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pure point spectrum, unbounded random potential, Anderson model
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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{Pure Point Spectrum\\
in the Anderson Model\\
with Unbounded Random Potential
\thanks{1999}
}
\author{V.(-D.) Grinshpun\thanks{no e-mail address, non-slavonic, myself alone, surname adopted, no relatives}
\thanks{alles ko-gb and kozakstan support ethnic (russian-"ukrainien"-based) genocide
of non-slavonic minorities}
}
\date{}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
The Anderson model with the unbounded random potential
(independent random variables
with identical probability disitributions of unbounded support
and bounded density) is established to have only pure point spectrum
(complete system of localized wave-functions) with probability one
in arbitrary dimension.
\noindent The respective result is deduced via trace-class perturbation
analysis as a consequence of the new original result on absence
of pure singular continuous spectrum of random perturbations \cite{G4,G6}.
\end{abstract}
% INTRODUCTION
\section {Introduction} \label{s:1}
\setcounter {equation} {0}
The Anderson model was initially introduced by P.Anderson
\cite{A} in 1958 to model physical processes of spin diffusion,
impurity conduction, and localization.
Rigorous study of the respective spectral properties
had been of essential importance in the recent years because of valuable
applications in physics.
For example, the energy states of a quantum system are described
in terms of self-adjoint operator (Hamilton operator) defined on the Hilbert space
of the corresponding wave functions.
Existence of the non-empty (pure) absolutely continuous component
in the spectrum of such an operator implies
non-zero value of conductivity within the certain energy zone, and metal properties
in the corresponding system.
Presence of only exponentially localized states
(e.g. the pure point spectrum "at high disorder" established previously)
means absence of diffusion and insulator properties in the respective
system (Anderson localization).
The Anderson model had been intensively
studied in the recent years, and it was rigorously established via different
approximation schemes ({\cite{FS,FMSS,DLS,STW,D,G2}) that the respective spectrum
exhibits exponential localization (i.e. it is pure point,
and corresponding eigenfunctions decay exponentially at infinity)
in the regions of impurity spectrum corresponding
to the certain values of disorder parameter
(the so-called "high disorder localization").
The conductivity spectral component
(corresponding to the region of the spectrum of non-perturbed Laplace
operator) was established recently (\cite{G4,G6}) as pure
absolutely continuous with probability 1 for the bounded random
potentials.
\noindent At the same time, it is well-known that self-adjoint
operators (on $\ell^2(\Z^d)$, or $\L^2(\R^d)$) with
growing potential have no absolutely continuous spectrum,
for example, if $\lim\limits_{|x|\rightarrow\infty}|q(x)|=\infty$,
then the spectrum is pure discrete (\cite{Gl}).
\noindent In the following paper there is proved the analogues
theorem for the multidimensional finite-difference operators
with random (ergodic) potential. All the spectrum is established to be
pure point with probability 1 assuming the single-site probability
distribution has unbounded support and bounded density.
\noindent The respective result
(Theorem \ref{t:3}, the main technical result involved
to establish Theorem \ref{t:1}) is the generalization for the case $d>1$
of the result on absence of the absolutely continuous spectrum
of the one-dimensional Jacoby matrix with the unbounded diagonal potential
(\cite{SS}, 1989). Extension of this result to the case
of finite-difference operator of infinite order had been found by \cite{G1}.
Similar result for one-dimensional finite-difference operator on half-line
with unbounded non-random potential and random boundary conditions
had been for the first time rigorously established by \cite{KMP}.
\noindent Consider the Anderson tight-binding model defined by the Hamiltonian
$H_U$ (\ref{1}) - (\ref{5})
with the random potential (independent identically distributed random variables
with single-site probability distribution of unbounded support
and bounded density).
\begin{theorem}\label{t:1} {\bf (Pure point spectrum in the Anderson model
with unbounded random potential)}:
$$
\sigma(H_U)\: =\: \sigma_{pp}.
$$
\end{theorem}
\noindent The proof is a consequence of Theorem \ref{t:3}
and Theorem \ref{t:2}.
Theorem \ref{t:3} (on absence of absolutely continuous spectrum)
permits arbitrary unbounded probability distributions, and
implies "absence of diffusion" in the corresponding disordered system
(\cite{A,FS}).
Theorem \ref{t:2} (on absence of singular continuous spectrum, \cite{G4,G6})
requires probability distributions to have bounded density,
and implies pure point spectrum.
However, the approximation method does not allow to control the
decay properties of the respective eigenfunctions.
The main condition imposed to deduce a.s. absence
of the singular continuous component in the spectrum of a random perturbation,
is either absence of its pure point, or absence of its absolutely continuous
spectral component established \' a priori.
Specifically, the pure absolutely continuous spectrum was recently
rigorously established (\cite{G4,G6})
to exist in the intervals free of the point spectrum of a random (ergodic)
Hamiltonian.
By the same way, the pure point spectrum fills the intervals
where the absolutely continuous spectrum is empty.
The basic result is represented via the following
\begin{theorem}\label{t:2} {\bf (The non-mixing property \cite{G4,G6})}
\noindent Consider the Anderson Hamiltonian $H$ defined by (\ref{1})-(\ref{4}).
\begin{enumerate}
\item[{\bf (A)}] Suppose
$$
\sigma_{pp}(H(\omega))\cap(a,b)\: =\: \emptyset,
$$
$(a,b)\subset\R$, with probability 1.
\noindent Then
$$
\sigma_{sc}(H(\omega))\cap (a,b)\: =\: \emptyset,
\;\;\hbox{and}\;\;\sigma(H(\omega))\cap (a,b)\: \subset\: \sigma_{ac}(H)
$$
(i.e. the spectrum of $H$ in $(a,b)$ is pure absolutely continuous),
with probability 1.
\medskip
\item[{\bf (B)}] Suppose
$$
\sigma_{ac}(H(\omega))\cap(a,b)\: =\: \emptyset,
$$
$(a,b)\subset\R$, with probability 1.
\noindent Then
$$
\sigma_{sc}(H(\omega))\cap (a,b)\: =\: \emptyset,
\;\;\hbox{and}\;\;\sigma(H(\omega))\cap (a,b)\: \subset\: \sigma_{pp}(H)
$$
(i.e. the spectrum of $H$ in $(a,b)$ is pure point),
with probability 1.
\end{enumerate}
\end{theorem}
For the unbounded random (strongly unbounded non-random)
potentials (cf. definition in Section \ref{s:2}), it is rigorously proved
(Theorems \ref{t:3}, \ref{t:4}) absence
of the absolutely continuous spectrum in multiple dimension
with probability 1:
$$
\sigma_{ac}(H_U)\: =\: \emptyset,
$$
which, according to the basic result (Theorem \ref{t:2}), implies
that the Anderson tight binding model (with unbounded random potential)
has only pure point spectrum, with probability 1.
\noindent {\bf Example}.
Consider the Anderson Hamiltonian $H_G$ on $\ell^2(\Z^d)$,
$d\geq 2$, with random potential formed by
the independent identically distributed random variables
with the Gauss probability distribution:
$$
g(q)\: =\: {1\over \sqrt{2\pi}\sigma}\, e^{-(q-m)^2\over 2\sigma^2},
$$
$g_0=\sqrt{2\pi}\sigma$. It had been previously established
(\cite{DLS,STW}, 1986), that
there exist $0\leq E_0=E_0(g_0)<\infty$,
and $0<\overline g_0<\infty$,
such that $E_0=0$ if $g_0>\overline{g_0}$, and
$$
\sigma(H_G)\: \cap\: (\pm E_0, \pm\infty)\: \subset\: \sigma_{pp}(H_G)
$$
(i.e. the impurity spectrum is pure point), with probability
one.
\noindent The new Theorem \ref{t:1} provides much stronger result,
proving all the spectrum of $H_G$ is pure point at arbitrary disorder
$g_0>0$.
\bigskip
\section {Pure point spectrum} \label{s:2}
\setcounter {equation} {0}
The Anderson Hamiltonian with the unbounded potential
is defined by the finite-difference
operator
\bn \label{1}
H_U(\omega)\: =\: H_0\: +\: Q(\omega),
\en
\noindent where $H_0$ is the Laplace operator
\bn \label{2}
H_0\psi(x)\: =\: \sum\limits_{\|x-y\|=1} (\psi(x)-\psi(y)), \;\; \psi\in\ell^2(\Z^d), \; x,y\in\Z^d,
\en
$\|x\|=\sum\limits_{1\leq j\leq d}|x_j|$, $Q_\omega$ is the random potential
\bn \label{3}
Q(\omega)\psi(x)\: =\: q_\omega (x)\psi(x), \;\; \psi\in\ell^2(\Z^d), \;\; x\in\Z^d,
\en
$\{q_\omega\}(x)_{x\in\Z^d}$ are independent random variables with
identical probability distributions having bounded density and unbounded support:
\bn\label{4}
dP(q)\: =\: {\rm Prob}\{q(0)\in dq\}\: =\: g(q)dq, \;\; g_0^{-1}=\sup\limits_q\, g(q)<\infty,
\en
\begin{eqnarray}\label{5}
\sup\limits_q\, {\rm supp}\, dP(q)\: & = &\: +\infty\;\;\;\;\hbox{\rm or}\nonumber\\
\inf\limits_q\, {\rm supp}\, dP(q)\: & = &\: -\infty.
\end{eqnarray}
\noindent The corresponding probability space
$$
(\Omega,\P)=\prod_{j\in\Z^d}(\Z_j, dP(q_j)).
$$
Operator $H_U$ is ergodic self-adjoint operator, which spectrum
$\sigma(H_U)$, as well as its corresponding pure point and (absolutely) continuous spectral
components $\sigma_{pp}(H_A)$,
$\sigma_{ac}(H_U)$,
$\sigma_{c}(H_U)$, are non-random subsets of $\R$:
$$
\sigma(H_U)\: =\: \sigma(H_0) \dot + {\rm supp}\, dP(q)\: =\: (-\infty,+\infty),
$$
where $\dot +$ denotes the algebraic sum of subsets of $\R$,
$\sigma(H_0)$ denotes the spectrum of the non-perturbed Laplace
operator, which is pure absolutely continuous:
$$
\sigma(H_0)\: =\: [0,4d]\: =\: \sigma_{ac}.
$$
\bigskip
\begin{theorem}\label{t:3} {\bf (Absence of absolutely continuous spectrum
for unbounded random potential)}:
$$
\sigma_{ac}(H_U)\: =\: \emptyset
$$
with probability 1.
\end{theorem}
\bigskip
\noindent {\bf Remark}. Theorem \ref{t:3} permits arbitrary unbounded
single-site probability distributions.
\bigskip
\noindent The proof follows from the result for the strongly unbounded non-random potentials
(Theorem \ref{t:4}).
\noindent Consider the operator $H$ defined on $\ell^2(\Z^d)$ by
(\ref{1})-(\ref{3}), where
\bn\label{6}
\limsup\limits_{j\rightarrow\infty}\:
\inf\limits_{x\in\partial\Lambda_{L_j}}\:
|q(x)|=\: \infty,
\en
where $\Lambda_{L_j}=\{x\in\Z^d|\: \|x\|\leq L_j\}$,
i.e. the potential $q(x)$ is unbounded over increasing to infinity
sequence of concentric spheres $\Lambda_{L_j}\subset\Lambda_{L_{j+1}}$
of radius $L_j$, $j\in\N$.
\bigskip
\noindent {\bf Definition}. {\it The potential satisfying (\ref{6})
is referred in the
following paper as the strongly unbounded.}
\bigskip
\begin{theorem}\label{t:4} {\bf (Absence of absolutely continuous spectrum
for the strongly unbounded potential)}:
$$
\sigma_{ac}(H)\: =\: \emptyset.
$$
\end{theorem}
\noindent {\it Proof of Theorem \ref{t:4}}.
\begin{lemma}\label{l:1} {\bf (The resolvent identity)}
\begin{eqnarray} \label{7}
(A+B-z)^{-1}\: -\: (A-z)^{-1}\: & = &\: -(A-z)^{-1}\, B\, (A+B-z)^{-1}\nonumber\\
& = &\: -(A+B-z)^{-1}\, B\,(A-z)^{-1},
\end{eqnarray}
where $A,B$ are arbitrary linear operators with bounded resolvents,
$z\not\in\sigma(A)\cup\sigma(B)$.
\end{lemma}
\noindent {\it Proof} follows multiplying both sides of (\ref{7})
by $(A+B-z), (A-z)$.
\begin{lemma}\label{l:2} Consider
$$
h\: =\: h_0\: +\: q,
$$
where $q$ is multiplication operator, and $h_0\in {\cal B}$,
where ${\cal B}$ denotes the Banach algebra
of bounded operators on $\ell^2(\Z^d)$. Denote by
$e(x)$ the unit vector in $\ell^2(\Z^d)$. Suppose
$z\not\in\sigma(h)$, $\Delta={\rm dist}\{z,\sigma(h)\}$. Then
$$
\|(h-z)^{-1}e(x)\|\: \leq\: {\Delta + \|h_0\| \over \Delta |q(x) - z|}.
$$
\end{lemma}
\medskip
{\it Proof.} By the resolvent identity (Lemma \ref{l:1}),
\begin{eqnarray*}
\|(h-z)^{-1}\: e(x)\|\:& \leq &\: \|(q-z)^{-1}e(x)\:\\
& - &\: (h-z)^{-1}h_0(q-z)^{-1}e(x)\|\\
& \leq &\: {\Delta\: +\: \|h_0\|\over \Delta\: |q(x)-z|^{-1} }.
\end{eqnarray*}
Lemma \ref{l:2} is proved.
\noindent Denote
$$
H_n\: =\: \cases{ H(x,y),\;\; x,y\in\Lambda_{L_n}\setminus\Lambda_{L_{n-1}}\cr
0,\;\; & otherwise,\cr }
$$
$n>1$ (finite-volume operator with the Dirichlet boundary conditions);
\bn\label{8}
H_\Lambda\: =\: \sum_n^\oplus H_n;
\en
$$
H\: = \: H_\Lambda\: +\: \AA.
$$
Denote
$$
\AA_n(x,y)\: =\: \cases{ 1,\;\;
&if $x\in\Lambda_n, y\not\in\Lambda_n$, or $x\not\in\Lambda_n, y\in\Lambda_n$\cr
0,\;\; &otherwise;\cr }
$$
Then
\begin{eqnarray}\label{9}
\AA\: & = &\: \sum_{n}^\oplus\AA_n\nonumber\\
& = &\: \sum\limits_{y\in\Lambda_{n-1}}\:
\sum\limits_{\scriptstyle x\in\Lambda_n\atop \|x-y\|=1}\:
\langle e(y),.\rangle\: e(x).
\end{eqnarray}
\noindent By the resolvent identity (Lemma \ref{l:1}), if $\Im\, z\, \ne 0$,
\begin{eqnarray}\label{10}
(H-z)^{-1}\: -\: (H_\Lambda-z)^{-1}\: & = &\:
-(H_\Lambda-z)^{-1}\: \AA\: (H-z)^{-1}\nonumber\\
& = &\:
(H_\Lambda-z)^{-1}\: \AA\: (H_\Lambda-z)^{-1}\:
(\AA (H-z)^{-1}-I).
\end{eqnarray}
\noindent Prove that
\bn\label{11}
(H-z)^{-1} - (H_\Lambda-z)^{-1}\: \in\: {\cal B}_1,
\en
where ${\cal B}_1$ denotes the normed space of trace-class operators on $\ell^2(\Z^d)$,
then by Theorem 16, ch.10 (\cite{Ka}),
\bn\label{12}
\sigma_{ac}(H)\: = \: \sigma_{ac}(H_\Lambda)\: =\: 0.
\en
Since $H_n\in {\cal B}_0$ ($H_n$ is of finite rank), (\ref{8}) implies
$(H_\Lambda-z)^{-1}\in {\cal B}_0$, where ${\cal B}_0$ denotes the Banach space
of compact operators on $\ell^2(\Z^d)$ which is a closed linear space
with respect to the ${\cal B}$- norm.
\noindent Since ${\cal B}_1$ is a closed ideal in ${\cal B}$,
and $(\AA(H-z)^{-1}-I)\in{\cal B}$, (\ref{9}) implies that
it is sufficient to prove
$$
(H_\Lambda-z)^{-1}\AA (H_\Lambda-z)^{-1}\: \in\: {\cal B}_1.
$$
Denote by $\|.\|_1$ the trace norm. If $A\in {\cal B}$ is of finite rank, then
$$
\|A\|_1\: \leq\: {\rm rank}(A)\, \|A\|.
$$
It follows by Lemma \ref{l:2} and (\ref{8})-(\ref{10})
\begin{eqnarray}\label{13}
& \; & \|(H_\Lambda-z)^{-1}\AA (H_\Lambda-z)^{-1}\|_1\nonumber\\
& \leq & \:
\sum\limits_{n}\:
\sum\limits_{\scriptstyle \|x-y\|=1\atop{ x\in\Lambda_n\atop y\in\Lambda_{n-1}} }\:
|\langle (H_n-\overline{z})^{-1} e(y),.\rangle|\: \|(H_n-z)^{-1}e(x)\|\nonumber\\
&\leq & \:
\sum\limits_{n}\:{2d(\Delta+2d)\over
\Delta^2\inf\limits_{x\in\partial\Lambda_{L_n}}\:|q(x)-z|}\: <\: \infty,
\end{eqnarray}
since condition (\ref{6}) implies that it is possible to choose
$\{x_n\}_{n\in\N}$ such that
$$
\sum\limits_n\: {1\over |q(x_n)|}\: <\: \infty.
$$
Hence (\ref{13}) imply (\ref{11}) and (\ref{12}).
Theorem \ref{t:4} is proved. $\Box$
\noindent {\it Proof of Theorem \ref{t:3}.}
Denote as before by $(\Omega, {\cal S}, \P)$ the probability
space of realizations of the random potential (\ref{3})-(\ref{5}),
where ${\cal S}$ denotes the $\sigma$- algebra of $\P$-
measurable subsets of $\Omega$.
Consider the sequence $L_{n+1}>L_n>0$, $n\in\N$, and denote
$$
\Omega(b,\Lambda_l)\: =\: \{q\in {\cal A}_Q|\:
\inf_{x\in\partial\Lambda_l}\:
|q(x)|\geq b\}\: \in\: {\cal S},
$$
$$
\Omega_n\: =\: \bigcup_{ \Lambda_{L_n}\in\Lambda_{L_{n+1}} }\:
\Omega(b_n,\Lambda_{L_n})\: \in\: {\cal S}.
$$
Then
\bn\label{14}
\P\{\Omega(b,\Lambda_l)\}\: =\: dP\{(\pm b,\pm\infty)\}^{|\partial\Lambda_l|},
\en
\bn\label{15}
\P\{\Omega_n\}\: \geq \: {L_{n+1}\over L_n}\:
\P\{\Omega(b_n,\Lambda_{L_n})\}.
\en
Condition (\ref{6}) implies that it is possible to choose $b_n>0$, $n\in\N$,
such that
\begin{eqnarray*}
\lim\limits_{n\rightarrow\infty}\: b_n\: =\: \infty,\\
dP\{(\pm b_n,\pm\infty)\}\: \ne\: 0.
\end{eqnarray*}
So choose $L_n$, $n\in\N$:
$$
L_{n+1}\: \geq\:{1\over |n|dP\{(\pm b_n,\pm\infty)\}^{C_dL_n^{d-1}}}\: L_n.
$$
Then by (\ref{14}), (\ref{15}):
\bn\label{16}
\sum\limits_{n\rightarrow\infty}\: \P\{\Omega_n\}\: =\: \infty.
\en
\noindent It follows by Borel-Kantelli lemma via (\ref{16}) that
\bn\label{17}
\P\{\overline{\Omega}=\bigcap_{n\geq 1}\bigcup_{k\geq n}\Omega_n\}\: =\: 1.
\en
(\ref{17}) implies the potential $Q_\omega$ satisfying (\ref{5})
is strongly unbounded with probability 1.
\noindent Theorem \ref{t:4} implies
$$
\sigma_{ac}(H_U)\: =\: \emptyset
$$
holds with probability 1.
Theorem \ref{t:3} is proved. $\Box$
\bigskip
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\noindent {\bf Acknowledgements}.
personal research
(1999, kharkov ("ukraine", kyev russe, warsaw pact, former-soviet union))
had been supported only by research grant "AMS"-1995 (\$ 150).
Typesetted in december 2005
in kozakstan (former-soviet union, warsaw pact since 1955)
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 illegal external access by former-soviet ko-gb.
VG would like to request excuse for
not answering to the e-mail correspondence
could had arrived to his previous e-mail address
(grinshpun@ilt.kharkov.ua):
mentioned e-mail box was closed, and permit for
entrance to the host-keeping institution
was denied (not prolonged) by
institute for low temperature physics, kharkov ("ukraine")
on December 31 (1999),
when the following paper had been under preparation
for submission for publication by author.
\noindent VG had had no opportunity to present his described in part personal
research results at ICMP XIII, the wrong reference in \cite{ICMP2000}.
\noindent His wrong postal address in the IAMP
(international association of Mathematical Physics since 1972)
internet database had not been corrected since 1997.
\bigskip
\newpage
%%%%%%%%%%%%%%%%%%%%%
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(1995)
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\end{thebibliography}
\end{document}
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