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localization, point spectrum, generalized eigenfunctions 
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\documentstyle[11pt]{article}

\textwidth 16.5truecm
\textheight 22.5truecm

\setcounter {section} {-1}

\font\newten=msbm10
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%\font\newsix=msbm5
\newfam\newfont
\textfont\newfont=\newten
\scriptfont\newfont=\newseight
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\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\point{-\Delta_{\alpha(\omega)}} 
\def\poin{-\Delta_{\alpha}} 
\def\po{-\Delta_{\overline\alpha,\Lambda}}
\def\gamr{\Gamma_{\alpha(\omega)}(\lambda)}
\def\gam{\Gamma_{\alpha}(\lambda)}
\def\Gaa{\Gamma_{\alpha(\omega)}}
\def\Ga{\Gamma_{\alpha}}
\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\ids{integrated density of states}
\def\po{P_{\alpha_0}}
\def\Dpoin{ {\cal D}(\poin) }
\def\D0{ {\cal D}_0}
\def\H02{H_0^{2,2}(\R^d\setminus\Z^d)}
\def\Lplus{\L_\delta^2}
\def\Lminus{\L_{-\delta}^2}
\def\lplus{\ell_\delta^2}
\def\lminus{\ell_{-\delta}^2}
\def\cH{\cal H}
\def\ef{F}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title{On Properties of Impurity Spectrum\\
	 in the Disordered Exactly Solvable Model\thanks{the following material
	  should had been published by Reviews in Mathematical Physics 
       {\bf 9}, N4, p.425 (1997)}
       }

\date{}
%\date{March 7, 2000}

\author{ V.Grinshpun\thanks{absolutely alone, non-committee's, non-slavonic, surname adopted, no relatives,
\newline never requested any third partie(s) neither to receive, nor to entrust
any of my correspondence, 
\newline nor to communicate on my behalf. Direct meaning in a non-cyrillic.}
}

\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}
The random point interaction Hamiltonian $\point$ is considered on
$\L^2(\R^d)$, $d=2$ or $d=3$.
Existence and certain bounds of the non-empty pure point component 
and exponential decay of the corresponding eigenfunctions 
with probability one, 
within region of impurity spectrum of $\point$,
are rigorously established. 
In order to prove the localization result, 
the structure of the generalized eigenfunctions of $\point$ is explicitly described,
and the relation between its spectral properties, and the properties of 
spectra of finite-difference infinite-order operators on $\ell^2(\Z^d)$, 
is established.
The multiscale analysis scheme is applied to investigate 
the point spectrum of finite-difference operators. 
In addition, the generalized spectral theorem and 
the absolute continuity of the integrated density 
of states of $\point$ at the negative (impurity) part of the spectrum,
rigorously proved. Applications of the new approximation scheme include
straightforward analysis of the absolutely continuous 
conductivity spectrum (\cite{G4} 1999),
subject to a (possible) separate publication by the author.
\end{abstract}


\tableofcontents

% INTRODUCTION

\newpage

\section {Introduction} \label{s:0}
\setcounter {equation} {0}

The following paper describes some important research results 
in Mathematical Physics
that were not available in the literature previously (new in 1993).

Consider the random point interaction Hamiltonian on $\L^2(\R^d)$, 
$d=2$, or $d=3$, formally defined by

\bn \label{0.1}
-\Delta _{\alpha (\omega )}=-\Delta - \sum_{j\in\Z^d} 
\alpha_j(\omega ) \delta_d
(.-j)
\en 
where $\Delta=\sum\limits_j {\partial^2\over \partial x_j^2}$ 
is the Laplace operator on $\L^2(\R^d)$, 
$\delta_d(.-j)$ denotes 
the $d$-dimensional point potential concentrated at $j\in \Z^d$, 
$\{\alpha_j(\omega)\}_{j\in\Z^d}$ are independent 
identically distributed random variables
with probability distributions of compact support and bounded density: 

\bn \label{0.2}P_{\alpha}\{\alpha\in d\alpha\}=
g(\alpha)d\alpha, \; \;
\delta^{-1}=\sup_{\alpha} g(\alpha)<\infty 
\en
($\delta $ characterizes the strength of the disorder). 
The probability-space $(\Omega,{\rm\bf P} )$ is the space of realization 
of random coefficients $\alpha$:
$\Omega =\prod\limits_{j\in\Z^d}(\R_j,dP_{\alpha_j})$. 
The operator (\ref{0.1}) may be rigorously defined for $d\leq 3$ as the 
strong resolvent limit of the selfadjoint extensions of operators 
$-\Delta_{|\C_0^\infty(\R^d\setminus\Lambda)}$, 
($\Z^d\supset\Lambda\to \Z^d$), e.g. \cite{AGHH} for a precise definition.
The resolvent  kernel $(-\Delta_\alpha-z)^{-1}(x,y)$ is defined by

\begin{eqnarray} \label{0.3}
(-\Delta _{\alpha }-z)^{-1}(x,y) & = & (-\Delta-z)^{-1}(x,y)  \nonumber \\
     & + & \sum\limits_{j,j^\prime\in \Z^d} (-\Delta -z)^{-1}(x,j)\: 
[\Gamma _\alpha(z)]^{-1}(j,j^\prime)\: (-\Delta -z)^{-1}(j^\prime,y),
\end{eqnarray}
where $z\not\in \sigma (-\Delta )$, and the operator
$\Gamma_\alpha(z):\ell^2(\Z^d)\rightarrow \ell^2(\Z^d)$ ($d=2,3$), 
is defined as follows :

\bn \label{0.4}
\Gamma _\alpha (z;j,j^\prime) = \cases{ \alpha _j + \beta(z),
&if $j=j^\prime$,\cr
  -(-\Delta -z)^{-1}(j,j^\prime),& otherwise, \cr}
\en
where $\beta(z) = \lim_{|x-j|\rightarrow 0} -((-\Delta-z)^{-1}(x,j)-G_0(x-j))$;
\ \ $G_0(x)=\cases{-(2\pi)^{-1}\ln(|x|) &if $d=2$, \cr 
(4\pi|x|)^{-1} &if $d=3$;\cr }$ 
$j,j^\prime\in\Z^d,\; \alpha_j\in\R,\; x,y\in\R^d$.

In the case of $d=1$, the single-site point potential corresponds to 
the Dirac $\delta$-function (\cite{DFP}), and the respective one-dimensional 
Hamiltonian represents the standard Schr\" odinger operator on $\L^2(\R)$ 
with $\delta$- potential (the Kronig-Penney model). 

\noindent If $d=2$, or $d=3$, the operator 
$-\Delta_{|\C_0^\infty(\R^d\setminus\Lambda)}$
initially defined on 
$\Lambda\subset\Z^d$
(the set of infinitely differentiable functions of $\C_0^\infty(\Lambda)$ 
vanishing at the points of the d-dimensional cell),
is symmetric, and its self-adjoint extensions 
(the point interaction Hamiltonians described by (\ref{0.1})-(\ref{0.4})) 
do not correspond to a formal sum of the Laplace operator and any multiplication 
operator on $\L^2(\R^d)$.

In what follows, we consider explicitly d=3 dimensional case, 
all the described proofs and results are also valid simultaneously 
in dimension $2$, and
the case $d=1$ had been studied already (\cite{DSS}).

Notice that all our results are valid for the operator with the
random point potential defined on arbitrary lattice

$$
\Lambda =\{n_1e_1+n_2e_2+n_3e_3\in \R^d|(n_1,n_2,n_3)\in \Z^d\},
$$
where $e_1,e_2,e_3$ is a basis in $\R^d$.

In the following denote as usual 
$\L^2(\R^d)\: =\: \{f(x):\R^d\to\C|\; x\in\R^d, f(x)\in\C,
\|f\|_{\L^2}=\int_{\R^d}|f(x)|^2\: {\rm dx}<\infty\}$,
$\ell^2(\Z^d)\: =\: \{f(j):\Z^d\to\C|\; j\in\Z^d, f(j)\in\C, 
\|f\|_{\ell^2}=\sum_{j\in\Z^d}\|f(j)\|^2<\infty\}$,
${\cal H}=\L^2$ ($\ell^2$) is called the Hilbert space with the scalar prodact
$\langle f,g\rangle_{\L^2}: =\: \int_{\R^d}\: \overline{f(x)}\: g(x)\: dx$
($\langle f,g\rangle_{\ell^2}: =\: \sum_{j\in\Z^d}\: \overline{f(j)}\: g(j)$). 
A number $\lambda\in\C$ is called the eigenvalue of operator 
$H:{\cal D}(H)\to {\cal I}(H)\subset {\cal H}$, if

\bn \label{0.5}
H\psi = \lambda \psi
\en
for some $\psi\in {\cal D}(H)$, where ${\cal D}(H)$ is
the domain of $H$, ${\cal I}(H)$ is the image of $H$,
$\psi$ is called the eigenfunction of $H$.
The closure of the set of all the eigenvalues of $H$ is called the point spectrum,
$\sigma_{pp}(H)$.

\noindent The set $\sigma\subset\C$, such that for arbitrary $\lambda\in\sigma$
there exists a sequence $f_n(\lambda)\in{\cal D}(H)$, $\|f_n\|=1$, 
$\lim_{n\to\infty}\|(H-\lambda)\|f_n\: =\: 0$, is called 
the continuous spectrum of $H$, $\sigma_c(H)$. 
$\sigma(H)=\sigma_c(H)\cup\sigma_{pp}(H)$ is called the spectrum of $H$.
$\sigma(H)$ is the closed subset of $\C$ and 
may be determined as the complement to the resolvent set 
of operator $H$, that is if $z\in\C\backslash \sigma(H)$
the resolvent $(H-z)^{-1}$ is a bounded operator on $H$
(i.e. $\|(H-z)^{-1}\|=\sup\limits_{\|f\|=1} \|(H-z)^{-1}f\|={\rm dist}\{z,\sigma(H)\}
<\infty$, $z\not\in\sigma(H)$).

Complex-valued function of several complex variables $f(z)$,
$z=(z_1,z_2,..,z_n)$, $z_i\in\C$, is called analytic (holomorphic)
function in an open $\Omega\subset\C^n$, $n\geq 1$, if in some vicinity $\Omega(z)$
of each $z\in\Omega$, $f$ expands in series 
$f=\sum\limits_{n_1,..,n_n}a_{n_1,..,n_n}(z_1-\omega_1)^{n_1}...(z_n-\omega_n)^{n_n}$
which converges $\forall\omega\in\Omega(z)$. 

The resolvent $(H-z)^{-1}$ is operator-valued
holomorphic function of $z\in\Omega\subset\C$, 
if $\Omega\cap\sigma(H)=\emptyset$.  
A set $S\subset\C^n$ is called zero-set for holomorphic function $f(z)$,
if $f(z)=0$, $z\in S$.

Remind that operator $H^*$, such that $\langle g,Hf\rangle=\langle H^*g,f \rangle$ 
for arbitrary $f\in {\cal D}(H)$, $g\in{\cal D}(H^*)$,
is called the operator adjoint to $H$.
Operator $A$ is called the symmetric operator, 
if $\overline{{\cal D}(A)}={\cal H}$, and 
$Af=A^*f$, $f\in{\cal D}(A)\cap{\cal D}(A^*)$.
Operator $H$ is called the self-adjoint operator, if $\sigma(H)\subseteq\R$
(or if $H=H^*$, which means $H$ is symmetric and 
${\cal D}(H^*)={\cal D}(H)$).
Self-adjoint operator $H$ is called a self-adjoint extension of a symmetric
operator $A$, if $H=A$ on ${\cal D}(A)\subset {\cal D}(H)$.  

\noindent Denote by $(\Omega,{\cal B},\P)$ the probability-space 
formed by $\sigma$-algebra ${\cal B}$ of $\P$-measurable subsets of $\Omega$. 
$\P$- measurable operator-valued random variable $H(\omega)$, 
$\omega\in\Omega$, ${\cal D}(H(\omega))\subset {\cal H}$, 
is called the random operator, self-adjoint, if $H(\omega)$, $\omega\in\Omega$ 
are self-adjoint operators.
$\Omega$ is called the realization space of the corresponding
random parameters.

\noindent The corresponding probability
measure $\P$ is called ergodic, if 1)it is invariant
with respect to the shift transformations: $\P(T_j(A))=\P(A)$,
for every $\P$-measurable $A\subset\Omega$, 
$T_j(A)=\{\omega(j^\prime-j)|\; j^\prime\in A\}$, 
2)$\P(A)=1$ or $\P(A)=0$
for arbitrary shift-invariant subset 
$A\subset\Omega$: $A=T_j(A)$, $j\in\Omega$.
For example, probability measure corresponding to the random field 
generated by independent identically distributed random variables 
$\{q_j(\omega)\}_{j\in\Z^d}$, defines
ergodic random operator. In many important cases the ergodic random operators
describe contensive and adopted to the rigorous study models.

The self-adjoint operators, and in particular the random ergodic operators,
are of special interest and their rigorous study
had been a problem of primary importance since the end of twenties of XX 
century, when the respective problem arised in context of the newly
developed quantum mechanics. In quantum mechanics, 
the self-adjoint operators are assigned to the multiple physical variables
in order to describe mathematically
their bihaviour, such as coordinate and impulse, determining 
current position 
of a quantum particle in space and time
(usually assigned with operators $x$, $-ih {\partial\over \partial x}$), 
energy (usually called the Hamiltonian operator $H$), 
determining ability of a quantum particle to change its location
in time in accordance with the corresponding driving equation
($H\psi(x,t)=ih{\partial\over \partial t}\psi(x,t)$, the Schr\" odinger equation
in the classical interpretation).
The rigorous study of the mathematical properties of the corresponding
operators allows to describe the properties of the respective
physical objects.
In particular, the knowledge of physical properties of an 
inhomogenuous material
(such as alloy, polymer, semiconductor, etc.) are important 
in radioelectronics, optics, aerospace industry, etc. 
The Hamilton operator studied in the following paper
represents in particular the energy operator
of a quantum particle moving with influence of point
impurities of the random strength in such a material.
If, under certain conditions, a particle has energy belonging
to a region of the pure point spectrum of the
Hamiltonian, it is described in terms of the square summable
wave function (associated with certain localized state),
and should remain within finite distance from
its initial position, which means localization,
absence of diffusion,
zero conductivity, and the corresponding 
material should have the insulator properties
(under respective conditions).

The corresponding region of the spectrum  of the
random Hamilton operator
(where the pure point spectrum could appear),
is called the impurity spectrum,
since it is actually produced by the impurity
sources.
The strength of the impurity influence
is described in terms of the explicit expression
for the corresponding "potential", or 
by the probability distributions 
of the corresponding real-valued random variables
$\alpha_j$.
In the model considered, the independent 
identically distributed random variables
have bounded density and
are chosen to construct the random operator
with ergodic properties, which permits
its rigorous analysis via the spectral
theory of random operators (\cite{CFKS,CL}).

The point spectrum of the Hamiltonians with ergodic
potential on $\L^2(\R^d)$  has been much studied recently
(cf. \cite{KS,CHM}).  There were usually studied random
potentials of the following type:

$$
V_\omega (x)=\sum_{j\in\Z^d} \lambda_j(\omega) \, f_j(x-x_j(\omega)),
$$
where the single-site potentials $f_j$ corresponding to the neighbouring 
points of the lattice $\Z^d$ have supports of non-zero Lebesgue measure. 

The detailed study of the spectral properties of operator (\ref{0.1}) 
with the so-called "zero-range point potential" of general type 
(i.e. corresponding to the arbitrary bounded set of values 
of the coefficients $\{\alpha_j\}_{j\in{\Z^d}}$), 
is a subject to many recent publications.  
A quite complete survey of the results may be 
found in \cite{AGHH}.

In the following paper we study mainly the impurity component of the spectrum 
of the operator 
(\ref{0.1}) -(\ref{0.4}), in dimensions $d=2$ or $3$. We prove (Theorem 1)
that while the disorder parameter $\delta$ increases to infinity, 
the pure point
component fills the negative part of the spectrum of $\point$,
and when $\delta$ reaches a determined value $\delta_0<\infty$, then all
the negative spectrum becomes pure point. We also prove
(Theorem 5) that the density of states of $\point$  
is absolutely continuous.

The explicit expression (\ref{0.3}) for the resolvent kernel 
of $\point$ implies that the spectral properties of this operator 
are related to the spectral properties of the operators $\Gaa (z)$, 
\ defined by (\ref{0.3}) for the fixed $z\not\in\sigma (\point )$, 
which are the bounded operators of infinite-order on $\ell^2(\Z^d)$ with random 
potential, selfadjoint if $z=\lambda <0$. 

The new approximation scheme developed to study the point spectrum 
of $\point$ is composed by the two separate parts. 
At first we establish the convenient relation between the spectral properties of
$\point$ and $\gamr$, 
and secondly we prove the relevant results for $\gamr$.

We prove (Theorem 2) that the spectral measure of $\poin$ is concentrated 
on a set of generalized eigenvalues which correspond to the  
"polynomially bounded" distributional solutions (Definition 1) of the equation
(\ref{0.5}).
This theorem is an analogue of the well-known theorem for finite-difference
operators (lemma B.1 \cite{Sch}), and for the Schr\" odinger operators
with potentials of the so-called $K_\nu$-class, \cite{Si1,KoSe}.
This theorem is valid for arbitrary selfadjoint operator with the following property:

\bn \label{0.5a}
T_\delta^{-1}E(\Delta)
T_\delta^{-1} \;\;\;\hbox{is a trace class operator},
\en
where \ $T_\delta$ is the multiplication operator by $ (1+|x|)^{\delta}$ 
on $\L^2(\R^d)$,
$\delta > d/2$, $\R\supset \Delta$ is arbitrary bounded measurable set, 
$E(\Delta)$ is the corresponding spectral projection 
of $-\Delta_\alpha$.

This implies the integral estimates for the respective generalized
solutions. The pointwise polynomial boundedness depends on the
continuity properties of the spectral kernel.

Using the explicit expression (\ref{0.3}) for the resolvent kernel,
we prove the statement (\ref{0.5a}) \ for $\poin$. 
The general properties of the point potential imply that the corresponding distributional
solution does not represent a continuous function in the points $j\in \Z^d$: 
its local asymptotics is determined by 
\ $a_jG^0_\lambda(x-j)$ \ for \ $x\to j\in \Z^d$, where 
\ $G^0_\lambda(x,y)$ denotes the integral kernel of $(-\Delta -\lambda)^{-1}$
for $\lambda<0$.

Theorem 3 proves that arbitrary distributional solution of (\ref{0.5}), corresponding
to $\lambda<0$, admits the following representation:

\bn \label{0.8}
\psi_\lambda(x) = \sum_{i\in\Z^d} a_i\, G_\lambda^0(x-i),
\en
where for a.e. $\lambda<0$ (with respect to $E(d\lambda)$)  
\ $\overline  a = \{a_i\}_{i\in \Z^d}$ \ is a (generalized) polynomially
bounded solution of the equation $\gam \overline a = 0$. Thus there is 
a one-to-one correspondence between the generalized eigenfunctions of operators
$\poin$ and $\gam$. This result is deterministic (i.e. is valid for
arbitrary admissible values of \ $\alpha$\ ).

In order to prove the localization result for $\poin$, it suffices
to verify that within the range of the impurity spectrum
(corresponding to the sufficiently low concentration of the random impurities), 
every polynomially bounded
generalized eigenfunction of $\gamr$ decays exponentially 
with probability one (Theorem 4).

Results of such a type were usually used to establish exponential
localization for the finite-difference operator with the random potential 
(the Anderson model, \cite{A}). These results involve the multiscale analysis
in order to estimate the corresponding Green function
(\cite{FS,SW,DLS,D}).
The infinite-order generalization of the multiscale analysis scheme 
(\cite{G1,G2}) is applied to establish the required result. We briefly sketch the 
proof showing that estimates of Sect. 4 are uniform in $\lambda$. 

Notice, that the new approximation scheme allows to avoid the straightforward  analysis
in $\L^2(\R^d)$ (however, this also remains possible: localization for $\point$ 
was first established by the author in that way, which however turned out to be
more technical and less natural than the scheme of the proof 
of the following paper).

We also establish that the integrated density of states of $\point$ at 
the negative spectrum is absolutely continuous,
describing the relation between the \ids \ of 
operators $\point$ and $\Gamma_{\alpha_{|\Lambda}(\omega)}(\lambda)$ 
by using monotonicity in $\lambda$ of
the corresponding operators.

 The new approximation scheme presented in the following paper is directly applicable 
(i.e. the respective proofs may be repeated)
to prove the localization at weak disorder (via applying suitable adaptation 
of the multiscale analysis) for the random point 
interaction hamiltonians on $\L^2(\R^d)$, 
as well as for the operators on $\ell^2(\Z^d)$ (\cite{G8}) 
(established previously
for operators with random potential on $\ell^2(\Z^d)$ 
(\cite{An}) via different approximation technics). 

The results presented are also applicable to study the delocalization properties
and to establish existence of the pure absolutely continuous
spectrum in multidimensional random models (\cite{G4,G6}), 
as well as in multiple other applications (\cite{ES}). 

In Section 1 there are listed some notations and definitions used in the text, 
stated main results and discussed some applications.

In Section 2 is proved Theorem 2 on the polynomial boundedness of the generalized
eigenfunctions.

In Section 3 is proved Theorem 3 on the structure of distributional solutions.

In Section 4 is proved Theorem 4 (localization for  
finite-difference infinite-order operators).

In Section 5 is proved Theorem 5 (result on the density of states).

In Section 6 is proved the generalized spectral theorem. 

In the Appendix there are proved some auxiliary technical results.
 
%SECTION 1.

\section {Results} \label{s:1}
\setcounter {equation} {0}

Denote by \ $G_z(x,y)=(\poin -z)^{-1}(x,y),\;\; z\not\in \sigma(\poin)$, 
\ and \ $G_z^0(x,y)=(-\Delta -z)^{-1}(x,y)={e^{i\sqrt{z}|x-y|}
\over 4\pi |x-y|},\;\; z\not\in [0,+\infty)$, 
\ the Green's functions (resolvent kernels) of the operators $\poin$ 
and $-\Delta$; by \ ${\cal D}(-\Delta)$ \ and \ ${\cal D}(\poin)$ the
domains of the operators $-\Delta$ and $\poin$ (for the fixed
value of $\alpha=\alpha(\omega)$):

\begin{eqnarray} \label{1.1}
 {\cal D}(-\Delta) & = & \left \{  f \; | \; f\in \L^2(\R^3) ,\;\; \Delta f\in  \L^2(\R^d) 
       \; \;\;\hbox{in the distributional sense} \right\},  \nonumber \\
 {\cal D}(\poin) &  = & \left\{
     \begin{array}{c}   \varphi \; | \; \varphi(x) =  \varphi_0(x) + 
     \sum_{i\in \Z^d}
     a_i\: G_z^0(x-i), \;\; x\in \R^d \setminus \Z^d,  \\
     \hbox{where }\; \;  \varphi_0(x)\in {\cal D}(-\Delta),
     \;\; a_i = \sum_{j\in \Z^d}\, \Gamma_z^{-1}(i,j)\: \varphi_0(j)
     \end{array}  
     \right\}.
\end{eqnarray}
If $\varphi\in\Dpoin$, 

\bn \label{1.2}
(\poin - z)\: \varphi = (-\Delta - z)\: \varphi_0.
\en
Also denote $\overline{a}=\{a_i\}_{i\in \Z^d}$, 
$\overline {\varphi}_0=\{\varphi_0 (i)\}_{i\in \Z^d}$.

It is known (\cite{AGHH}) that selfadjoint operators
$\{\point\}_{\omega\in\Omega}$ are ergodic, the spectrum 
$\Sigma = \sigma(\point )$,
as well as the point spectrum ($\sigma_{pp}$), the absolutely continuous 
spectrum ($\sigma_{ac}$) and the singular continuous spectrum 
($\sigma_{sc}$) are non-random subsets
of $\R$ (i.e. they coincide with non-random sets with probability one),
and the discrete spectrum is empty (with probability one).

We say that \ $\alpha = \{\alpha(i)\in {\rm supp}\, 
P_{\alpha_i}\}_{i\in\Z^d}$ \ is an admissible potential.
\ Denote by $A$ the set of all admissible potentials. 

\begin{proposition} \label{pr:0} 

\begin{enumerate}

\item[{\bf 1}]. $\sigma\: (\poin)\: \subseteq \: \Sigma, \;\;\alpha\in A$;

\item[{\bf 2}]. $\Sigma = \bigcup_{\alpha\in A}\: \sigma\, (\poin)$.
\end{enumerate}
\end{proposition}

\begin{proposition} \label{pr:1}
Suppose ${\rm supp}\, \po \subset [\mu,\nu]$. Then

$$
\sigma(\poin) = \overline { \bigcup_{m\in \N} \; [a_m, b_m]},
$$
where $-\infty < a_0 < 0$, $a_m<b_m$, $b_m\leq \infty$, $m\in \N$
(the corresponding intervals may intersect).
Furthermore, there exists $\lambda_0<\infty$ such that if $\nu < \lambda_0$,
and ${\rm supp} \, \po = [\mu,\nu]$, then

$$
\sigma(\poin) = [a_0(\mu), b_0(\nu)] \cup [A(\mu),\infty) =
		\sigma(-\Delta_\mu) \cup \sigma(-\Delta_\nu),
$$
where $b_0(\nu)<0$, $A(\mu)\geq 0$.
\end{proposition} 

\noindent The proofs of Propositions \ref{pr:0}, \ref{pr:1} 
could be found in \cite{AGHH}.

The following is the result on localization.


%THEOREM 1
\begin{theorem} \label {t:1}

Given arbitrary $r,\; 0<r<1$, there exist $\delta_0=\delta_0(r,\po)$ and
$\delta_1=\delta_1(r,\po)$, $\delta_0>\delta_1>0$, such that:

\begin{enumerate}

\item[{\bf 1}]. if $\delta_1\leq\delta<\delta_0$, there exists an increasing
function $E_0=E_0(\delta) \leq -r_0<0$ determining the pure point spectrum 
of $\point$ in
the interval $(-\infty,E_0)$:

$$
\sigma(\point) \cap (-\infty,E_0) \subset \sigma_{pp};
$$

\item[{\bf 2}]. if $\delta\geq \delta_0(r)$, then 
$\sigma(\point) \cap (-\infty,-r) \subset \sigma_{pp}$. Moreover, 
if ${\rm supp} \, \po = [\mu,\nu]$, and $\nu< \lambda_0$
($\lambda_0$ is defined by Prop.\ref{pr:1}), then $\delta_0$ 
may be chosen such that for $\delta > \delta_0$:

$$ 
\sigma(\point) \cap (-\infty,0) \subset \sigma_{pp}. 
$$

\item[{\bf 3}]. The eigenfunctions of $\poin$, corresponding to the 
eigenvalues of the point spectrum in the cases 1 and 2 decay exponentially 
at infinity, with probability one:

$$  
\psi_{\lambda,\omega}(x) = \sum _{i\in \Z^d}\: a_{\lambda,\omega}(i)\:
					 G^0_\lambda (x-i),
$$
where

$$
|a_{\lambda,\omega}(i)|\leq C(\lambda,\omega)\; e^{-m_\lambda |i|},
	\;\;\; 0<m_\lambda<\sqrt{|\lambda|}.
$$
\end{enumerate}
\end{theorem}

\bigskip

\noindent {\bf Remark.} Eigenfunction corresponding to an eigenvalue
$\lambda <0$, decays exponentially at infinity with 
rate at least $\sqrt{|\lambda|}$.

\bigskip

\noindent {\bf Remark.}  Theorem \ref{t:1} holds for the
H\" older continuous distributions $\po$. In this case one considers the
potential $\{\lambda\, \alpha_i\}_{i\in \Z^d}$ and $\lambda$ measures the
strength of the disorder.

\bigskip

Theorem 1 follows by Theorems 2,3,4. In order to prove Theorem 1,
we study the properties of the spectral kernel
of $\poin$ and show that its spectral measure is concentrated on the set 
of generalized eigenvalues, corresponding to the solutions of equation 
(\ref{0.5}) that are "polynomially bounded" (Definition 1).

Denote by $E_{\alpha(\omega)}(\Delta)$ the spectral projection
of $\point$ (we usually omit dependence on $\omega$ in our notation
when this does not lead to contradiction).

We say that the statement $S(\lambda)$ is true for a.e.
$\lambda\in I\subset \R$ with respect to the spectral measure of $\poin$,
if \ $E_\alpha(\Delta)=0$ \ for \ $\Delta = \{\lambda\in I\: |\: S(\lambda)\;
\hbox{is not true}\}$.  

 Denote 
\  $\Lminus = \L^2(\R^d, d\mu_{-\delta})$,  where 
\ $d\mu_{-\delta}(x) = (1+|x|)^{-2\delta}\, {\rm d}x$;  and
\ $\Lplus = \L^2(\R^d, d\mu_{\delta})$,  where \ $d\mu_{\delta}(x)
= (1+|x|)^{2\delta}\, {\rm d}x, \;\; \delta>d/2$.

Denote \ $\lminus = \ell^2(\Z^d, d\nu_{-\delta})$, where
$\nu_{-\delta}(i)= (1+|i|)^{-2\delta}, \;\; i\in \Z^d$,
\ and \ $\lplus = \ell^2(\Z^d, d\nu_{+\delta})$,  
\ where $\nu_{+\delta}(i)=(1+|i|)^{2\delta}, \;\; i\in \Z^d, \;\; \delta > d/2$.


\begin{definition} \label{d:1} 
A number $\lambda\in \R$ is called 
the generalized eigenvalue, and a function $\psi_\lambda\in \Lminus$ the generalized
eigenfunction of $\poin$, if

\bn \label{1.3}
\langle\psi_\lambda,\, (\poin - \lambda)\varphi\rangle = 0
\en
for arbitrary $\varphi$ of type (\ref{1.1}) with $\varphi_0\in \C_0^\infty$,
and $z\not\in\sigma(-\Delta_\alpha)$. 
\end{definition}

Denote 
$$
{\cal D}_0(\poin)\: =\: 
\{\varphi\in {\cal D}|\; \varphi_0\in \C_0^\infty,\; 
z\leq E_0(\alpha,\lambda)<0\}.
$$

\noindent The following remarks mention that Definition \ref{d:1} is correct.

\bigskip

\noindent {\bf Remark.} Lemma A1 of Appendix implies that
it is possible to choose $E_0$ in such a way that for every admissible 
$\alpha (\omega )$,\ $\omega \in\Omega $, 
\ $\poin ({\cal D}_0(\poin))\, \subset \, \Lplus$.

\bigskip

\noindent {\bf Remark}. 
$\overline{{\cal D}_0(-\Delta_\alpha)}=\overline{{\cal D}(-\Delta_\alpha)}$,
where $\overline{A}$ denotes the closure of $A\subset {\cal H}$ in $\L^2$-norm (Lemma \ref{l:6.3}).

\bigskip

The following theorem is deterministic 
(i.e. is valid for arbitrary admissible \ $\alpha\in A$\ ).

\begin{theorem} \label{t:2}
Almost every $ \lambda\in \R$ with respect to the spectral measure of
$\poin$ is a generalized eigenvalue.
\end{theorem}

The following theorem describes explicitly the structure of the 
generalized eigenfunctions of $-\Delta_\alpha$.

A function $f(x), \; x\in \Z^d$ is said to decay exponentially
at infinity with rate $m>0$, if
$\limsup\limits_{|x|\to\infty} {\ln{|f(x)|} \over |x|}\leq -m$.

\begin{theorem} \label{t:3}
1. $f_\lambda(x)$ is a generalized eigenfunction of $\poin$,
relative to a negative generalized eigenvalue $\lambda < 0$, if
and only if

\bn \label{1.3a}
f_\lambda (x) = \sum_{i\in \Z^d}\: f_i\; G_\lambda^0(x-i),
\en
where  $\overline{f}_\lambda = \{ f_i\}_{i\in \Z^d}\; \in \lminus$ 
\ is a generalized solution of the equation

\bn \label{1.4}
\gam\; \overline{f}_\lambda = 0.
\en

2. $\lambda < 0$ \ is a proper eigenvalue of $\poin$ if and only if
$0$ is an eigenvalue of \ $\Gamma_\alpha(\lambda)$.
\end{theorem}

\bigskip

\noindent {\bf Theorem 3b} {\it $F_\lambda(x)$ is a generalized
eigenfunction of $-\Delta_\alpha$, relative to a positive generalized
eigenvalue $\lambda>0$, if and only if

\begin{equation} \label{1.3b}
F_\lambda(x) = \phi_\lambda(x) + \sum_{i\in \Z^d}\: f_i\: 
G_\lambda^0(x-i)\: \in \: \L^2_{-\delta},
\end{equation}
where $\phi_\lambda(x)$ is the distributional solution of the equation 
\ $(-\Delta-\lambda)
\phi(x) = 0$, \ and $\overline f = \{ f_j\}_{j\in\Z^d}\in \ell^2_{-\delta}$
satisfies 

$$
\sum_{j\in \Z^d}\: \Gamma_\alpha (\lambda; k,j)
\: f_j \: = \: \phi_\lambda(k)\;\;\; k\in \Z^d,
$$
where $\Gamma_\alpha (\lambda; k,j)$ is determined by (\ref{0.4}).}

Theorems 2 and 3 imply that there is the direct correspondence between 
the spectral properties of operator $\point$ and   
spectral properties of operators $\Gamma_\alpha(\lambda)$.
Lemma A2 of Appendix implies that generalized solutions of the equations
(\ref{0.5}) and (\ref{1.4}) exist simultaneously, and obey the same
asymptotics at infinity.

The point spectrum of finite-difference operators of infinite-order, with
the random potential was studied by \cite{G1}. The respective 
results are formulated by the following theorem:

\begin{theorem} \label{t:4}
Given an interval $I\subset (-\infty,-r)$, $r>0$, there exist
increasing function $E_0=E_0(\delta)$ \ ($E_0\leq -r$ 
$\forall\delta>0$), and 
$0<\delta_0=\delta_0(r,|I|)<\infty$, 
such that each generalized solution $\overline{f}_\lambda$ of (\ref{1.4}) decays
exponentially at infinity with probability 1, if $\lambda$ satisfy
one of the following conditions a), b):

{\em a)} $\lambda\in I$ if $\delta\geq\delta_0(r,|I|)$,

or

{\em b)} $\lambda\in (-\infty,-E_0(\delta))\: \cap \: I$ if
$0< \delta<\delta_0(r)$.
\end{theorem}

Theorem \ref{t:4} is proved in Section \ref{s:4}.

\noindent {\bf Remak}. Results presented by \cite{G1,G2} permit 
arbitrary complex-valued off-diagonal terms

$$
\Gamma_\alpha(\lambda;j,j^\prime)\: =\: 
\overline{\Gamma_\alpha(\lambda;j^\prime,j)},\;\;\; j,j^\prime\in\Z^d.
$$


The following is the result on the density of states.

\noindent Consider $N(\lambda)$, the \ids\ of $\point$ (the discussion
is presented in Sect. \ref{s:5}).

\begin{theorem} \label{t:5}
Suppose $\po$ has a density bounded by $\delta^{-1}$.
Given $r>0$, there exists $C=C(r)$ such that if $a<b$,
$(a,b)\subset (-\infty, -r)$, then

$$
|N(b)-N(a)|\leq 2C\delta ^{-1}\; |b-a|.
$$
\end{theorem}

\noindent{\bf Remark.} If $P_{\alpha_0}$ is H\"older continuous
of order $0<\gamma <1$, then $N(\lambda)$ is also H\"older continuous
of the same order.

Theorem 5 follows by Lemmas 5.1 and 5.2.
The following statements are some corollaries of the results above.

\bigskip

\noindent{\bf Corollary 1.} It is not hard to develop the arguments of
Theorem 2, and to prove the theorem on expansion in the generalized eigenfunctions
for \ $\poin$, as it was done by
\cite{Ti,Si1} for the wide class of selfadjoint operators,
which however does not concern the point interaction Hamiltonians (Sect. 2, 6). 

\bigskip

\noindent{\bf Corollary 2.} If $\lambda$ is a generalized eigenvalue of
$\poin$, then $\lambda\in\sigma(\poin)$.

Corollary 2 follows by Theorem 3, Lemma \ref{l:3.1} (Sect. \ref{s:3})
since the generalized eigenvalues of a finite-difference operator
belong to its spectrum.

\bigskip

\noindent{\bf Corollary 3.} $\sigma(\poin)$ is the closure of the set of
generalized eigenvalues.

\noindent Corollary 3 follows by Theorem 2, Corollary 2 since if
$\lambda_0\in \sigma(\poin)$ and \ $\varepsilon > 0$, then
$E_\alpha(\lambda_0-\varepsilon, \lambda_0+\varepsilon)>0$.

\bigskip

\noindent{\bf Corollary 4.}{\it (Location of the negative  spectrum)} 
\begin{eqnarray} \label{1.6}
\;\; \sigma\: (\point)\: \cap\: (-\infty,0)\: & = & \:
		\left\{ \lambda<0\: |\:
		0\in\sigma\, ( \Gamma_{\alpha(\omega)} (\lambda))\right\},  \\
\;\; \sigma_{pp}\: (\point)\: \cap\: (-\infty,0)\: & = & \:
		\left\{ \lambda<0\: |\:
		0\in\sigma_{pp}\, ( \Gamma_{\alpha(\omega)} (\lambda))\right\}.
\end{eqnarray}

\noindent {\it Proof}. Suppose $\lambda\in\sigma(\point)$. Then
Proposition 1 implies \ $\lambda\in \sigma(-\Delta_\alpha)$, where
$\alpha$ is some admissible potential.  
Hence there exists a sequence \ $F_n\in \Dpoin$, such that $\|F_n\|_{\L^2}=1$,
$\|(\poin -\lambda)F_n\|_{\L^2}\to 0$. Lemma \ref{l:3.1} implies
that there exists a sequence $f_n\in \ell^2(\Z^d)$, such that 
$\|f_n\|_{\ell^2}= 1$, $\|\Gamma_\alpha\, f_n\|_{\ell^2}\to 0$, hence
$0\in\sigma(\Gamma_\alpha)\subset \sigma(\Gamma_{\alpha(\omega)})$ 
(since Proposition 1 holds for finite-difference ergodic operators).
Now assume $\lambda\in\sigma_{pp}(\point)$. Then there exists a set
$\Omega_0\subset\Omega$, such that ${\rm\bf P}\, (\Omega_0)=1$, and
$\lambda\in\sigma_{pp}(-\Delta_{\omega_0})$ for $\omega_0\in\Omega_0$.
According to Theorem 3 this implies that $0 \in\sigma_{pp}(\Gamma_{\omega_0})$
 $\forall\omega_0\in\Omega_0$, and hence
$0\in\sigma_{pp}(\Gamma_{\alpha(\omega)})$. The converse is proved
by the same way.

\bigskip

\noindent The generalized spectral theorem (Theorem 6)
is proved in Section 6.

\bigskip

\noindent {\bf Theorem 7. (Absence of the mixed point spectrum)} 

$$
\sigma_{pp}(-\Delta_\alpha)\cap (0,+\infty)=\emptyset.
$$

\bigskip

\noindent The proof of Theorem 7 is analogues to the proof
of absence of the mixed point spectrum in the Anderson model 
(\cite{G4}).

\bigskip

\noindent {\bf Remark.}

$$
\sigma(-\Delta_{\alpha(\omega)})\:\cap\: (a,+\infty) \subset\: \sigma_{ac},
$$
for some $a\geq 0$ 
(i.e. the conductivity spectrum is pure absolutely continuous (1999)).

\bigskip

\noindent The proof is a part of possible separate publication by the author.

Approximation scheme presented by the following paper is applicable
to describe spectral properties of different other models important in physics.

\bigskip

\noindent {\bf Example. (Pure point spectrum for random point interaction  
Hamiltonian in the presence of magnetic field)}

\noindent Consider the random point interaction Hamiltonian on $\L^2(\R^d)$,
$d=2$ or $d=3$, in the presence of magnetic field 
{${\bf B} = \nabla\times {\bf A} =  {\rm const} \ne 0$}:

$$
H_{\alpha(\omega)}\: =\: (-i\nabla-{\bf A})^2\: 
+\: \sum_{j\in\Z^d}\: \lambda\alpha_\omega(j)\delta_d(.-j),
$$
where $\alpha(\omega)$ are i.i.d.v. with bounded distribution
density of compact support. 

\noindent Applying methods and results described by Theorems 1-7,
it is not hard to prove that the spectrum of $H$ is pure point,
and corresponding eigenfunctions are exponentially localized
(with probability 1) in any closed interval 
$I\subset (E_n,E_{n+1})\cap\sigma(H_{\alpha(\omega)})$,
where $\{E_n\}_{n\in \N}$ are the corresponding Landau levels,
if $\lambda >\lambda_0({\bf B},|I|,d,P_\alpha)$ (at large disorder).

\noindent The rigorous poof of the latter result requires
to prove the exponential decay of $((-i\nabla -A)^2-z)^{-1}(x,y)$
as function of $|x-y|$, to verify polynomial boundedness
of the generalized eigenfunctions (presented proof
of Theorem 6 requires only semiboundedness of $H$), and
to confirm validity of the initial estimates $(P1)$,
$(P2)$ at high disorder (appropriate values of $\lambda,{\bf B}$), 
which according to Theorem 4, ensure exponential localization 
in the corresponding spectral bands.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%SECTION 2.

\section {On properties of the spectral kernel} \label{s:2}
\setcounter {equation} {0}

\noindent{\bf Proof of Theorem 2.} 

\noindent Denote by $E_\alpha$ the spectral projection 
of \ $H=-\Delta_\alpha$, \  $G_E=(-\Delta_\alpha-E)^{-1}$,
$E<\inf\sigma(-\Delta_\alpha)$,
\ $T_\delta=(1+|x|)^{\delta}$ (multiplication operator), $\delta>d/2$.

\begin{lemma} \label{l:2.1}
If $\Delta \subset \R$ is a bounded measurable set,
$A(\Delta)=T_\delta^{-1}E(\Delta)T_\delta^{-1}$ is a trace class operator.
\end{lemma}

\noindent{\it Proof.} 
Suppose  $E< \inf \sigma(-\Delta_\alpha)$ and consider the function  

$$
S_0(x)={1 \over |x|} \chi_{\Lambda_{1/2}}(x),
$$
where $\chi_{\Lambda_{1/2}}(x)$ is the characteristic function of the
cube $\Lambda_{1/2}(0)$, centered at 0, with side-length $1/2$. Denote

$$
S(x)=\sum_{i\in \Z^d} S_0 (x-i),
$$
then $ S(x)>0$ and $|S^{-1}(x)|<1/2$. Denote by $S$ the
corresponding multiplication operator.
\noindent   Operator on $\L^2(\R^d)$
with the integral kernel $a(x,y)$: 

$$
\sup_x\int\limits_{\R^d} |a(x,y)|^2\; dy <\infty,
$$
is called a Carleman operator.

The resolvent formula
(\ref{0.3})-(\ref{0.4}) implies that $S^{-1}G_E$ is a Carleman operator,
since 

$$
\sup_x
\int\limits_{\R^d} S^{-2}(x)\; |G_E(x,y)|^2\; dy <\infty
$$
(apply Lemma A2 of Appendix and the dominated convergence theorem). Hence 
$S^{-1}G_E$ is a bounded mapping from $\L^2$ to
$\L^\infty$ (\cite{Kor,Si1}). Now prove that if a real 
function $f(x)$ satisfies $f(x)\leq \frac C {1+|x|}$, 
$x\in\sigma(-\Delta_\alpha)$, then the 
operator $S^{-1}f(-\Delta_\alpha)$ maps $\L^2$ to $\L^\infty$ 
(i.e. is of Carleman type (\cite {Kor})). For this purpose consider 
the function

$$
g(x)=(x-E)f(x), \;\;\; E<\inf \sigma(-\Delta_\alpha).
$$
Then $f(-\Delta_\alpha)=(-\Delta_\alpha-E)^{-1}g(-\Delta_\alpha)$, i.e.

$$
S^{-1}f(-\Delta_\alpha)=[S^{-1}\; (-\Delta_\alpha-E)^{-1}]\; g(-\Delta_\alpha).
$$
Since
$g(-\Delta_\alpha)$ is bounded in $\L^2$, and it is proved that 
$S^{-1}(-\Delta_\alpha-E)^{-1}$ is bounded from $\L^2$ to $\L^\infty$, then
$S^{-1}f(-\Delta_\alpha)$ maps $\L^2$ to $\L^\infty$. Thus $S^{-1}E(\Delta)$ 
is a Carleman operator. Denote its kernel by $[S^{-1}E(\Delta)](x,y)$, 
then 

$$
\sup_x\int\limits_{\R^d}[S^{-1}\; E(\Delta)]^2(x,y)\; dy=B<\infty.
$$
Now prove that $T_\delta^{-1}E(\Delta)$ is a Hilbert-Schmidt operator. 
Indeed, \ $T_\delta^{-1}\:
E(\Delta)=T_\delta^{-1}S\; [S^{-1}E(\Delta)]$, and

$$
\int\limits_{\R^d\times \R^d}
(1+|x|)^{-2\delta}\; S^2(x)\;[S^{-1}E(\Delta)]^2(x,y)\; dxdy
$$
$$
\leq B\;
\int\limits_{\R^d} (1+|x|)^{-2\delta}\; S^2(x)\; dx<\infty.
$$
Hence 
$$
T_\delta^{-1}E(\Delta), \;\;\; 
E(\Delta)T_\delta^{-1}=(T_\delta^{-1}E(\Delta))^*
$$ 
are the Hilbert-Schmidt operators, and 
$A(\Delta)=T_\delta^{-1}E(\Delta)T_\delta^{-1}$ is of trace class. Lemma 
\ref{l:2.1} is proved.

Now follow the standard argument (\cite{Si1}). Clearly,
since $A(\Delta)$ is nonnegative, 
given arbitrary bounded set $\Delta=
\cup_{n=1}^\infty \Delta_n$, such that $\Delta_i\cap \Delta_j=\emptyset$, 

$$
A(\Delta)=s-\lim\limits_{m\to\infty}\sum_{n=1}^m
A(\Delta_n).
$$
It follows that the operator-valued measure $A(\Delta)$ is
absolutely continuous with respect to the scalar measure

$$
\rho(\Delta)={\rm Tr}(T_\delta^{-1}\; E(\Delta)\; T_\delta^{-1}),
$$
so, by the Radon-Nikodym theorem, for $d\rho$-a.e. $\lambda$, there exists a
positive measurable trace class operator-valued function $a(\lambda)$
such that 

$$
A(\Delta)=\int\limits_\Delta a(\lambda)\: d\rho(\lambda),\;\;\;
{\rm Tr}(a(\lambda))=1.
$$
It is easy to check that $d\rho$ is equivalent to $dE(\Delta)$
($d\rho$ is the spectral measure). By $a(x,y;\lambda)$ denote the 
integral kernel of $a(\lambda)$. Consider the function

$$
F(\lambda;x,y)=(1+|x|)^\delta\; a(x,y;\lambda)\; (1+|y|)^\delta.
$$

\begin{lemma} \label{l:2.2}
There exist a spectral measure $d\rho(\lambda)$, and a function 
$\ef(\lambda;x,y)$, measurable in $x,y$ and  $\lambda$, which has 
the following properties (all the statements are valid for $d\rho$-almost all 
$\lambda$):

\begin{enumerate}

\item[{\bf (1)}]  

\bn \label{2.1}
\int\limits_{\R^d\times \R^d} |\ef(\lambda;x,y)|^2\;
(1+|x|)^{-2\delta}\;(1+|y|)^{-2\delta}\; dxdy\leq 1.  
\en

\medskip

\item[{\bf (2)}] Given arbitrary bounded measurable function $g(\lambda), \;
\lambda\in\sigma(-\Delta_\alpha)$, \ and arbitrary 
\ $\psi\in \D0 (-\Delta_\alpha),\; \varphi\in \L^2_\delta$:

\bn\label{2.2} 
\langle \varphi,g(-\Delta_\alpha)\psi\rangle=
\int\limits_{\sigma(-\Delta_\alpha)} g(\lambda)\; 
[\int\limits_{\R^d\times \R^d}\ef(\lambda;x,y)\; \overline{\varphi(x)}\;
\psi(y)\; dx dy]\; d\rho(\lambda).
\en

\medskip

\item[{\bf (3)}] 
$\ef(\lambda;x,y)$ is continuous in 
$(x,y)\in (\R^d\times\R^d)\backslash (\Z^d\times \Z^d)$.

\medskip

\item[{\bf (4)}] Consider $\widetilde{H_0}=-\Delta_x - \Delta_y$,  $x,y \in \R^d$, 
then for arbitrary $\widetilde{f_0}\in 
\widetilde C_0^{\infty}(\R^d\times \R^d\setminus 
\Z^d\times \Z^d)$ (the set of infinitely differentiable functions on 
$\L^2(\R^6)$ with compact support containing no point of $\Z^d\times \Z^d$):

\bn \label{2.3}
\langle \ef(\lambda;\cdot,\cdot),
(\widetilde{H_0}-2 \lambda)\widetilde{f_0}\rangle =0.
\en

\medskip

\item[{\bf (5)}] 
$\ef(\lambda;\cdot,y)$ 
is a generalized eigenfunction of $-\Delta_\alpha$.
\end{enumerate}
\end{lemma}

\noindent {\it Proof.} 
1. Follows since ${\rm Tr}(a(\lambda))=1$. 

\medskip

2. (\ref{2.1}) implies that the integrand in [.] is 
absolutely integrable, and the integral is a bounded function of $\lambda$. 
Thus (\ref{2.2}) follows by the definition of $E(\Delta)$.) 

\medskip

3. Since
${-\Delta_\alpha}_{|\C_0^\infty(\R^d\setminus \Z^d)}\: = \:
-\Delta_{|\C_0^{\infty}(\R^d\setminus \Z^d)}$,
$\widetilde{H_0}$ is a proper
Laplacian on $\widetilde \C_0^{\infty}(\R^d\times \R^d\setminus 
\Z^d\times \Z^d)$, the statement 3) follows by 4).

4. Suppose $\varphi_x,\psi_y \in \C_0^{\infty}(\R^d\setminus \Z^d)$, then
\begin{eqnarray*}
[\widetilde{H_0}\: \varphi_x\: \psi_y](x,y)  
		  & = & [-\Delta_x\varphi](x)\;  \psi(y) + \varphi(x)\; 
[-\Delta_y\psi](y) \\
		  & = & [H_x\varphi](x)\; \psi(y) + \varphi(x)\; [H_y\psi](y).
\end{eqnarray*}
Hence by (\ref{2.2}) for arbitrary $g\in \C_0^{\infty}(\sigma(-\Delta_\alpha))$, 
$\phi,\psi\in \C_0^\infty(\R^d\setminus \Z^d)$:

\begin{eqnarray*} 
  &    & \int_{\sigma(-\Delta_\alpha)} g(\lambda)\; \int_{\R^d\times\R^d}        
  F(\lambda;x,y)\; [\widetilde{H_0}\overline{\varphi_x}\; 
     \psi_y](x,y)\; dxdy\;        d\rho(\lambda)   \\ 
  & =  & \int_{\sigma(-\Delta_\alpha)} g(\lambda)\;
	 [\int_{\R^d\times \R^d} 2\lambda\;         
   F(\lambda;x,y)\; \overline{\varphi(x)}\; \psi(y)\;dx dy]\; 
	d\rho(\lambda), 
\end{eqnarray*}
which implies (\ref{2.3}).
 
5. Relation (\ref{2.2}) and Fubini theorem imply
that for arbitrary $\varphi\in\D0(-\Delta_\alpha),\; \psi\in \C_0^\infty$

$$
\int_{\R^d}\; \left( \int_{\R^d} F(\lambda;x,y)\: (-\Delta_\alpha-\lambda)
\varphi(x)\: dx \right) \; \psi(y)\: dy = 0,
$$
hence 
\bn \label{2.3a}
\int_{\R^d} F(\lambda;x,y)\: (-\Delta_\alpha-\lambda)\varphi(x)\: dx = 0
\en
in every point $y$ of continuity of $F$. Now fix arbitrary $y\not\in \Z^d$ and
consider functions $f_\lambda(x) = F(\lambda;x,y)$. Then for $d\rho$ -
a.e. $\lambda$, \ $f_\lambda \in\Lminus$  ((\ref{2.1})), and $f_\lambda$
satisfies (\ref{2.3a}). Lemma \ref{l:2.2} is proved.
The functions constructed are the generalized eigenfunctions of 
$-\Delta_\alpha$ for
$dE$ - almost all $\lambda\in\Delta$, Theorem 2 is proved. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%SECTION 3.

\section {Explicit expressions for the generalized eigenfunctions} \label{s:3}
\setcounter {equation} {0}

\noindent{\bf Proof of Theorem 3.}  

Suppose $f_\lambda(x)$ is a generalized
eigenfunction of $\poin$, \  $\lambda<0$,  
then $f\in \Lminus,\;\; \delta>d/2$,
and

\bn \label{3.1}
\langle f,(\poin -\lambda)\: \varphi\rangle = 0
\en
for arbitrary $\varphi$ of type:

\bn \label{3.2}
\varphi(x)= \varphi_0(x) + \sum_{i\in \Z^d} \: a_i \: G_E(x-i),
\en
for some $E< \inf {\rm supp } \, \po $, \  $a_i = \sum_{j\in \Z^d}\: 
\Gamma_\alpha^{-1}(E;i,j)\: \varphi(j)$, \ $\varphi_0\in \C_0^\infty$.

\noindent  (\ref{3.1}) and the definition of $\poin$ imply, that if 

$$
\phi\in\H02 := \left\{ \varphi\in H_0^{2,2}(\R^d)\: |\: \varphi(j)=0,\; 
   j\in \Z^d, \;\; {\rm supp}\,\varphi \hbox{ is finite } \right\},
$$
then
\bn \label{3.3}
\langle f,\, (-\Delta-\lambda)\varphi\rangle = 0.
\en

Now prove that if $g\in\Lminus$ satisfies (\ref{3.3}) for arbitrary
$\varphi\in\H02$, then 

$$
g\in \overline { {\rm Lin} \{ G_\lambda^0(.-j)\} }_{j\in \Z^d} 
$$ 
(i.e. $g$ belongs to the closure of the linear span of 
$\{G_\lambda^0(.-j)\}_{j\in \Z^d}$ in $\Lminus$). 

\noindent First prove that there exists a sequence $\{\gamma_i\}_{i\in
\Z^d}$, depending only on $g$ and $\lambda$, such that for arbitrary
$\psi\in \C_0^\infty\subset {\cal D}(-\Delta)$ with supp$\,\psi
\subset\Lambda\subset \R^d$:

$$
\langle g,\, (-\Delta-\lambda)\psi\rangle = \sum_{i\in\Lambda}\: 
\gamma_i\: \psi(i).
$$
For this purpose consider the function
$$
\widetilde{\psi}(x) = \psi(x) - \sum_{i\in\Lambda}\: \psi(i)\: \theta(x-i),
$$
where the function $\theta\subset
\C_0^\infty$ is such that ${\rm supp}\, \theta\subset\Lambda_{1/2}(0)$,
$\theta(0)=1, \; |(-\Delta -\lambda)\theta(x)|\leq \tau_\lambda<\infty$.
Then $\widetilde{\psi}\in\H02$, and

$$
\langle g,\, (-\Delta-\lambda)\widetilde\psi)\rangle = 0.
$$
It follows that
\begin{eqnarray*}
  \langle g,\, (-\Delta-\lambda)\psi\rangle & = & \sum_{i\in \Z^d}\: 
  \langle g,\, (-\Delta-\lambda)\: \theta(x-i)\rangle\: \psi(i) \\
	& = & \sum_{i\in \Z^d}\: \gamma_i\: \psi(i)
\end{eqnarray*}
(the r.h.s. is finite since the sum is taken over a finite number
of points $i\in \Z^d\cap {\rm supp}\, \psi$). 
Hence $g\in \L^2_{loc}$, and

\begin{eqnarray} \label{3.4}
\gamma_i & =    & \langle g,\, 
      (-\Delta - \lambda)\theta(x-i)\rangle \nonumber \\
	 & \leq & C_\lambda\: \left ( \int_{\Lambda_{1/2}(i)}
	 |g(x)|^2\: (1+|x|)^{-2\delta}\: d x \right )^{1/2}
	 \; \left ( \int_{\Lambda_{1/2}(i)}
      (1+|x|)^{2\delta}\: d x \right )^{1/2}  \nonumber \\
       & \leq & C_\lambda\: g_i\: (2+|i|)^\delta,
\end{eqnarray}
where $\sum\limits_{i\in \Z^d} g_i^2<C<\infty$.

\noindent Notice that $\gamma_i$ are uniquely
defined by $g$ and $\lambda$. Indeed, if there exist
$\{\gamma_i\}_{i\in \Z^d}$ and $\{\beta_i\}_{i\in \Z^d}$ such that for
arbitrary $\psi\in \C_0^\infty$

$$
\langle g, \, (-\Delta-\lambda)\psi\rangle = \sum_{i\in \Z^d}\: 
\gamma_i\: \psi(i) = \sum_{i\in \Z^d}\: \beta_i\: \psi(i),
$$
choose $\psi$ such that ${\rm supp}\, \psi \subset \Lambda_{1/2}(i)$,
then $\gamma_i = \beta_i$,  $i\in \Z^d$. Suppose

\bn \label{3.5}
h(x) = \sum_{i\in \Z^d}\: \gamma_i\: G_\lambda^0(x-i).
\en
Since $\gamma_i$ satisfy (\ref{3.4}) then for $\lambda<0$ the series 
(\ref{3.5}) is absolutely convergent for arbitrary $x\not\in \Z^d$ and 
$h\in\Lminus$ (Lemma A2 of Appendix); if $\lambda > 0$, define
(\ref{3.5}) in the distributional sense.

It follows that

$$
\langle g-h,\, (-\Delta-\lambda)\psi\rangle = 0
$$
if $\psi\in \C_0^\infty$, so  $\phi=(g-h)$ is a distributional solution
of the equation

\bn \label{3.6}
(-\Delta-\lambda)\phi = 0.
\en
The following paper mainly concerns with the case $\lambda <0$. 
Then $\phi\in\Lminus$, and
(\ref{3.6}) implies that $\phi = 0$.

It is proved, that if $f\in \Lminus$ satisfies (\ref{3.3}), then

\bn \label{3.7}
f = \sum_{i\in \Z^d} \: f_i\: G_\lambda^0(x-i),
\en
where the series converges in $\|.\|_{\Lminus}$ and
$\{f_i\}_{i\in \Z^d}$ satisfy (\ref{3.4}).

\noindent So far there had only been used the property that $\poin$ is a selfadjoint extension
of $-\Delta_{|\C_0^{\infty}(\R^d\setminus \Z^d)}$. Now consider the explicit
expression of this extension in order to determine the coefficients of the expansion
(\ref{3.7}). Consider an arbitrary function $\varphi$ of type (\ref{3.2}),
with $\varphi_0\in \C_0^\infty$. Set 
$\overline{\varphi}_0=\{\varphi_0(j)\}_{j\in \Z^d}$, 
$\overline c = \{c(j)\}_{j\in \Z^d}$,
$\overline c = \Gamma_E^{-1}\overline\varphi_0$; $E<\inf {\rm supp}\, \po$, 
and \ $|E|$ is chosen large enough, so that
$\varphi\in\Lplus$ and $(\poin-\lambda)\varphi\in\Lplus$ \ (Lemma A1).

Since
\bn \label{3.8}
 (\poin-\lambda)\varphi(x) = (-\Delta - \lambda)\varphi_0(x)
		+ (E-\lambda)\: \sum_{i\in \Z^d}\: c_i\: G_E(x-i),
\en
substituting (\ref{3.7}) and (\ref{3.8}) in (\ref{3.1}), and taking into
account 

\begin{eqnarray*}
&\; & \langle G_\lambda^0(x-i), (-\Delta-\lambda)\varphi_0\rangle = \varphi_0(i), \\
&\; & \langle G_\lambda^0(x-i), G_E^0(x-j)\rangle = {1\over E-\lambda}
			(\Gamma_\lambda(i,j) - \Gamma_E(i,j)),
\end{eqnarray*}
it follows 

$$
\sum_{i\in {\rm supp}\, \phi_0} \;f_i\: \varphi_0(i)\: +\:
\sum_{i,j\in \Z^d}\: f_i\: c_j\: \left [ \Gamma_\lambda(i,j)-\Gamma_E(i,j) 
\right ] = 0.
$$
Since $\overline\varphi_0 = \Gamma_E \overline c$, the latter equality 
could be written as

\bn \label{3.9}
\sum_{k\in \Z^d}\: c_k\: \{\: \sum_{j\in \Z^d}\: \Gamma_\lambda(k,j)\: f_j 
\: +\: (E-\lambda)\int\limits_{\R^d}\phi_\lambda(x)G_E^0(x-k)\:dx\: \}\: =\: 0.
\en

The statement of Theorem 3a 1) follows immediately from (\ref{3.9}) since
$\phi_\lambda=0$, and $\Gamma_E^{-1}:\:
\ell^2(\Z^d)\rightarrow\ell^2(\Z^d)$ is a bijection, if $\lambda<0$
and $|E|>>0$. 

\noindent By choosing the sequence \ $\overline\varphi_{0,n} = \{\delta(i-n)\}_{i\in \Z^d}$
\ determine the sequence $\overline {c}_n = \{\Gamma_E^{-1}(j,n)\}_{j\in
\Z^d}$, where each $\overline {c}_n$ satisfies (\ref{3.9}). This implies


\bn \label{3.10}
\Gamma_E^{-1}\: \left [ \sum_{j\in \Z^d} \: 
\Gamma_\lambda (k,j)\: f_i\right ] = 0, \;\;\;\; k\in \Z^d.
\en

\noindent If \ $\lambda < 0$, then

\bn \label{3.11}
\Gamma_\lambda\: \overline f\: =\: 0, \hbox{\enspace} \;\;
|f_j|\leq(1+|j|)^\delta,\;\;\; j\in \Z^d.
\en
Consider the function $f(x)$ \ defined by (\ref{3.7}). It follows 
from Lemma A2, that \ $f\in \Lminus$ \ and the argument above implies 
that $f$ satisfies (\ref{3.1}) since (\ref{3.9}) is valid. This proves 
the statement of Theorem 3a 1).

Suppose that $0$ is an eigenvalue of $\Gamma_\lambda$, $\lambda<0$,
i.e. (\ref{3.11}) is valid and \ $\overline f\in \ell^2 (\Z^d)$.
\ Lemma A2 implies
that $f$, defined by (\ref{3.7}), belongs to \ $\L^2(\R^d)$. \ Define

\bn \label{3.12}
\varphi(x)\: =\: \varphi_0(x)\: +\: \sum_{j\in \Z^d}\: f_j\, G_E^0(x-j)
\en
for some \ $E<0$, $E\ne\lambda$, \ where

\bn \label{3.13}
\varphi_0(x) = (\lambda - E)\: (-\Delta - E)^{-1}\: \sum_{j\in \Z^d}
\: f_j\: G_\lambda^0(x-j).
\en
Clearly 

\begin{eqnarray*}
\varphi_0(x) & = & (\lambda - E)\: \sum_{j\in \Z^d}\: 
	f_j\: G_E^0\, G_\lambda^0(x-j) \\
   & = & \sum_{j\in \Z^d} \: f_j\: \left \{ G_\lambda^0(x-j) - G_E^0(x-j)\right\},
\end{eqnarray*}
and hence

$$
\varphi(x) \: = \: f(x) \: = \: \sum_{i\in \Z^d}\: f_i\: G_\lambda^0 (x-i).
$$
Notice that \ $\varphi\in \Dpoin$, \ since it follows by (\ref{3.11})
and (\ref{3.13}) that 

$$
\varphi_0(j)\:  = \: \sum_{i\in \Z^d}\: f_i\: \left [\Gamma_E(j,i) -
		       \Gamma_\lambda(j,i) \right ]
	    \:  = \: \sum_{i\in \Z^d}\: f_i\: \Gamma_E(j,i).
$$
It remains to check that $f$ is the eigenfunction:

$$
\poin\: f\:  =  \: (-\Delta-E)\, f \:  +  \: E\, f \: 
	     =  \: (\lambda - E)\, f \:  + \: E\, f\: = \: \lambda\, f.
$$
This proves Theorem 3a.

{\it Proof of Theorem 3b.}
Notice that (\ref{3.1}) is valid, by limiting agument, for abitrary $\varphi\in \Dpoin$
such that $(-\Delta_\alpha-\lambda)\varphi\in \L^2_\delta$. 
Consider the sequence $\varphi_n(x)$ of functions of type (\ref{3.2}),
$n\in \Z^d$, where $\varphi_n^0(x)\in \Dpoin$ satisfy

$$
\varphi_n^0(x) = \Gamma_E(x-n),
$$
$\overline{a}_n = \Gamma_E^{-1}\, \overline{\varphi}_n^0
\, = \, \{\delta(n-k)\}_{k\in \Z^d}$, i.e. 

$$
\overline{a}_n(k)\, = \,\cases{ 1, &if $k=n$,\cr 0, &if $k\ne n$\cr},
$$ 
then $\varphi_n$ satisfies (\ref{3.1}), and (\ref{3.9})
implies 

\bn \label{eq3.b}
\sum_{j\in \Z^d}\: \Gamma_\alpha(\lambda;k,j)\: f_j\: = \:
\phi_\lambda(k), \;\;\; k\in\Z^d.
\en
Theorem 3 is proved.

\medskip

Modifying slightly the latter argument, prove the following
useful statement.

\begin{lemma} \label{l:3.1}

\medskip
1. Suppose that a sequence
\  $f_n\in \ell^2(\Z^d)$, \ $0 < \|f_n\|_{\ell^2}\leq C <\infty$ 
is such that for some \ $\lambda<0$,

\bn \label{l1}
\|\Gamma_\lambda\, f_n\|_{\ell^2}\to 0 \; \hbox{ for } \; n\to\infty, 
\en
then there exists the sequence \ $F_n\in \Dpoin$ \ of functions which satisfy

\bn \label{l2}
\|(\poin - \lambda)\: F_n\|_{\L^2} \to 0 \;\; \hbox { for } \;\; n\to \infty
\;\; \hbox{ and }\;\; 0< \|F_n\|_{\L^2}\leq C_1 <\infty.
\en

2. Suppose that a sequence $F_n\in \L^2(\R^d)$
\ satisfies (\ref{l2}). Then there exists the corresponding sequence 
\ $f_n\in\ell^2(\Z^d)$ \ such that (\ref{l1}) is valid.

In both cases one may choose the functions of norm 1 for the second 
sequence if the first one consists of functions of norm 1.

\end{lemma}

\noindent {\it Proof.} 
1. Define

\begin{eqnarray*}
F_n(x) \: & = &\: \varphi_n(x) \: + \:  \sum_{i\in \Z^d}\: f_n(i)\, 
       G_E^0(x-i), \\
  \varphi_n(x)\: & = & \: (\lambda - E)\: (-\Delta-E)^{-1}\: 
	\sum_{i\in \Z^d}\: f_n(i)\: G_\lambda^0(x-i)\: + \: g_n(x),
\end{eqnarray*}
where \ $g_n(x)\, =\, \sum_{i\in \Z^d}\: a_{n,i}\, \theta(x-i)$, the function 
 \ $\theta\in \C_0^\infty$ is such that \ ${\rm supp}\, \theta\subset 
\Lambda_{1/2}$,  $\theta(0)=1$, $a_{n,i}=\sum_{i\in \Z^d}\: \Gamma_\lambda(i,j)
\, f_n(j)\: =\: \Gamma_\lambda\, \overline f_n\, (i),\; E<0$.

\noindent Then  

$$
F_n\: =\: \sum_{i\in \Z^d}\: f_n(i)\: G^0_\lambda (x-i)\: + g_n(x) 
\in\Dpoin,
$$
and

$$
(\poin - \lambda)\: F_n(x) =  (-\Delta-\lambda)\, g_n(x).
$$
It follows that \ $\|\overline a_n\|_{\ell^2} \: = \|\Gamma_\lambda\: 
\overline f_n\|_{\ell^2}
\to 0$ as $n\to \infty$, \ so \ $\|g_n\|_{\L^2}\to 0$ \ and 
\ $\|(-\Delta-\lambda)
g_n\|_{\L^2}\to 0$ as $n\to\infty$. The argument of Lemma A2 
implies

$$
C_1^\prime\|f_n\|_{\ell^2}\leq \|F_n\|_{\L^2}\leq C_1^{\prime\prime}\|f_n\|_{\ell^2}.
$$

This implies that one may take normalized functions 
\ $\widetilde F_n = {F_n\over \|F_n\|}$, if $f_n$ are normalized.

\noindent 2. Suppose \ $(-\Delta_\alpha-\lambda)\: F_n(x)\, =\, H_n(x)$. 
\ Since \ $F_n\in\Dpoin$, \ it has the form

\bn \label{3.16}
F_n(x) = \psi_n(x) + \sum_{i\in \Z^d}\: a_{n,i}\: G^0_{E_n}(x-i),
\en
for some $E_n<0$, $\psi_n\in {\cal D}(-\Delta)$,  
$\overline a_n  = \{a_{n,i}\}_{i\in\Z^d}$, $\overline\psi_n = 
\{\psi_n(i)\}_{i\in\Z^d}$, such that $\Gamma_{E_n}\overline a_n\, =\,
\overline\psi_n$.
\noindent Define \ $\overline f_n\: = \: \overline a_n$. 
It follows by (\ref{3.16}):

$$
(-\Delta_\alpha-\lambda)F_n(x) 
	 \: = \:  (-\Delta-E_n)\psi_n(x) \: + \: (E_n-\lambda)F_n(x),
$$
and hence
$$
 (-\Delta-\lambda)\psi_n(x) \: = \: H_n(x)\: + 
	    \: (\lambda-E_n)  \sum_{i\in \Z^d}\, a_{n,i}\, G_{E_n}^0(x-i).
$$
The resolvent identity implies

\bn \label{3.17}
\psi_n(x) = G_\lambda^0\, H_n(x) \: + \: \sum_{i\in\Z^d} 
    (G_\lambda^0 - G_{E_n}^0)(x-i)\, a_{n,i}.
\en
Substituting $x=j$ into (\ref{3.17}), 

$$
\psi_n(j)\: = \: G_\lambda^0\, H_n(j)\: + 
	      \: \sum_{i\in \Z^d}\, a_{n,i}\, [\Gamma_{E_n}(i-j) -
\Gamma_\lambda (i-j)].
$$
Since $\Gamma_{E_n}\overline a_n\, =\, \overline\psi_n$, it follows that $a_n$
satisfies the equation:

\bn \label{3.18}
  (-\Delta-\lambda)^{-1}\: H_n(j)\: = \: \Gamma_\lambda\: \overline a_n(j), 
  \;\;\;\;  j\in \Z^d.
\en
Observe that the r.h.s. of (\ref{3.17}) is a continuous function of 
$x$ since it belongs to
${\cal D}(-\Delta)$, i.e. it is well defined in the points of $\Z^d$ and
tends to $0$ as $n\to\infty$. It follows by (\ref{3.16}) and 
(\ref{3.17}) 

$$
F_n(x) = \sum_{i\in \Z^d}\: a_{n,i}\: G_\lambda^0(x-i)\: + \: 
(-\Delta-\lambda)^{-1}\, H_n(x).
$$
Since \ $F_n$ \ are uniformly bounded in \ $\L^2(\R^d)$,  
\ $\|(-\Delta-\lambda)^{-1}\, H_n\|_{\L^2}$ \ tends to zero as $n\to \infty$, 
($\lambda <0$),  and the argument of Lemma A2 implies 

$$
C_2^\prime \|F_n\|_{\L^2}\leq \|a_n\|_{\ell^2} \leq C_2^{\prime\prime} 
\|F_n\|_{\L^2},
$$
which implies that $a_n$ may be chosen normalized if $F_n$
are normalized. 

\noindent{\bf Remark.} This statement implies that there is the geometrical
relation between the spectral sets of the corresponding operators (Sect.1,
Corollary 4).


%SECTION 4.

\section {Localization} \label{s:4}
\setcounter {equation} {0}

In this section sketch the proof of Theorem 4 (\cite{G2}). By \ $x,y,z...$ \ 
denote integers of $\Z^d$.

 Consider selfadjoint operators on $\ell^2(\Z^d)$ defined as
follows:

\begin{equation}\label{4.1}
A(\lambda)=A_0(\lambda)+Q_\lambda,
\end{equation}
where $A_0(\lambda)$ is a non-random finite-difference  
operator of infinite order

\begin{equation}\label{4.2}
(A_0(\lambda)\Psi )(x) = \sum\limits_{|x-y|\geq1}a_\lambda(x-y)\Psi (y),
\;\;\; x\in \Z^d,\;\; \Psi \in \ell^2(\Z^d), 
\end{equation} 
and for each $\lambda$ the sequence $\{a_\lambda(x)\}_{x\in \Z^d}$ 
satisfy the conditions

\begin{equation}\label{4.3}
a_\lambda(x)=\overline{a_\lambda(|x|)},\;\;
|a_\lambda(x)|\leq Ce^{- \sqrt{|\lambda|}|x|}, \;\;  C>0.
\end{equation}
The  potential $Q_\lambda$ is the random multiplication operator:

\begin{equation}\label{4.4}
(Q_\lambda\Psi )(x)=\alpha(x)\Psi (x) + {\sqrt{|\lambda|}\over
4\pi}\Psi (x), 
\end{equation}
where $\{\alpha(x)\}_{x\in \Z^d}$ are  
independent random variables with identical distribution and bounded 
(by $\delta^{-1}$) density  ((\ref{0.2})).

Operators $A(\lambda)$ correspond to the single probability space, permitting
use of their explicit dependence on $\lambda$ to verify that the multiscale
analysis scheme may be applied (\cite{G2}).
Denote by $B_L(x)$ the cube centered at $x\in\Z^d$, with sides
of length $L$, and by $A_B(\lambda) $ the restriction of the operator 
$A(\lambda)$ to $\ell^2(B)$  (Dirichlet boundary conditions), by $R_B
(z,\lambda)=(A_B(\lambda)-z)^{-1}$  the resolvent of the operator $A_B(\lambda)$ 
($z\not\in \sigma (A_B(\lambda))$), by \ $R_B (z,\lambda;x,y)$ the corresponding
Green function, continued by zero to $\Z^d\times \Z^d$

$$
R_B (z,\lambda;x,y)=\cases{ (A_B(\lambda) -z)^{-1}(x,y), & if $x,y\in B$;
\cr 0, & otherwise.\cr} 
$$ 
Consider $m>0$, $E\in J\subset \R, \; \lambda\in I\subset
(-\infty,-r)$.

\medskip
\noindent{\bf Definition 4.1}
{\it The cube $B_L(x)$ is called $(h,m,E,\lambda)$ -regular (for a
fixed $\omega\in\Omega$) if \newline 
\noindent $E\not\in\sigma (A_{B _L(x)}(\lambda))$, and} 

$$
|R_{B _L(x)}(E,\lambda;x,y)|\leq e^{-m|y-x|} \;\;  \forall y:  
h/2\leq |y-x|\leq L/2.
$$
{\it If this inequality holds for
all $E\in J\subset\R, \;\; \lambda\in I$ and for $h=0$, say that $B_L(x)$ 
is $(m,E\in J, \lambda\in I$)-regular.}

\noindent{\bf Definition 4.2}
{\it The cube $B _l$ is $(\theta,E,\lambda)$
non-resonant (for a fixed $\omega\in\Omega$) if}
$$
\mbox{\rm dist}(E,\sigma (A_{B_L}(\lambda))\geq
e^{-l^\theta } \;\; 
\mbox{\it for some } \theta,\  \ 0<\theta <1.
$$

\begin{lemma} \label{l:4.1}
Given intervals  $J\subset \R, \;I\subset
(-\infty,-r)$, suppose that for $0<L_0<L_1<\infty$ 

\begin{enumerate}
\item[(P1)]  
${\rm\bf P} \bigl \{ \hbox{ the cube } B _{L_i} \hbox{ is } 
(m_0,E\in J,\lambda\in I)\hbox{- regular } \bigr \} \geq 1-L_i^{-p},\;\; 
i=0,1;$ 

for some $p>d$ and $0<m_0\leq \rho$;
\item[(P2)]  ${\rm\bf P} \bigl 
\{\hbox{ the cube } B_L\hbox{ is }(\theta ,E, \lambda)\hbox{non-resonant }
\bigr \} \geq 1 - L^{-q}$

for  all $L\geq L_0$,\ $E\in J$,\
$\lambda\in I$, for some $\theta $ $(0<\theta \leq 1/2)$ and $q>q_0(p)$.
\end{enumerate}
Then given $m$ $(0<m<m_0)$ there exist  $\alpha =\alpha(p)$ 
$(1<\alpha <\sqrt{2})$ and 
$B_0=B_0(p,\theta,\alpha,q,m,m_0)<$ $\infty $ such that if $L_0>B_0$ 
and $L_n=(L_0)^{(\alpha )^n}$, then  $\forall n\in\N$  
the following inequality

\begin{equation}\label{4.5}
{\rm\bf P} \left \{ \begin{array}{c}\hbox{ if }
  (E,\lambda)\in J\times I \hbox{ one of two cubes } \\  
   B _{L_n}(x) \hbox{ or } B _{L_n}(y)
   \hbox{ is }  \\ 
    (L_n^\beta ,m,E,\lambda) \hbox{ - regular } 
  \end{array}\right\} \geq 1-(L_n)^{-2p}
\end{equation}
holds for $x,y\in \Z^d:$ $|x-y|\geq L_n$, if $\beta =\frac2{1+\alpha ^3}$.
\end{lemma}

\begin{lemma}\label{l:4.2}
Suppose that operators \ $A_\lambda$ \ satisfy assumption (P2) of
Lemma \ref{l:4.1}, and that (\ref{4.5}) holds for every $(E,\lambda)
\in J\times I$, \ $I\subset(-\infty, -r)$.

\noindent Then with probability 1 every generalized solution $f\in\lminus$
of the equation
$$
(A(\lambda)-E)\; f = 0
$$
decays exponentially at infinity with rate $m<m_0$.
\end{lemma}

The proofs of Lemmas \ref{l:4.1}-\ref{l:4.2} do not much differ from
those of \cite{G2}, where the corresponding results were established
for the single operator (\ref{4.1}) - (\ref{4.4}) (i.e. for the fixed 
value of $\lambda$). In order to complete the proof of Theorem 4,
it is sufficient to verify validity of assumptions (P1) and (P2)
of Lemma \ref{l:4.1}. (P1) holds in one of the two following cases.

\begin{lemma} (``Low energy'' case)
\label {l:4.3}
Given $0<m<\sqrt{r}$, $L_1>L_0>0$, there exists
$\lambda_0 = \lambda_0(\po,m,r$,  $p,L_0,L_1)\leq -r<0$
such that for arbitrary interval $I\subset (-\infty,-\lambda_0)$

\bn \label{4.6}
{\rm\bf P}\bigl \{ \hbox{ cube } B_{L_i}  
       \hbox{ is } (m,E\in (-1,1),\lambda\in I)
	\hbox{ - regular} \bigr \} \geq 1-L_i^{-p}, \;\;i=0,1.
\en
\end{lemma} 

\begin{lemma} (``High disorder'' case)
\label {l:4.4}
Given $l>0, \; 0<m<\sqrt{r}$, $L_1>L_0>0$, there exists $\delta_0 = \delta_0
(\po,m,r,p,L_0,L_1,l)<\infty$ such that if $\delta>\delta_0$, then inequalities
(\ref{4.6}) hold for arbitrary interval $I\subset (-\infty,-r)$ of length $|I|=l$.
\end{lemma} 

The proofs of the Lemmas \ref{l:4.3} - \ref{4.4} are almost the same as in
\cite{G2}. Validity of Assumption (P2) follows from the bound on the
density of states (\cite{W}) for the finite-difference operators. 
Reproduce the result in the convenient form.

Consider \ $N_{\Gamma_B(E)}(\lambda) = {1\over |B|}\: \# \: \{\lambda_i(\omega)
<\lambda | \lambda_i \in \sigma(\Gamma_B(E))\}$, \ the normalized
counting function of $\Gamma_B(E)$, \ $B\subseteq \Z^d$.

\begin{lemma} \label{l:4.5} 
(Estimate on the density of states in $\ell^2(\Z^d)$)

\noindent Given arbitrary $E<0$,\ $I=(a,b)\subset \R$,

$$
{\rm\bf E} \: \left\{ N_{\Gamma_B(E)}(I)\right\}
\: =\: {\rm\bf E} \: \left\{ N_{\Gamma_B(E)}(b) - N_{\Gamma_B(E)}(a)\right\}
\: \leq\: 2|b-a|\delta^{-1}.
$$
\end{lemma}

\noindent {\bf Remark.} Lemma \ref{l:4.5} for $E<0$
is proved in \cite{G3}. 
If $\po $ is H\" older continuous of order $\gamma$, then
$N(\lambda )$ is also H\" older continuous of the same order.

\noindent {\bf Remark.} Lemmas \ref{l:4.1} - \ref{l:4.5} prove more than 
is stated by Theorem 4. They imply that all the operators $A_\lambda$ (with
$\lambda$ in the interval corresponding to the high disorder situation) have pure
point spectrum in $(-1,1)$ with probability 1.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%SECTION 5.

\section {Density of states} \label{s:5}
\setcounter {equation} {0}

In this section consider the integrated density of states
of $\point$. Since the proof of existence of the \ids\ for the operator
$-\Delta_\alpha$ did not appear in the literature as yet, at first briefly
discuss a possible way to define it.  As before, denote by \ $E_\omega
(\Delta)$
\ the spectral projection of $\point$,\ $\chi_\Lambda$ the characteristic
function of the cube \ $\Lambda\in \R^d$, \ which sides do not cross
 sites of \ $\Z^d$. Define the measure

$$
d\, K_{\Lambda,\omega }(\Delta)\: = \: {1\over |\Lambda |}\;
{\rm Tr}\: \left ( \chi_\Lambda\: E_\omega(\Delta)\: \chi_\Lambda\right ).
$$
The standard argument based on the ergodic theorem (cf. \cite{CFKS}) implies
that \ $dK_{\Lambda,\omega }$ converges, 
in the distributional sense  for {\rm\bf P}-almost all 
$\omega\in\Omega $, to the non-random limiting measure

\bn \label{5.1a}
dN(\Delta)\: =\: {\rm\bf E}\: {\rm Tr}\: 
\left (\chi_1\: E_\omega(\Delta)\: \chi_1\right ),
\en
where $\chi_1$ is the unit cube in $\R^d$,\ and ${\rm\bf E}\, \{.\}$ denotes 
the expectation value on the probability space $(\Omega, {\rm\bf P})$. 
$N(\lambda)$\ is called the \ids\ of $\point$.  
The following is the rigorous analysis of the continuity properties of 
$N(\lambda)$\ for \ $\lambda < 0$.

Theorems 2-3 and Lemma A2 (cf. (\ref{5.1a})) imply that $N(\lambda)$ is a continuous
function for $\lambda < 0$. Indeed,  
\begin{eqnarray} \label{5.1b}
&   & {\rm\bf P}\: \bigl\{ \;\lambda<0\;\; \hbox{ is an eigenvalue of }\;\; 
 \point \bigr\} \nonumber \\
 & = &  {\rm\bf P}\: \bigl\{ \;0\;\; \hbox{ is an eigenvalue of }\;\; 
 \Gamma_{\alpha(\omega)}(\lambda)\bigr\}\; =\; 0.
\end{eqnarray}

The first equality is a consequence of Theorem 3. The second one follows from
the continuity of the \ids \ $N_{\Gamma(\lambda)}$ \ (Lemma \ref{l:4.5}).
 In order to prove this,
denote by \ $E_{\Gamma_\omega(\lambda)}\{\Delta\}(x,y)$ \ the kernel
of the spectral projection of operator
\ $\Gamma_{\omega}(\lambda)$, \ then 

$$
dN_{\Gamma(\lambda)}\{0\} \: = {\rm\bf E}\: \left\{ E_{\Gamma_\omega(\lambda)}
\{0\}(0,0)\right\}.
$$
It follows by Lemma 4.5 that $N_{\Gamma(\lambda)}(t)$ 
is  continuous in $t=0$, therefore 
$E_{\Gamma_\omega(\lambda)}\{0\}(0,0)=0$ \ with probability 1, which implies
(since $E_{\Gamma_\omega(\lambda)}\{0\}$ \ is ergodic projection) that
 $E_{\Gamma_\omega(\lambda)}\{0\}(x,x)=0$  $\forall x\in \Z^d$ with
 probability 1. Hence $E_{\Gamma_\omega(\lambda)}\{0\} = 0$ with probability
1, and (\ref{5.1b}) follows. 

\noindent Since (\ref{5.1b}) is zero, it follows that
$E_\omega\{ \lambda\} = 0$ \ with probability 1, and the statement
\ $dN\{\lambda \} = 0$  is implied by (\ref{5.1a}). 
A more detailed analysis is required to prove the absolute continuity of
$N(\lambda)$.

Consider a cube $B\subset \R^d$, $\Lambda = B\cap \Z^d$,
the operator $-\Delta_{\overline\alpha,\Lambda}$, where
$\overline\alpha$ is the restriction of the function $\alpha(j): \Z^d\to \R$ 
to a finite subset $\Lambda\subset \Z^d$ 
(i.e. the interactions at the points $\Z^d\setminus\Lambda$ are "switched off").
The resolvent kernel of this operator 
(\cite{AGHH}) is given by

\begin{eqnarray} \label{5.1}  
(-\Delta_{\overline{\alpha},\Lambda} -z)^{-1}(x,y) & = & (-\Delta-z)^{-1}(x,y)  \nonumber \\
     & + & \sum\limits_{j,j^\prime\in \Lambda} (-\Delta -z)^{-1}(x,j)\: 
[\Gamma_{\overline\alpha(\omega)} (z)]^{-1}(j,j^\prime)\: 
(-\Delta -z)^{-1}(j^\prime,y),
\end{eqnarray}
where $\Im z>0$ and operator \ $\Gamma_{\overline\alpha(\omega)} 
(z):\ell^2(\Lambda)\rightarrow \ell^2(\Lambda )$ \ is defined as follows :

\bn \label{5.2}\Gamma _{\overline\alpha(\omega)} (z;j,j^\prime)=
\cases{ \alpha _j-\frac{i\sqrt{z}}{4\pi },
&if $j=j^\prime$,\cr
   -(-\Delta -z)^{-1}(j,j^\prime)& otherwise \cr}, \; \; j,j^\prime\in\Lambda.
\en 

It is known (\cite{AGHH}) that for arbitrary fixed $\omega$, the negative spectrum
of $\Gamma_{\overline\alpha(\omega)}$ is discrete and consists of not
more than $N=|\Lambda|$ eigenvalues (number of points in 
$\Lambda\subset \Z^d)$.
It follows that for  $E<0$, the normalized counting
function of $\Gamma_{\overline\alpha (\omega )}$ is well-defined:

$$
N_{\Lambda,\omega}(E) = {1\over |\Lambda|} 
\#\left\{ \lambda_i(\omega)<E \right\}
$$
If $z<0$ and $|z|$  is
sufficiently large, then the sequence of resolvents
$(-\Delta_{\overline\alpha,\Lambda}-z)^{-1}$ is monotone increasing as
$\Lambda\to \Z^d$, and strongly converges over the filter of finite
subsets \ $\Lambda\subset \Z^d$:

$$
s-\lim_{\Lambda\nearrow \Z^d} \:
(-\Delta_{\overline\alpha,\Lambda}-z)^{-1} \: = \: (\poin -z)^{-1}.
$$
Thus the corresponding sequence of operators $-\Delta_{\overline\alpha}$ is 
monotone decreasing and converges in the strong resolvent sense to $\poin$ as 
$\Lambda\to\ \Z^d$ (this is the way to define $\poin$, \cite{AGHH}).
It follows that the corresponding sequence $N_{\Lambda,\omega}(E)$ is
a superadditive stochastic process, that is given closed cubes
\ $\Lambda_1,\: \Lambda_2$,\ such that \ $\Lambda_1\cap \Lambda_2$\ has
empty interior:   
$$
N_{\Lambda_1\cup \Lambda_2,\omega}(E)\:
\geq N_{\Lambda_1,\omega}(E) + N_{\Lambda_2,\omega}(E).
$$
The standard arguments (\cite{Ki}) based on the superadditive ergodic
theorem show that the sequence of distribution functions
$\{ N_{\Lambda_n,\omega}(E)\}$ converges in the distributional sense 
with probabability 1 to
a non-random limiting distribution function $\widetilde N(E)$, if

\bn \label{5.3}
\sup_n {1\over |\Lambda_n|}\: {\rm\bf E}\:
\left\{N_{\Lambda_n,\omega}(E)\right\} < \infty.
\en
In order to prove the formulated theorem, it is enough to observe the following: 
since $\Delta_\alpha$ is
the strong resolvent limit of a decreasing sequence $-\Delta_{\overline\alpha,
\Lambda}$, then for arbitrary interval  $I_\eta\subset(-\infty,-r)$  of length
$\eta>0$, which endpoints are not atoms of  $E_\omega(\Delta)$, it follows

\bn \label{5.3d}
s-\lim_{\Lambda\nearrow \Z^d} \; E_{\Lambda,\omega}(I_\eta) \: = 
\: E_\omega(I_\eta).
\en
Since \ $\point $ \ a.s. has no fixed (at least negative) eigenvalues 
((\ref{5.1b})),  it follows that (\ref{5.3d}) is valid for arbitrary fixed interval,
with probability 1. (\ref{5.1a}) and (\ref{5.3d}) imply 

\begin{eqnarray} \label{5.3a}
N(I_\eta) & \leq &\sup_\Lambda {1\over |\Lambda|} \: {\rm\bf E} \:
		     \left \{ {\rm Tr} \: 
		  (E_{\Lambda,\omega}(I_\eta))\right\}  \nonumber \\
	  & = & \sup_\Lambda\: {\rm\bf E} \left\{ N_{\Lambda,\omega}
		   (I_\eta)\right\}.
\end{eqnarray}

Since the set $\Lambda\subset \Z^d$ is finite, then for every fixed $\omega$

\begin{eqnarray} \label{5.3b}
 N_{\Lambda}(I_\eta) & = & {1\over |\Lambda|}\: \#\: \left\{ E\in I_\eta\;|\; 
		       0\in\sigma(\Gamma_\Lambda,(E))\right\} \nonumber \\
		     & = & {1\over |\Lambda|}\: \#\: \left\{ \lambda\; |\;
      -\eta\leq\lambda\leq0,
	       \;\; 0\in\sigma(\Gamma_\Lambda(E_0+\lambda))\right\},
\end{eqnarray}
where $E_0\leq -r<0$ is the right endpoint of $I_\eta$, and for
simplicity denote $\Gamma_\Lambda = \Gamma_{\overline\alpha,\Lambda}$ 
((\ref{5.2})).

\begin{lemma} \label{l:5.1}
Assume that $0\in\sigma(\Gamma_\Lambda(E+\lambda))$, 
$\; -\eta\leq\lambda <0$, $ E\leq -r<0$. Then 
${\rm dist}(0,\sigma(\Gamma_\Lambda(E)))<C|\lambda|$, and hence there exists 
$x\in\sigma(\Gamma_\Lambda(E))$ such that $|x|<C|\lambda|$\  (with $C=C(r)$).
\end{lemma}

\noindent {\it Proof.} It follows
	  
$$
\Gamma_\Lambda(E+\lambda)=\Gamma_\Lambda(E)+\Delta(E,\lambda).
$$
From the resolvent identity:
$$
\Gamma_\Lambda^{-1}(E+\lambda)=\Gamma_\Lambda^{-1}(E)-\Gamma_\Lambda^{-1}(E)
\;\Delta(E,\lambda) \;
\Gamma_\Lambda^{-1}(E+\lambda),
$$
hence
\begin{equation}\label{5.4}
\Gamma_\Lambda^{-1}(E+\lambda)=(1+\Gamma_\Lambda^{-1}(E)
\; \Delta(E,\lambda))^{-1}\; \Gamma_\Lambda^{-1}(E),
\end{equation}
if 
\ $1+\Gamma_\Lambda^{-1}(E) \, \Delta(E,\lambda)$ \ is invertible. 
From the Hilbert equation 

$$
\Delta(E,\lambda)=G_E^0-G_{E+\lambda}^0=-\lambda G_E^0 \, G_{E+\lambda}^0,
$$
i.e. if \ $\vert E\vert \leq -r< 0$, \ then

$$
\|\Delta(E,\lambda)\| \leq {|\lambda | \over r^2}.
$$
Suppose that \ $\|\Gamma_\Lambda^{-1}(E)||\leq {1\over C|\lambda|}$ \ (i.e. 
${\rm dist}(0,\sigma(\Gamma_\Lambda(E)))\, > \, Ct$)  \ then 

\bn\label{5.5}
\|\Gamma_\Lambda^{-1}(E) \; \Delta(E,\lambda)\|\leq {1\over r^2C},
\en
from which, by choosing \ $C\geq {2\over r^2}$, it follows:

$$
\|\Gamma_\Lambda^{-1}(E) \; \Delta(E,\lambda)\|\leq {1\over  2}.
$$
This implies that the operator \ $1+\Gamma_\Lambda^{-1}(E)
\; \Delta(E,\lambda)$\ is invertible and

\begin{equation}\label{5.6}
\|(1+\Gamma_\Lambda^{-1}(E) \; \Delta(E,\lambda))^{-1}\|\leq 2.
\end{equation}
It follows by (\ref{5.4}) and (\ref{5.6}) that

$$
\|\Gamma_\Lambda^{-1}(E+\lambda)\|\leq 2\|\Gamma_\Lambda^{-1}(E)\|.
$$ 
It is proved that if \ $C\geq {2\over r^2}$ \ and \  ${\rm dist}(0,\sigma
(\Gamma_\Lambda(E)))\geq C|\lambda|$, \ then

$$
{\rm dist}(0,\sigma(\Gamma_\Lambda(E+\lambda)))\geq  {C|\lambda|\over 2}>0.
$$ 
Lemma \ref{l:5.1} is proved.

\begin{lemma} \label{l:5.2}
Denote by \ $\gamma_1(E),..., \gamma_k(E), \;\;\; 1\leq k\leq |\Lambda|$ 
the eigenvalues of matrix $\Gamma_\Lambda(E)$, $E<0$. Then  $\gamma_j(E)$
are strictly increasing functions of $|E|$.
\end{lemma}

\noindent{\it Proof} \ follows by verifying the matrix
${\partial \over \partial\sqrt{|E|}}\, \Gamma_\Lambda\, = 
\, \left [{1\over 4\pi}\: e^{-\sqrt{|E|}|j-j\prime | } 
  \right ] _{j,j\prime = 1}
  ^{|\Lambda|}$ \ is positive-defined (\cite{Si2}).

Now finish the proof of Theorem 5. Return to (\ref{5.3b}) 
and suppose that 

$$
0\in\sigma(\Gamma_\Lambda(E_0+\lambda_i)),\;\;\; -\eta\leq\lambda_i<0,\;\;
i=1,..,k.
$$
It follows by Lemma \ref{l:5.1} that there exist $\{x_i\}_{i=1,...,m}$
such that $|x_i|\leq C\eta$, and $x_i\in\sigma(\Gamma_\Lambda(E_0))$.

\noindent Denote by  $x_i(\lambda)$ the corresponding eigenvalues
of $\Gamma_\Lambda(E_0+\lambda)$. According to Lemma \ref{l:5.2},
each $x_i(\lambda)$ increases as $\lambda$ decreases to $-\eta$, and hence
cannot cross change of sign more than once, i.e. $m\geq k$. It follows:

$$
{\rm\bf E}\left\{ N_{\Lambda}(I_\eta)\right\}\leq {\rm\bf E}\left
\{ N_{\Gamma_\Lambda(E_0)} (E_0-C\eta ,E_0+C\eta)\right\},
$$
where $N_{\Gamma_\Lambda(E_0)}(\lambda)$ is the normalized counting function
of $\Gamma_\Lambda(E_0)$. The estimate on the density of states for 
finite-difference operators $\Gamma_\Lambda$ (Lemma \ref{l:4.5}) 
implies that there is the bound

$$
{\rm\bf E}\left\{ N_{\Gamma_\Lambda(E_0)}(E_0-C\eta ,E_0+C\eta)\right\}
\leq 2C\eta\delta^{-1}.
$$
Theorem 5 is proved.

\section {Generalized spectral theorem}  \label{s:6}
\setcounter {equation}{0}

\begin{theorem} [Spectral Theorem] \label{t:6}
There exist a spectral measure $d\rho(\lambda)$, an integer valued function
$N(\lambda)$, $\sup\limits_\lambda N(\lambda)\equiv N\leq\infty$,  
and disjoint $d\rho$-measurable sets $\{ \Delta_n\}_{n=1}^N$: 

$$
\Delta_m\cap\Delta_n = \emptyset,\; 
  \cup_{n=1}^N\Delta_n = {\rm supp} \{d\rho\},
$$
such that for a.e. $\lambda\in\Delta_n$ (with respect to $d\rho$),  
there are $n$ functions 
$\{ F_j(\lambda;x)\}_{j=1}^n$ satisfying the following conditions:

\begin{enumerate}

\item[{\bf (i)}] For $\lambda$ fixed, $\{F_j(\lambda;\cdot)\}_{j=1}^n$ are 
linearly independent and $F_j(\lambda;\cdot)\in\Lminus$.

\item[{\bf (ii)}] $F_j(\lambda)$ are the generalized eigenfunctions of $-\Delta_\alpha$,
relative to $\lambda$.

\item[{\bf (iii)}] Consider the mapping 
$\widetilde U\: \Lplus \to {\cal H}$, where

$$
{\cal H} = \oplus_{n=1}^N \: \L^2(\R, \C^n; \chi_n\, d\rho), 
$$
$\chi_n$ is the characteristic function of $\Delta_n$:

$$
(\widetilde U\, f)_j(\lambda)\: = \: \int\limits_{\R^d} \: 
\overline {F_j(\lambda;x)}\, f(x)\, dx.
$$
Then $\widetilde U$ is a unitary map, i.e.

$$
\|\widetilde U\, f\|_{\cH}^2\: = \: \int\limits_{\sigma(H)}\, 
\{ \sum_{j=1}^N |(\widetilde{U}f)_j(\lambda)|^2\}
\: d\rho(\lambda) \: = \: \|f\|_{\L^2}^2.
$$

\item[{\bf (iv)}] $\widetilde U$ extends to the unitary map $U:\: \L^2(\R^d) 
\stackrel{\rm onto}\longrightarrow \cH$
defined for arbitrary $f\in \L^2(\R^d)$ by

$$
(U\, f)_j(\lambda)\: = \: {\cal H}- \lim_{R\to\infty} \,
 \int\limits_{|x|\leq R}\, F_j(\lambda;x)f(x)\, dx,
$$
where $\lim$ means the limit in $\cH$ - norm.

\item[{\bf (v)}] If $g$ is a bounded measurable function on $\sigma(H)$,
then for arbitrary $f\in \L^2(\R^d)$:

$$
U\, [g(H)\, f] \: = \: g(\lambda) [U\, f].
$$

\item[{\bf (vi)}] If $f\in \Dpoin$ then

$$
U(Hf) = \lambda (Uf).
$$

\item[{\bf (vii)}] If $h\in \cH$, the inverse map $U^{-1}$ is defined by

\bn \label{6.1}
(U^{-1}\, h)(x) \: = \: \L^2- \lim_{\scriptstyle E\to\infty\atop
 N\to \infty}\: \int\limits_{|\lambda|\leq E}\: 
\sum_{j=1}^{\min (N, N(\lambda))}
 \: h_j(\lambda)\,  F_j(\lambda;x)\, d\rho(\lambda).
\en
\end{enumerate}
\end{theorem}

\noindent {\it Proof.} Denote by $H=-\Delta_\alpha$ the point interaction Hamiltonian    
on $\L^2(\R^d)$, $1<d<4$,  $|\alpha_j|\leq {\rm sup}\{\alpha_j\}<\infty$,
\ $T_\delta=(1+|x|)^{\delta}$ (multiplication operator), $\delta>d/2$,
$E(\Delta)$ is the spectral projection of $-\Delta_\alpha$.

\begin{lemma} \label{l:6.1} For arbitrary bounded measurable set $\Delta \subset \R$, 
$A(\Delta)=T_\delta^{-1}E(\Delta)T_\delta^{-1}$ is a trace class operator.
\end{lemma}
Lemma 6.1 is proved in section \ref{s:2}. 
It follows by the Radon-Nikodym theorem that there exists a positive measurable 
trace class operator-valued
function $a(\lambda)$, ${\rm Tr}\: a(\lambda)=1$, such that

$$
A(\Delta)\: = \: \int_{\Delta}\: a(\lambda)\: d\rho(\lambda),
$$
where the scalar measure (which is the spectral measure of $H$)
is defined by

$$
\rho(\Delta)\: = \: {\rm Tr}\: \{T_\delta^{-1}\: E(\Delta)\: T_\delta^{-1}\}.
$$
Define

$$
\ef(\lambda;x,y)\: = \: (1+|x|)^\delta\: a(\lambda;x,y)\: (1+|y|)^\delta,
$$
where $a(\lambda;x,y)$ is the integral kernel of $a(\lambda)$, i.e.
$\ef(\lambda;x,y)$ can be regarded as the integral kernel of the
operator ${dE(\lambda)\over d\rho(\lambda)}$ on $\L^2(\R^d)$.

The kernel of the Hilbert-Schmidt operator $a(\lambda)$:

\bn\label{6.2}
a(\lambda;x,y) \: = \: \sum_{j=1}^{N(\lambda)}\: \overline{\varphi_j(\lambda;x)}\,
\varphi_j(\lambda;y),
\en
where $\varphi_j(\lambda)$ are (non-normalized) eigenfunctions of $a(\lambda)$.
They are orthogonal, may be chosen jointly measurable in $x,\lambda$ 
and their norms satisfy the relation:

\bn \label{6.3}
\sum_{j=1}^{N(\lambda)}\: 
\|\varphi_j(\lambda)\|_{\L^2}^2 \: = \: {\rm Tr}\: a(\lambda) \: = \: 1.
\en
Define 

$$
F_j(\lambda;x) \: = \: (1+|x|)^\delta\, \varphi_j(\lambda; x),\;\; x\in\Delta_n,
$$
$j=1,..,n$, $\Delta_n \: = \: \{\lambda| N(\lambda) = n\}$.

\noindent (i) follows by definition of $F_j$ since 
$\|\varphi_j\|\leq 1$.

\noindent (ii) 
By definition of $F_j$, since
$\varphi_j(\lambda)$ are eigenfunctions of $a(\lambda)$:

$$
\int\limits_{|x-y|\leq 1}\: |F_j(\lambda;x)|^2 \: dx\: \leq\:  
C(1+|y|)^\delta,
$$
and

\bn \label{6.4}
\lambda\: F_j(\lambda;x)\: = \: \int\limits_{\R^d}\:
\ef(\lambda;x,y)\: (1+|y|)^{-2\delta}
\: F_j(\lambda;y)\: dy
\en
Given arbitrary $\psi_0\in \Dpoin$, 

\begin{eqnarray*}
\langle F_j(\lambda),H\psi_0\rangle & \: = \: & {1\over \lambda}
\langle \int\limits_{\R^d} \ef (\lambda;x,y)\, (1+|y|)^{-2\delta}\: 
  F_j(\lambda;y)dy, H\psi_0\rangle \\
    & \: = \: & \int\limits_{\R^d} \: 
	  \langle \ef (\cdot,y;\lambda),H\psi_0\rangle\:
       (1+|y|)^{-2\delta} F_j(\lambda;y)\, dy \\
    & \: = \: & \lambda\: \langle F_j(\lambda; \cdot),
\psi_0(\cdot)\rangle,
\end{eqnarray*}
this proves (ii).

Let us prove (iii). Since $F_j(\lambda)\in \L^2_{-\delta}$, the mapping
$\widetilde{U}\psi$ is correctly defined if $\psi\in \L^2_\delta$.

By definition of $\ef$  and (\ref{6.2}),

\begin{eqnarray} \label{6.5}
     \int\limits_{\R^d\times\R^d}\: 
	 \ef(\lambda;x,y)\overline{\psi(x)}\psi(y)\: dxdy\: 
 & = & \: \sum_{j=1}^{N(\lambda)}\: \int\limits_{\R^d\times\R^d}\:
\overline{F_j(\lambda;x)}F_j(\lambda;y)\:
 \overline{\psi(x)}\psi(y)\: dx dy \nonumber\\
 \: & = & \: \sum_{j=1}^{N(\lambda)}\: |\langle F_j(\lambda),\psi\rangle |^2
 \: = \: \sum_{j=1}^{N(\lambda)}\: |(\widetilde{U}\psi)_j(\lambda)|^2,
\end{eqnarray}
Relations (\ref{2.2}) and (\ref{6.5}) imply that  
given arbitrary $\psi\in \Lplus$, and bounded measurable function $g$ on
$\sigma(H)$:

\bn\label{6.6}
\langle \psi, g(H)\psi\rangle \: = \: \int\limits_{\sigma(H)}\: g(\lambda)
\sum_{j=1}^{N(\lambda)}\: |(\widetilde{U}\psi)_j (\lambda)|^2\: d\rho(\lambda).
\en
This implies the required result (iii) by substituting $g\equiv 1$.

The statement (iv) for arbitrary $g\in \L^2(\R^d)$ follows
via limiting argument by (iii), and it remains to prove
that $U$ maps $\L^2(\R^d,dx)$ onto $\cH$.


\begin{lemma}\label{l:6.2}
\bn \label{6.7}
S(\lambda;x) \stackrel{\rm def}{=} \sum_{j=1}^{N(\lambda)}|\, 
     F_j(\lambda;x)|^2\leq C(\lambda,dist(x,\Z^d))\: (1+|x|)^{2\delta},
\en
for arbitrary $x\not\in \Z^d$,
$d\rho$- a.e. $\lambda\in\sigma(-\Delta_\alpha)$,
$C(\lambda,t)> 0$, $t>0$.
\end{lemma}

\noindent {\it Proof}.
Relation (\ref{6.3}) implies $S(\lambda)\in \L^2_{-\delta}$
for $d\rho$- a.e. $\lambda\in\sigma(-\Delta_\alpha)$, and

\bn \label{6.8}
 \int |S(\lambda;x)|\: (1+|x|)^{-2\delta}\: dx \: = 
  \: {\rm Tr} \: a(\lambda)\: = \: 1.
\en
The function $S(\lambda;x)$ is continuous in $A_\varepsilon = \{ x\in \R^d|\:
{\rm dist}(x,\Z^d)\geq \varepsilon\}$ if $\varepsilon>0$.
Denote $A_\varepsilon\supset B_r(x_0)=\{x\in A_\varepsilon| |x-x_0|\leq r\}$,  
$r>0$, ${\rm dist}(x_0,\Z^d)\geq \varepsilon>0$. 
By the Harnack inequality 

$$
|F_j(\lambda;y)|\geq C(\lambda,r,\varepsilon)\, |F_j(\lambda;x_0)|,
\;\;\;y\in B_r(x_0).
$$
Suppose $S(\lambda;x)(1+|x|)^{-2\delta}$ is unbounded in
$A_{\varepsilon}$, then there exists sequence $\{x_n\}_{n\in\N}$,
$|x_n|\to\infty$,
$x_n\in A_{\varepsilon}$, ${\rm dist}(x_i,x_j)>1$, $i\ne j$, such that 
$\forall x\in B_{\varepsilon/2}(x_n)$, $n\in\Z^d$, 

$$
S(\lambda;x)\:\geq C^2  (1+|x|)^{2\delta}, \;\;\; 
$$
which contradicts to (\ref{6.8}), so (\ref{6.7}) follows.
Lemma \ref{l:6.2} is proved.

\bigskip

Suppose $h\in\cH$ is orthogonal to the range of $U$. 
Since $\sum_{j=1}^{N(\lambda)}\, 
|h_j(\lambda)|^2 <\infty$ for $d\rho$ - a.e. $\lambda$, 
one may define 

$$
K(\lambda;x) = \sum_{j=1}^{N(\lambda)}\: \overline{h_j(\lambda)}\: F_j(\lambda;x)\:
\in \: \L^2_{-\delta}.
$$
Then if $f_0\in C_0^\infty(\R^d),\; g_0\in C_0^\infty(\sigma(H))$,

\bn\label{6.9}
\langle h, g_0(\lambda)(Uf_0)\rangle_{\cal H} \: = \:
 \int\: f_0(x)\: g_0(\lambda)\: \sum_{j=1}^{N(\lambda)}\: 
 \overline{h_j(\lambda)}
F_j(\lambda;x) \; dx d\rho(\lambda).  
\en
At first prove (v) 
for arbitrary  $f_0\in C_0^\infty(\R^d)$, 
 $g_0\in C_0^\infty(\sigma(H))$. 

Lemma \ref{l:2.2} (2) implies that $\ef(.,y;\lambda)$ is the distributional 
solution of the equation

$$
[g(H)-g(\lambda)]\psi\: = \: 0,
$$
and by the argument similar to one applied above to prove
(ii) it follows that 
$F_j(\lambda;x)$ is the distributional solution of the same equation.
Notice, that if $f_0\in\C_0^\infty$, then 
$U\, g_0(H)\, f_0$
is correctly defined since $g_0(-\Delta_\alpha)$ is the local operator simultaneously
with $-\Delta_\alpha$. This implies

\begin{eqnarray} \label{6.10}
(U\, g_0(H)\, f_0)_j(\lambda) &\: =\: & \int\: \overline{F_j(\lambda;x)}
\: g_0(H)\: f_0(x)\: dx \nonumber\\
     & \: = \: & g_0(\lambda)\: \int\: \overline{F_j(x;\lambda)}\: 
	f_0(x)\: dx
   \: = \: g_0(\lambda)\: (Uf_0)_j\: (\lambda).
\end{eqnarray}
It follows by (\ref{6.10}) and Lemma 2.2 (2), that if
$g_0\in C_0^\infty(\sigma(H))$, 

$$
\int\: f_0(x)\: g_0(\lambda)\: K(\lambda;x)\: d\rho(\lambda)\: dx
\: = \: \langle h, Ug_0(H)f_0\rangle_{\cal H}\: = \: 0,
$$
hence $K(\lambda;x)\equiv 0$ for a.e. $x$, $d\rho$- a.e. $\lambda$.
Thus $(1+|x|)^{-2\delta}K(\lambda;x) = 0$ as an element of $\L^2(\R^d)$.
Since $\varphi_j(\lambda;x)=(1+|x|)^{-\delta}\, F_j(\lambda;x)$ are
orthogonal and non-zero, it follows that $K(\lambda;x)=0$ implies 
$h_j(\lambda) = 0$, $j=1,...,N(\lambda)$, for a.e. $\lambda$
(with respect to $d\rho$). This proves (iv), and (v) supposing
$f_0\in\C_0^\infty(\R^d)$, $g_0\in\C_0^\infty(\sigma(H))$.

The statement (v) in case  
$f\in \L^2(\R^d)$ follows by the limiting argument (similar to the proof 
of (vi) below).
Now assume that $g$ is a bounded measurable function on $\sigma(H)$.
Then there exist functions $g_n\in C_0^\infty$,
such that $g_n(\lambda)\to g(\lambda)$  $\forall\lambda\in \sigma(H)$, and
$\|g_n\|_\infty\: \leq\: \|g\|_\infty\: = \: \sup_x\, |g(x)|$.
Then $g(H)\: = \: {\rm s}-\lim\limits_{n\to\infty}\, g_n(H)$, and 
if $f\in \L^2$

\begin{eqnarray*}
U[g(H)f]\: & = & \: {\cal H}- \lim_{n\to\infty}\: Ug_n(H)f \\
    \: & = & \: {\cal H}- \lim_{n\to\infty}\: g_n(\lambda)Uf \: = \: g(\lambda)Uf.
\end{eqnarray*}
This proves (v).

\begin{lemma}\label{l:6.3}
Given $f\in {\cal D}(-\Delta_\alpha)$, there exist 
$\{f_n\}_{n\in\N}\subset {\cal D}_0(-\Delta_\alpha)$,
such that 

\begin{eqnarray}\label{6.11}
\L^2-\lim\limits_{n\rightarrow\infty}\, f_n\: =\: f,\nonumber\\
\L^2-\lim\limits_{n\rightarrow\infty}\, (-\Delta_\alpha - \lambda)f_n\: =\: 
(-\Delta_\alpha - \lambda)f.
\end{eqnarray}
\end{lemma}

\noindent {\t Proof}. 
Given arbitrary $f\in {\cal D}(H)$,

\bn \label{6.12}
f(x) = \varphi(x)\: + \: \sum_{j\in\Z^d}\: a_j\: G_z^0(x-j),
\en
where $\overline{a} = \{a_j\}_{j\in\Z^d}\: =\:
  \Gamma_\alpha(z)^{-1}\overline{\varphi},
  \;\;\; \overline{\varphi}\: = \: \{\varphi(j)\}_{j\in \Z^d}$, 
  $z\not\in\sigma(-\Delta_\alpha)$,
$\varphi\in {\cal D}(-\Delta)$,
consider 

\bn\label{6.13}
f_n\: = \: \varphi_n\: + \: \sum_{j\in\Z^d}\: b_n^j\: G_z^0(x-j)
\in {\cal D}_0(H),
\en
where $\overline{b_n} = \Gamma_\alpha(z)^{-1}\overline{\varphi_n}$, and

\bn\label{6.14}
\varphi_n\in C_0^\infty, \;\;\; \L^2- \lim\limits_{n\rightarrow\infty}\varphi_n\: =\: \varphi,
\;\;\; \L^2- \lim\limits_{n\rightarrow\infty}(-\Delta-z)\varphi_n\: =\: 
(-\Delta-z)\varphi.
\en
A sequence $\{\varphi_n\}_{n\in\N}$ satisfying (\ref{6.14}) exists
for $z\in\C$, $\varphi\in \L^2$. Given arbitrary $z\not\in (0,+\infty)$, 
this follows since $(-\Delta-z)^{-1}$ 
is bounded for such $z$, and 
$(-\Delta-z)(C_0^\infty)$ is dense in $\L^2$. 
The same sequence $\varphi_n$ satisfies
(\ref{6.14})  $\forall z\in\C$.  

Prove that $\ell^2- \lim\limits_{n\rightarrow\infty}
\overline{\varphi}_n\: =\: \overline{\varphi}$.

\noindent If $\psi\in{\cal D}(-\Delta)$, then $\psi=(-\Delta-\lambda)^{-1}h$ for some
$\lambda\in\C\setminus\sigma(-\Delta_\alpha)$, $h\in \L^2(\R^d)$. Hence

$$
\overline{\psi}_j\: = \: \psi(j) \: = \: 
\int\limits_{\R^d}\: G_\lambda^0(j,x)\, h(x)\: dx\: = \: 
\sum_{k\in \Z^d}\:\int\limits_{\Lambda(k)}\: 
G_\lambda^0(j,x)\, h(x)\: dx,\;\;\; j\in \Z^d,
$$
where $\Lambda(k)=\Lambda_{1/2}(k)$ is the cube of sidelength $1/2$ centered
at $k\in \Z^d$, 

\bn\label{6.15}
|\overline{\psi}_j|\: \leq \: \sum_{k\in\Z^d}\: B(j,k)\: \overline{h}_k\: <\infty,
\en

$$   
   B(j,k;x)  \: = \: 
    (\int\limits_{\Lambda(k)\subset\Z^d} |G_\lambda^0(j,x)|^2dx )^{1/2},    
$$

$$
\overline{h}_k  \: =\:  ( \int\limits_{\Lambda(k)}\: 
|h(x)|^2\: dx ) ^{1/2},
$$
i.e. $\overline{h}\in \ell^2(\Z^d)$, and $\{B(j,k)\}_{j,k\in\Z^d}$ 
define bounded
operator $B$: $\ell^2(\Z^d)\to \ell^2(\Z^d)$:

$$
\|B\|_{\ell^2}\: \leq \: C(\lambda)\: <\: \infty.
$$
It follows that $\overline{\psi}\in \ell^2(\Z^d)$:

\bn\label{6.16}
\|\overline{\psi}\|_{\ell^2}\leq \|B\|_{\ell^2}\: \|h\|_{\L^2}.
\en
Now consider functions $\{\varphi_n\}_{n\in\N}, \varphi\subset\Dpoin$,
satisfying (\ref{6.14}), i.e. for some $z<0$:

$$
h_n\: \stackrel{\rm def}{=} \: (-\Delta-z)\varphi_n;\;\;\; 
h\: \stackrel{\rm def}{=}\: (-\Delta-z)\varphi;
$$
$$
\L^2- \lim\limits_{n\rightarrow\infty}\: h_n\: =\: h.
$$

Relation (\ref{6.16}) implies
$$
\|\overline{\varphi}_n-\overline{\varphi}\|_{\ell^2}\leq \|B\|_{\ell^2}\: 
\|h^n - h\|_{\L^2},
$$
i.e. 

\bn\label{6.17}
\ell^2- \lim\limits_{n\rightarrow\infty}\overline{\varphi}_n\: =\:
\overline{\varphi}.
\en

\noindent Since $\Gamma_\alpha(z)^{-1}$ is bounded, 

\bn\label{6.18}
\ell^2- \lim\limits_{n\rightarrow\infty} \overline{b}_n \: =\: 
\overline{a},
\en
where $\overline{a}\: = \: \Gamma_\alpha(z)^{-1}\, \overline{\varphi}$,
$\overline{b}_n\: = \: \Gamma_\alpha(z)^{-1}\, \overline{\varphi}_n$.

Relations (\ref{6.12}), (\ref{6.13}) imply

$$
\|f-f_n\|_{\L^2}\: \leq \: \|\varphi-\varphi_n\|_{\L^2}\: + \:
C_z\|\overline{a}-\overline{b}_n\|_{\ell^2},
$$
hence (\ref{6.18}) imply

\bn\label{6.19}
 \L^2- \lim\limits_{n\rightarrow\infty}f_n\: =\: f.
\en
Given arbitrary $\lambda>0$, it follows by (\ref{6.14})

\bn\label{6.20}
\|(H-\lambda)(f-f_n)\|_{\L^2}\: \leq\:
\|(-\Delta-z)(\varphi-\varphi_n)\|_{\L^2} \: + \:
      |z-\lambda|\|f - f_n\|_{\L^2}.
\en
Relations (\ref{6.14}), (\ref{6.19}), (\ref{6.20}) imply 

\bn\label{6.21}
\L^2-\lim\limits_{n\rightarrow\infty} (H-\lambda)f_n\: =\:
(H-\lambda)f\: 
\en
Lemma \ref{l:6.3} is proved.

\bigskip

Now turn to the proof of (vi). Suppose $f_0\in {\cal D}_0(H)$.
Lemma A1 (Appendix A) and (ii) imply that $(H-\lambda)f_0\in L_\delta^2$, and

$$
\int\: \overline{F_j(\lambda;x)}\, (H-\lambda)\, f_0(x)\: dx\: = \: 0,\;\;\; 
j=1,..,n;
$$
which implies

\bn\label{6.22}
U[(H-\lambda)f_0]\: = \: 0.
\en
Now given  $f\in{\cal D}(H)$, consider $f_n\in{\cal D}_0(H)$ such that

\begin{eqnarray*}
& \L^2- & \lim\limits_{n\rightarrow\infty} f_n\:  =  \: f, \\
& \L^2- & \lim\limits_{n\rightarrow\infty} (H-\lambda)f_n\:  =  \: (H-\lambda)f
\end{eqnarray*}
(cf. Lemma \ref{l:6.3}). Relations
(iv), (v) and (\ref{6.22}) imply

\begin{eqnarray}U[Hf]\: & = & \: {\cal H}- \lim_{n\to\infty}\: U[Hf_n] \nonumber \\     
\: & = & \: {\cal H}-\lim_{n\to\infty}\: \lambda[Uf_n]     
\: = \: \lambda\: [Uf],
\end{eqnarray}
(vi) is proved.
Let us prove (vii). Suppose $f\in\L^2_\delta(\R^d)$. Consider 

\begin{eqnarray}\label{6.23}
\widetilde{f(x)} \: & = & \: \L^2- \lim_{\scriptstyle
E\to\infty\atop N\to\infty}\: \int\limits_{|\lambda|\leq E}\:
\left \{\sum_{j=1}^{\min(N,N(\lambda))}\: \int\limits_{\R^d}\:
\overline{F_j(\lambda;y)}F_j(\lambda;x)f(y)\: dy\:\right\}\: 
d\rho(\lambda) \nonumber \\
\: & = & \: \lim_{E\to\infty}\: \int\limits_{|\lambda|\leq E}\:
\int\limits_{\R^d}\: \ef(\lambda;x,y)\, f(y) \: dy\: d\rho(\lambda),
\end{eqnarray}
then for arbitrary $g\in L_\delta^2$:

\bn\label{6.24}
\langle g,\widetilde{f}\rangle\: = \: \lim_{E\to \infty}
\int\limits_{|\lambda|\leq E}\: \int\limits_{\R^d\times\R^d} 
\ef(\lambda;x,y)\overline{g(x)} f(y)\:dx\, dy \: d\rho(\lambda) 
\: = \: \langle g, f\rangle.
\en
It follows that $\|\widetilde{f}\| = \|f\|$,  
hence the limit at the r.h.s. of (\ref{6.23}) exists.
Since $\L^2_\delta$ is dense in $\L^2$, (\ref{6.24}) implies
$\widetilde f = f =U^{-1}h$,
where $U^{-1}$ is defined by (\ref{6.1}).

\noindent Now suppose $f\in \L^2$, 
denote
$f_R(x)=f(x)\chi_{(-R,R)}(x)$. Consider

\begin{eqnarray}\label{6.25}
\widetilde{f(x)} \: & = & \: \L^2-\lim_{\scriptstyle
E\to\infty\atop N\to\infty}\: \int\limits_{|\lambda|\leq E}\:
\left \{\sum_{j=1}^{\min(N,N(\lambda))}\:
F_j(\lambda;x)\: {\cal H}-\lim_{R\to\infty} \int\limits_{|y|\leq R}\:
\overline{F_j(\lambda;y)}f(y)\: dy\:\right\}\: d\rho(\lambda) \nonumber \\
\: & = & \: \L^2-\lim_{E\to\infty}\: \int\limits_{|\lambda|\leq E}\:
\:\left \{{\cal H}- \lim_{R\to\infty}
\int\limits_{\R^d}\: \ef(\lambda;x,y)\, f_R(y) \: dy\:\right\} 
d\rho(\lambda),
\end{eqnarray}
The limits  
in (\ref{6.25}) exist, since for arbitrary $g\in \L^2_\delta$,

\begin{eqnarray*}
\langle g,f\rangle \: & = & \: \lim_{\R\to\infty}\:\langle g,f_R\rangle \\
\: & = &\: \langle g, U^{-1}[\L^2-\lim_{\R\to\infty}\: Uf_R]\: = \:
\langle g,\widetilde{f}\rangle,
\end{eqnarray*}
where the latter $\lim$ means the limit in $\L^2$-norm, and it follows 
$\widetilde{f} = f$. Theorem 6 is proved. $\Box$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%APPENDIX.

\section*{Appendix A}

\noindent{\bf Lemma A1.}
{\it Suppose \ $\varphi\in \D0(\poin)$,  i.e.}

$$
\varphi(x) = \varphi_0(x) + \sum_{i\in \Z^d}\: a_i\: G_E^0(x-i),
$$
{\it where \ $\varphi_0\in \C_0^\infty$, $E\leq\inf {\rm supp}\, \po  -d$, 
$a_i = \sum_{j\in \Z^d}\: \Gamma_E^{-1}(i,j)\: \varphi_0(j)$.
Then there exists $d_0$ such that if \  $d>d_0$, then \
$(-\Delta_\alpha - \lambda)\varphi\in \Lplus$. }

\medskip

\noindent {\it Proof.} Denote $\Gamma_E^0 = \Gamma_E -
{\sqrt{|E|}\over 4\pi}$. Then  (cf. (\ref{0.4})):

$$
\| \Gamma_E^0\|\leq \sum_{i\in \Z^d\setminus 0} {1\over 4\pi\, |i|}\:
e^{-\sqrt{|E|}\,|i|}\: =\: \varepsilon (E) \to 0 \;\;\; 
E\to -\infty.
$$
It follows from the ergodicity that if $E<0$ 

$$
\sigma(\Gamma_E) \: =\: {\rm supp}\, \po  \dot{+} \sigma(\Gamma_E^0) +
{\sqrt{|E|}\over 4 \pi}
$$
(the sum above is the algebraic sum of subsets of $\R$), so 

$$
{\rm dist} (0,\sigma(\Gamma_E))\geq {\rm dist} \left ( -{\sqrt{|E|}\over 4 \pi},
\: \inf {\rm supp}\, \po - \varepsilon (|E|)\right )\: \geq d/2,
$$
if $|E|$ is large enough. The Combes-Thomas argument implies

$$
|\Gamma_E^{-1}(i,j)|\leq {2\over d(E)}\: e^{-m(E)\, |i-j|},
$$
hence if \ ${\rm supp}\, \varphi_0\subset \Lambda$, then

$$
|a_i| \leq \sum_{j\in \Z^d\cap \Lambda}\: \Gamma_E^{-1}(i,j)\: |\varphi_0(j)|
\leq \|\varphi_0\|_\infty\, C_1(E,|\Lambda|)\: e^{-m(|E|)|i|}.
$$
The next Lemma A2 implies that \ $\varphi\in\Lplus$ \ and \ $(\poin -
\lambda)\varphi\in\Lplus$.

\bigskip

\noindent{\bf Lemma A2.}
{\it Suppose \ $\lambda < 0$, consider}

$$
%A.1
F(x)=\sum_{i\in \Z^d} \, f_i\: G_\lambda^0(x-i),\qquad
\overline f= \{f_i\}_{i\in \Z^d}.
\eqno(A.1)
$$

{\it 1. If \ $F\in \L^2(\R^d,d\mu)$ \ (i.e. the series converges in
$\|.\|_{\L^2(d\mu)}$),
\ where \ $\mu(x) = g^2(x)\, dx$ \ ($g(x)=g(|x|)>0$\ is a monotone, 
continuous function of $|x|$), and $\varepsilon > 0$, then

\noindent $\overline f \in \ell^2(\Z^d,d\widetilde\mu )$,  where 
$\widetilde\mu (i) = g^2(|i|-t)$, $i\in \Z^d$,   
for some  $t$:  $|t|=\varepsilon$.

2. If \ $\overline f \in \ \ell^2(\Z^d, d\widetilde\mu)$,  
    where \ $\widetilde\mu
(i) = \widetilde g^2(i)$, $i\in \Z^d$ \
($e^{-\varepsilon\sqrt{|\lambda|}\, |i|}\, \leq \, \widetilde g(i)
\, \leq \,  e^{\varepsilon\sqrt{|\lambda|}\, |i|}$ \ for some \ $0<\varepsilon
< 1$,  $\widetilde g$  is a monotone function of  $|i|$, $i\in \Z^d$), then
$F\in \L^2(\R^d, d\mu)$,
\ with \ $d\mu(x) = g^2(x)\, dx$, and $g$  is monotone, continuous
function of $|x|$,  $g(i)=\widetilde g(i)$. }

\medskip

\noindent {\it Proof.} 1. Apply once again the argument described in the proof
of Theorem 3 (Sect. \ref{s:3}). Given \ $\psi\in \C_0^\infty$

$$
\langle F, (-\Delta-\lambda)\psi\rangle\: = \: \sum_{i\in \Z^3\cap 
{\rm supp}\, \psi}\: f_i\, \psi (i). 
$$
Consider the function \ $\theta\in \C_0^\infty$, \ ${\rm supp}\,
\theta \subset\Lambda_{1/2}(0)$, \
$\theta(0)=1, \; |(-\Delta -\lambda)\theta(x)|\leq \tau_\lambda<\infty$, \
and denote \ $\theta_i(x) = \theta(x-i)$. Then

$$
\langle F,\, (-\Delta-\lambda)\theta_i\rangle = f_i.
$$
It follows that $f_i$ obey 

$$
|f_i|  \leq  \left ( \int_{\Lambda_{1/2}(i)} 
	     |F(x)|^2\: g(x)^2\: dx \right )^{1/2} \;
		   \tau_\lambda\: \left ( \int_{\Lambda_{1/2}(i)} 
		   g(x)^{-2}\: dx \right )^{1/2}.
\eqno(A.2)
$$
Since \ $g(x)=g(|x|)$ \ is monotone and continuous,  
$\sup_{x\in\Lambda_{1/2}} |g(x)|^{-2} \leq |g(|i|-t)|^{-2}$, for some  
$|t|=1/2$, \ and it follows by (A.2) 

$$
\sum_{i\in \Z^d} |f_i\, g(|i|-t)|^2\: \leq \: \| F\|_{\L^2(d\mu)}\: \leq \infty.
$$
To get the result with \ $0<|t|<\varepsilon$, for arbitrary $\varepsilon >0$ 
just
consider the function  $\theta_\varepsilon(x)$ with  ${\rm supp}\,
\theta \subset \Lambda_\varepsilon$.  Statement 1 is proved.

2. Since $\widetilde g(i)\geq e^{\varepsilon\sqrt{|\lambda|}\, |i|}$,
for arbitrary fixed  $x\not\in \Z^d$, series (A.1) converges
absolutely. Moreover, if function  $g(x)=g(|x|)$  is monotone,
continuous and  $g(i)=\widetilde g(i)$,  then

$$
\| F\|_{\L^2(d\mu)}^2 = \sum_{i\in \Z^d} f_i \: \sum_{j\in \Z^d} f_j
\: \langle G_\lambda^0(x-i),G_\lambda^0(x-j)\rangle _{\L^2(d\mu)},
$$
where

\begin{eqnarray*}
\langle G_\lambda^0(x-i), G_\lambda^0(x-j)\rangle_{\L^2(d\mu)}
     & = & {1\over 16\pi^2}
	   \int_{\R^d} {e^{-\sqrt{|\lambda|}\, |x-i|}
\: e^{-\sqrt{|\lambda|}\, |x-j|}
     \over |x-i|\: |x-j|} \: g(x)^2\: dx  \\
	& \leq & C\: e^{-m\sqrt{|\lambda|}\, |i-j|}
\: g(i)g(j), \;\;\;\; 0<m<1.
\end{eqnarray*}
Hence

\begin{eqnarray*}
\| F\|_{\L^2(d\mu)}^2 & \leq &
 C  \sum_{i\in \Z^d} f_i g(i) \: \sum_{j\in \Z^d} f_j g(j) e^{-m\sqrt{|\lambda|}\,
 |i-j|} \\
 & = & C \sum_{i\in \Z^d} f_i g(i)\: \phi_i \: <
\: C_1\|f\|_{\ell^2(d\widetilde\mu)}^2,
\end{eqnarray*}
since $\{\phi_i\}_{i\in \Z^d}\in \ell^2(\Z^d)$.
Lemma A2 is proved. $\Box$


\section*{Appendix B}

\noindent{\bf Lemma B.1.}

\noindent {\it Suppose\ $h$\ is arbitrary selfadjoint operator in\ $\ell^2(\Z^d)$\ with simple
spectrum,\ $\Psi(\lambda,x)$ is the generalized eigenfunction of\ $h$,\  
$\rho(\lambda)$\ is the spectral measure of\ $h$.\ Then  
for\ $\rho$ - a.e. $\lambda\in\sigma(h)$:

$$
\sum_{x\in\Z^d}\, |\Psi(\lambda,x)|^2(1+|x|)^{-\nu}\: \leq C\: <\infty,
\eqno(B.1)
$$
where\ $\nu>d$, $C=C(\lambda)<\infty$.}

\noindent {\it Proof.}
Consider\ $I=(a,b)\subset\R$,\ $R(i)=(h-i)^{-1}$.\ By the spectral theorem,

$$ 
\Im\, R(i;x,x)\: = \: \int\limits_\R\, 
|\Psi(\lambda,x)|^2 (\lambda^2+1)^{-1}\, d\rho(\lambda)\:\geq \: (b^2+1)^{-1}\: 
\int\limits_{(a,b)}\, |\Psi(\lambda,x)|^2 d\rho(\lambda).
\eqno(B.2)
$$
Multiplying both sides of (B.2) by\ $(1+x)^{-\nu}$\ and summing over\ $x$,
\ for \ $d<\nu$,\ one gets:

\begin{eqnarray*} 
\infty\: & \geq &\: \sum_{x\in\Z^d}\, (1+|x|)^{-\nu}|\Im\,R(i;x,x)|\nonumber\\
& \geq &\: (b^2+1)^{-1}\, \sum_{x\in\Z^d}\, (1+|x|)^{-\nu}\int\limits_{(a,b)}
|\Psi(\lambda,x)|^2d\rho(\lambda). 
\end{eqnarray*}
It follows that the series (B.1) is absolutely integrable, so Lemma B.1 is proved. 
$\Box$.

%%%%%%%%%%%%%%
\bigskip{\bf Acknowledgments}. 

\bigskip
 
V.G. thanks the Laboratory of Mathematical Physics and Geometry,
University Paris 7 for invitation and financial support, 
Prof. A.-M.Bertier for discussion of Lemma 5.1.

\noindent Text was re-typesetted in December 2005 in 
kozakstan (former-soviet union, warsaw pact) 
on private PC (Intel Pentium II (korea), OS Windows XP, Home Edition, 
certificate authenticity (Microsoft corp) 00049-120-546-750, N09-01178, X10-60277, 
no internet access)  
with possible unauthorized external illegal access by unated former-soviet ko-gb. 

\noindent "LMS", "EPSRC", imperial college, and 
{\bf warsaw pact's union} 
(including ukraine (kyev russe) and kozakstan),
as well as "NASA" ("national aeronauthics and space administration" since 1997), 
and "eu" space research agency (since 2000)
have no rights in any use of the following research

\noindent because of support of inhuman slavonic ("ukrainien"-russian-based) 
national communism
held by unated former-soviet ko-gb.

\noindent The author's wrong postal address in the internet database 
had not been corrected by IAMP
(international association of Mathematical Physics since 1972)
since 1997.

%%%%%%%%%%%%%%%%%%%%%
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