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\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
%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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