigma(H) = \sigma_{pp}(H)$ if $\lambda \geq \bar{\lambda}(d,P)$, and $\sigma(H) \setminus (-\bar{E}, \bar{E}) \subset \sigma_{pp}(H)$ for $\bar{E} = \bar{E}(d,P)$. Thus it is proved that there occurs the transition from the absolutely continuous spectrum to the pure point spectrum, when the disorder increases, and the bounds on the respective mobility edges are found. The rate of decay of the localized states at the weak disorder is estimated, and the representation of the localized and extended states is obtained. \bigskip \bigskip \noindent{\bf Remark}. More extended abstracts were sent to the official representatives of the conference (Profs. V.Bach and J.Br\"uning) on December 10, 1999 via e-mail (computer room N 205, Math. Dept., ILTP). Entrance to eu was not permitted in german embassy on Febr 29, 2000. Prepublication was refused because of "absence of presentation" via e-mail on March 14-15, 2000. \bigskip \bigskip \noindent ---------------------------------------------------- \noindent FROM: \noindent TO: najanas@cyf-kr.edu.pl, stolz@math.uab.edu, laptev@math.kth.se, carleson@math.kth.se \noindent DATE: 23 December˙2003, 21:06:15 \noindent SUBJECT: From: Dr. V.Grinshpun %%%%%%%%%%%% \bigskip \bigskip %%%%%%%%%%%% \noindent Official application for personal presentation at Conference OTAMP-2004. \noindent Possible titles (Generalized Eigenfunctions and Absence of Singular Spectrum). \noindent (no relation to ILTP, "INTAS" since 1999, no e-mail address, undelivery, postal address is to be changed). %%%%%%%%%%%% \bigskip %%%%%%%%%%%% \noindent Conference "Operator Theory \& Applications in Mathematical Physics" \noindent Mathematical Research \& Conference Center \noindent July 6-11, 2004 \noindent Bedlewo, Poland %%%%%%%%%%%% \bigskip %%%%%%%%%%%% \noindent Chairman Scientific Programme \noindent Fourth ECM-2004, Stockholm, Sweden %%%%%%%%%%%% \bigskip %%%%%%%%%%%% This is my application for personal presentation of my new (1999) research results (1)-(2): %%%%%%%%%%%% \bigskip %%%%%%%%%%%% \noindent (1) "Generalized Eigenfunctions and Absence of Singular Spectrum of Some Multidimensional Operators", and/or \noindent (2) "On Properties of Essential Spectrum of Schroedinger Operator with Zero-range (Surface) Random Potential in Dimensions Two and Three". %%%%%%%%%%%% \bigskip %%%%%%%%%%%% I suppose to provide my new postal address (required for letter of invitation) with my official registration form later. I suppose to pay my conference fees via onsite registration, if possible. %%%%%%%%%%%% \bigskip %%%%%%%%%%%% \noindent Yours sincerely, Dr. V.Grinshpun \noindent ---------------------------------------------------- %%%%%%%%%%%% \bigskip %%%%%%%%%%%% \noindent{\bf Remark}. Abstracts were sent (officially submitted) on May 4, 2004. There had not been received any correspondence neither from organizers of ECM-2004 (sweden), nor from organizers of OTAMP (satellite conference, poland). \bigskip \bigskip \noindent{\bf B}. \noindent From vgr@online.kharkiv.com Wed Jan 5 16:52:53 2000 +0200 \noindent Status: \noindent Date: Wed, 5 Jan 2000 16:52:01 +0200 (EET) \noindent From: "V.Grinshpun" \noindent To: pastur@ilt.kharkov.ua, lpastur@ilt.kharkov.ua \noindent cc: grinshpun@ilt.kharkov.ua \noindent Subject: absence of permition \noindent Message-ID: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dear Prof. L.Pastur, as I wrote in the e-mail letter of November 23, I proved existence of the continuous (pure absolutely continuous) spectrum in the tight binding Anderson model, and also for some other random operators. I promised you the abstract of the corresponding results. Please, let me know where and when it could be done. I am enclosing a copy of my request for permition to enter the institute, in order to be able to finish the research (i.e. to submit the papers for publication). If you agree in principle to sign it, then I will have the possibility to follow further official procedure. If you do not accept it, please, let me know your official reason. Please, let me know if it is required to offer the Curriculum Vitae, list of publications, a copy of the PhD thesis, letter from INTAS, research report, research project, abstract of my recent reasults concerning absolutely continuous spectrum of some random operators, etc. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent Yours sincerely, Dr. V.Grinshpun %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent Professor L.Pastur \noindent Chief of Department N24 \noindent Institute of Low Temperature Physics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I would like to request to issue me a permition to enter the Institute, from January 4, till September 30, 2000, to finish the research concerning spectral properties of some random operators. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dr. V.Grinshpun %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent From lpastur@fint.ilt.kharkov.ua Thu Jan 6 12:24:38 2000 for ; Thu, 6 Jan 2000 12:24:38 +0200 (EET) for ; Thu, 6 Jan 2000 12:24:31 +0200 (EET) \noindent Date: Thu, 6 Jan 2000 12:24:21 +0200 (EET) \noindent From: "Leonid A. Pastur" \noindent To: "V.Grinshpun" \noindent Subject: Re: absence of permition \noindent In-Reply-To: \noindent Message-ID: \noindent Status: RO %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dear Dima, \noindent I believe that the question concerning your pass to the Institute for January 2000 is settled in the sense that Khruslov and myself made all necessary official moves. As I have explained you while giving you the "zayavka", signed by Khruslov and myself, the only thing that you have to do now is to give me your passport and the "zayavka" and I ask somebody from our Department to contact respective person, issuing passes. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent Best regards L.Pastur \bigskip \bigskip {\bf C}. \noindent Date:Thu, 30 Mar 2006 21:10:14 +0100 (BST) \noindent From:"V. Grinshpun" \noindent Subject: paper submission \noindent To:CommMathPhys@caltech.edu \noindent CC:CommMathPhys@caltech.edu \noindent The Editor \noindent "Communications in Mathematical Physics" \noindent Springer Dear Editor, \noindent the following is my letter of submission of my paper "On Absence of Pure Singular Spectrum of Multidimensional Random Hamiltonians at Low Disorder". I announced my new results via e-mail on November 24, 1999, and via telephone call on March 2, 2001 (Prof. E.Lieb, IAMP). I have had no opportunity to submit for publication my mentioned research results since 1999: when my (enclosed) paper had been under preparation for submission, my e-mail box at Institute for Low Temperature Physics (Kharkov, Ukraine, Warsaw Pact) was suddenly closed, and permit for entrance was denied (not prolonged). Later my invited exit to the international conference (Berlin, EU, March 2000) was not permitted without any reason explained. I also had no financial opportunity to present my research at ICMP-2000 (UK), the corresponding proceedings contain the wrong reference to my attendance (XIII ICMP proceedings, UK (International Press), p.490). This is why I would be grateful if You could confirm receipt of my letter of submission via current (temporary) e-mail address. The paper is enclosed as file-attachments (LaTeX 73.571 Kb, dvi 97.68 Kb) The printed copy of the paper should be received via air-mail in April. \noindent Sincerely yours, Dr V.Grinshpun %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip \bigskip \noindent{\bf Remark}. The printed copy of paper \cite{G6} with very few corrections in the text (supplied by all required detailed original proofs) is supposed to be sent via air-mail upon receipt of respective inquire of the referee. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent {\bf Remark}. IAMP dues were paid in euro in 2005. \noindent ICMP-2006 fees were paid at the minimal rate (\$65) officially permitting author's participation. The only possible transfer via private author's account was made on May 22, 2006 to the official bank address. \noindent Comments in russian were demanded to make the transfers from the private usd-accounts in local banks (karaganda, kazakhstan). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent {\bf Acknowledgements}. Personal research (1999, kharkov \- ("ukraine", \- kyev-russe, \- former-soviet \- union, warsaw pact since 1955)), typesetted in December 2005 in former-soviet union (pact of May 9-11, 2005) on private PC (intel pentium II (korea), OS Windows XP Certificate Authenticity (Microsoft corp) 00049-120-546-750, N09-01178, X10-60277, no internet access), with possible unauthorized illegal external access by united former-soviet ko-gb, was supported only by research grant "AMS-1995" (\$150). \noindent United former-soviet {\bf KO-gb (former-"cossacks", slavonic ([Ph]) military forces in XV-XX centuries)}, \newline "INTAS", and hannover have no rights in any use of the following research because of support of slavonic-based national communism and "political" repressions against minoritie(s) of non-cyrillic alphabet(s) of standard orientation. \noindent The author's education within USSR was made possible in part thanks to the efforts of the Communist Party of Ukrainian Soviet Socialist Republic, which had been employed in the USSR. \noindent VG would like to request excuse for not answering to the e-mail correspondence could had arrived to his previous e-mail address (grinshpun@ilt.kharkov.ua): mentioned e-mail box was closed, and permit for entrance to the host-keeping institution was denied (not prolonged) by institute for low temperature physics, kharkov ("ukraine") on December 31 (1999), when the following paper had been under preparation for submission for publication by author. \noindent He had had no opportunity to present his described in part personal research results at ICMP XIII, the wrong reference in \cite{ICMP2000}. \noindent Exit to the international conference (berlin, "eu", March 2000) was not permitted in "eu" embassy, kyev ("ukraine"), on Febr 29, 2000. \bigskip \bigskip %%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{MMMMMMMM} \bibitem[A]{A} Anderson, P., \newblock Absence of diffusion in certain random lattices, Physical Review {\bf 109}, 1492-1505 (1958). \bibitem[BG]{BG} Bertier, A.M. and Grinshpun, V., \newblock R. Math. Phys. {\bf 9}, 4, (1997). \bibitem[CL]{CL} Carmona, R. and Lacroix, J., \newblock {\it Spectral theory of random Schr\" odinger operators},\\ Birkh\" auser, Boston, 1990. \bibitem[CFKS]{CFKS} Cycon, H.L., Froese, R.G., Kirsch, W., and Simon B., \newblock{\it Schr\" odinger operators with application to quantum mechanics and global geometry}, Springer-Verlag, Berlin Heidelberg New York, 1987. \bibitem[DLS]{DLS} Delyon, F., Levi, Y., and Souillard B., \newblock Anderson localization for multidimensional systems at large disorder or low energy, Comm. Math. Phys. {\bf 100}, 463-470 (1985). \bibitem[AD]{AD} Donoghue, W., \newblock On the perturbations of spectra, Comm. Pure Appl. Math. 18, 559-579 (1965). \bibitem[D]{D} Dreifus, H., \newblock On the effects of randomness in ferromagnetic models and Schr\" odinger operators, NYU Ph.D. Thesis (1987). \bibitem[ET]{ET} Edwards, J.T. and Thouless, D.J., \newblock Numerical studies of localization in disordered systems, J. Phys. C: Solid St. Phys. {\bf 5}, 8, 807-820 (1972). \bibitem[FS]{FS} Fr\" ohlich, J. and Spencer, T.X., \newblock Absence of diffusion in Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. {\bf 88}, 151-184 (1983). \bibitem[Gl]{Gl} Glasman, I.M., \newblock {\it Direct methods of qualitative spectral analysis of singular differential operators}, Israel program of scientific translations, 1965. \bibitem[ICMP2000]{ICMP2000} \newblock {\it XIII International Congress Of Mathematical Physics}, proceedings, July 17-22, Imperial College, London, UK, Fokas, Grigoryan, Zegarlinski Eds., p.490 2000. \bibitem[G1]{G1} Grinshpun, V., \newblock On the structure of spectrum of one-dimensional finite-difference operators with unbounded potential, in proc. XXII physical-technical conference of young scientists, p.77-78, Kharkov, USSR (May 1991, in russian) \bibitem[G2]{G2} Grinshpun, V., \newblock Point spectrum of some multidimensional random operators, PhD Thesis, University Paris 7 (1994). \bibitem[G3]{G3} Grinshpun, V., \newblock Localization for random potentials supported on a subspace, Lett. Math. Phys. {\bf 34}, N2, 103-117 (1995). \bibitem[G4]{G4} Grinshpun, V., \newblock On properties of impurity spectrum in the disordered exactly solvable model, preprint (1997). \bibitem[G6]{G6} Grinshpun, V., \newblock On absence of pure singular spectrum of random Hamiltonians at low disorder (1999), HAD BEEN SUBMITTED FOR PUBLICATION (CMP, california, "SPRINGER") \bibitem[G8]{G8} Grinshpun, V., \newblock Localization for low disorder: the new proof, Unpublished (1997) \bibitem[G10]{G10} Grinshpun, V., \newblock On absence of singular spectrum at low disorder, Preprint (1999) \bibitem[G11]{G11} Grinshpun, V., \newblock Pure Point Spectrum for Unbounded Random Potential, Preprint (1999), \bibitem[DFP]{DFP} P.A.Dirac, V.A.Fock, B.X.Podolsky: \newblock Phys. Ztschr. Sowjetunion {\bf 2}, 468-479 (1932) \bibitem[GRo]{GRo} Gunning R. and Rossi H.: \newblock {\it Analytic functions of several complex variables}, Englewood, 1965. \bibitem[Hd]{Hd} Hardy G.: \newblock On double Fourier series, Quart. J. of Math., {\bf 37}, 53-79 (1905). \bibitem[ICMP]{ICMP} \newblock {\it XI International Congress of Mathematical Physics} Proceedings, International Press, France (1994). \bibitem[Ph]{Ph} \newblock Philip's Encyclopedia, popular edition, london, 1999 (p.758) \bibitem[JMP]{JMP} Jak\v si\' c, V.X., Molchanov, S.A., and Pastur, L.A., \newblock On the propagation properties of surface waves, Preprint, IMA 1995-1996. \bibitem[Ka]{Ka} Kato, T., \newblock {\it Perturbation theory for linear operators}, Springer-Verlag, Berlin Heidelberg New York, 1966. \bibitem[KS]{KS} Kunz, H. and Souillard, B., \newblock Sur le spectre des op\' erateurs aux diff\' erences finies al\' eatoires, Comm. Math. Phys. {\bf 78}, 201 (1980). \bibitem[Il]{Il} Lifshitz, I.M., \newblock On the structure of the energy spectrum and quantum states in disordered condensed systems, Adv. Soviet Physics {\bf 83}, 617-663 (1964). \bibitem[M]{M} Mott, N., \newblock {\it Metall-insulator transitions}, London 1974 \bibitem[MT]{MT} Mott, N. and Twose, W., \newblock The theory of impurity conduction, Adv. Phys. {\bf 10}, 107-163 (1961). \bibitem[p1999]{p1999} Pastur, L.A., \newblock Spectral properties of random selfadjoint operators and matrices (a survey), American Math. Soc. Transl. {\bf 188}, p.169, line N14 (1999). \bibitem[p1986]{p1986} Pastur, L.A., \newblock On the pure point spectrum of one-dimensional Anderson model with the Gaussian potential, Preprint Karl-Marx University, DDR (1986). \bibitem[RF]{RF} Riesz, F., \newblock \" Uber orthogonale funktionensysteme, G\" ottinger Nachrichten 116-122 (1907). \bibitem[S]{S} Simon, B.X., \newblock Schr\" odinger operators in the twentieth century, J. Math. Phys. {\bf 41}, 6, 3523-3555 (2000). \bibitem[SS]{SS} Simon, B.X. and Spencer, T.X., \newblock Trace class perturbations and absence of absolutely continuous spectra, Comm. Math. Phys. {\bf 125}, 113-125 (1989). \bibitem[STW]{STW} Simon B.X., Taylor, W., and Wolff, T., \newblock Phys. Rev. Lett. {\bf 54}, 1589 (1986). \bibitem[SW]{SW} Simon, B.X. and Wolff, T., \newblock Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. {\bf 39}, 75-90 (1986). \bibitem[YO]{YO} Yoshino, S. and Okazaki, M., \newblock Numerical study of electron localization in Anderson model for disordered systems: spatial extension of wavefunctions, Journ. Phys. Soc. Japan {\bf 43}, 415-423 (1977). \end{thebibliography} \end{document} ---------------0606011335584--