# Absolutely continuous and pure point spectra of discrete operators with
sparse potentials

S. Molchanov, O. Safronov, and B. Vainberg  S. Molchanov: Dept. of Math. and
Statistics, UNCC and Higher School of Economics, Russia; O. Safronov, and B.
Vainberg: Dept. of Math. and Statistics, UNCC<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider the discrete Schrödinger operator $H=-\Delta+V$ with a sparse
potential $V$ and find conditions guaranteeing either existence of wave
operators for the pair $H$ and $H_{0}=-\Delta$, or presence of dense purely
point spectrum of the operator $H$ on some interval $[\lambda_{0},0]$ with
$\lambda_{0}<0$.

###### 2010 Mathematics Subject Classification:

81Q10, 39A14, 47A10

##  1\. introduction

The class of sparse potentials was introduced in the spectral theory of one-
dimensional Schrödinger operators by D. Pearson [10]. Such potentials are
intermediate between compactly supported (or fast decaying) potentials
associated to the scattering theory, and the generic bounded potentials ( such
as periodic, almost periodic, random ergodic potentials) which are studied in
the solid state physics.

Typical sparse potentials have the form

(1.1) $V(x)=\sum_{n\in{\mathbb{N}}}a_{n}\phi(x-x_{n})$

where $\phi\in C_{0}^{\infty}$ is a fixed bump function and $x_{n}$ is a
sequence for which the quantity $d(n)={\rm dist}\,(x_{n},\bigcup_{j\neq
n}\\{x_{j}\\})$ grows as $n\to\infty$. The amplitudes $a_{n}$ either slowly
tend to zero, or are of order $O(1)$ (as in the case where they are
independent identically distributed random variables).

The paper by Kiselev, Last and Simon [4] as well as the paper by Molchanov [5]
extend the results of [10] in different directions:

Let $H=-\frac{d^{2}}{dx^{2}}+V$ be the operator with a potential of the form
(1.1) acting in the space $L^{2}[0,\infty)$. Assume that the boundary
condition at the origin is the Dirichlet condition. Suppose also that
$\phi\geq 0$, the sequence $x_{n}$ is monotone and $a_{n}\to 0$ as
$n\to\infty.$

1) If $\sum_{n}a_{n}^{2}<\infty$ and $x_{n}/x_{n+1}=o(1)$ as $n\to\infty$,
then the spectrum is pure absolutely continuous on $(0,\infty)$.

2) If $\sum_{n}a_{n}^{2}=\infty$ and $x_{n}/x_{n+1}=o(1)$ as $n\to\infty$,
then the spectrum is purely singular continuous on $(0,\infty)$.

These statements could be compared with the results of the paper by Kotani and
Ushiroya [8] where the authors study the transition from the absolutely
continuous spectrum to the pure point spectrum for the operator with a random
potential.

Technically, the analysis of the absolutely continuous spectrum of the
operator $H$ under the condition $\sum_{n}a_{n}^{2}<\infty$ resembles the
theory of lacunary Fourier series and is based on the idea of the
stochastization of the phase of the solution of the equation
$H\psi=k^{2}\psi$, $k\in{\mathbb{R}}\setminus\\{0\\}$, as one passes from one
bump to another.

One should also notice that the spectral properties of the Hamiltonian $H$
depend on the nature of the elementary bump $\phi$. For instance, if all
$a_{n}=1$ and $\phi$ is a reflectionless potential, then the spectrum of $H$
is pure absolutely continuous on $(0,\infty)$ provided the sequence
$\\{x_{n}\\}$ is exponentially sparse (see [5]). At the same time, if the
reflection coefficient $r(k)$ constructed for the bump $\phi$ is not
identically zero, then the spectrum of $H$ is purely singular continuous on
$(0,\infty)$ provided some additional mild technical assumptions are fulfilled
(see [5]).

One expects the multi-dimensional spectral theory of Schrödinger operators
with sparse potentials to be different from the one-dimensional theory,
because the waves ”reflected” from one individual bump decay at infinity, if
$d\geq 2$. More precisely, the solution of the equation
$-\Delta\psi+a_{n}\phi(x-x_{n})\psi=k^{2}\psi$ constructed for one bump decays
as $O(|x|^{-(d-1)/2})$ as $|x|\to\infty$. The latter leads to the fact that
the absolutely continuous spectrum of the milti-dimensional Schrödinger
operator with a sparse potential can cover the positive half-line even in the
case where the coefficients $a_{n}$ do not decay.

Indeed, one of the results of the paper by Safronov [11] says that if
$V_{\xi}$ is a function of the form

$V_{\xi}(x)=\sum_{n\in{\mathbb{Z}}^{d}}a_{n}\xi_{n}\chi(x-n),$

where $\xi_{n}$ are independent identically distributed bounded random
variables such that ${\mathbb{E}}(\xi_{n})=0$ and $\chi$ is the characteristic
function of the cube $[0,1)^{d}$, then the absolutely continuous spectrum of
the operator $-\Delta+tV_{\xi}$ almost surely covers the positive half-line
for almost every $t\in{\mathbb{R}}$ provided

(1.2) $\sum_{n\neq 0}\frac{a_{n}^{2}}{|n|^{d-1}}<\infty.$

This statement should be compared with the results of Bourgain [1] and Denisov
[3] in which the authors assume that $|a_{n}|\leq C(1+|n|)^{-1/2-\varepsilon}$
with $\varepsilon>0$ and prove the corresponding claim for every $t$. The main
difference between [1] and [3] is that the first paper deals with the discrete
operator on the lattice ${\mathbb{Z}}^{2}$ while the second one handles the
continuous operator on ${\mathbb{R}}^{d}$. The potentials considered in the
paper [7] are also decaying at infinity.

One should mention that the corresponding statement about the absolutely
continuous spectrum of the discrete Schrödinger operator on ${\mathbb{Z}}^{d}$
under the condition similar to (1.2) has not been proved. Neither has been
proved a discrete analogue the Laptev-Naboko-Safronov (see [9]) theorem saying
that if $V\geq 0$ and

(1.3) $\int_{{\mathbb{R}}^{d}}\frac{V(x)}{(1+|x|)^{d-1}}dx<\infty,$

then the absolutely continuous spectrum of the operator $-\Delta+V$ covers the
half-line $[0,\infty)$. Note that both conditions (1.2) and (1.3) are
fulfilled for some sparse potentials that do not decay at infinity.

The operator $H$ with a sparse potential that does not decay at infinity can
have pure point spectrum outside of the spectrum of the free Laplace operator.
As a result, one can expect the discrete operator $H$ with a negative random
sparse potential to have absolutely continuous spectrum on $[0,4d]$ and dense
pure point spectrum on some interval $[-a,0)$ with $a>0$. This phenomenon was
proved in the paper [6] by Molchanov but without the specification of the
physical nature of the absolutely continuous component of the spectrum. In the
present paper, we develop the scattering theory for a class of discrete
operators on the lattice ${\mathbb{Z}}^{d}$ with sparse potentials $V$ and
prove existence of wave operators.

##  2\. Statement of the main results

Let $H_{0}=-\Delta$ be the ”free” operator on the lattice ${\mathbb{Z}}^{d}$
whose action is defined by the formula

$\bigl{[}H_{0}u\bigr{]}(n)=\sum_{|n-j|=1}\Bigl{(}u(n)-u(j)\Bigr{)}.$

Let $H=-\Delta+V$, where $V$ is the operator of multiplication by a bounded
real-valued function $V:{\mathbb{Z}}^{d}\to{\mathbb{R}}$. We are interested in
the question of existence of the wave operators

(2.1) $\text{s -}\lim_{t\to\pm\infty}e^{-itH}e^{itH_{0}}=:W_{\pm}.$

######  Theorem 2.1.

Assume that

$\sum_{n\in{\mathbb{Z}}^{d}\setminus\\{0\\}}\frac{|V(n)|}{|n|^{(d-1)/2}}<\infty.$

Then the wave operators (2.1) exist.

Note that Theorem 2.1 is applicable to operators with sparse potentials, for
which the quantity

(2.2) $d(n)=\text{dist}\,\bigl{(}n,\,\,\text{supp}(V)\setminus\\{n\\}\bigr{)}$

tends to infinity sufficiently fast as $|n|\to\infty$. Another theorem that
deals with sparse potentials is the followings statement, in which $\chi_{n}$
is the characteristic function of the one point set
$\\{n\\}\subset{\mathbb{Z}}^{d}$.

######  Theorem 2.2.

For an arbitrary number $\lambda_{0}<0$, let

$a=\frac{1}{\bigl{(}(H_{0}-\lambda_{0})^{-1}\chi_{0},\chi_{0}\bigr{)}}.$

Let $\Omega$ be a fixed subset of the lattice ${\mathbb{Z}}^{d}$, such that

$V_{\xi}(n)=\begin{cases}\xi_{n},\qquad\text{if}\qquad n\in\Omega,\\\
0,\qquad\text{if}\qquad n\notin\Omega,\end{cases}$

where $\\{\xi_{n}\\}_{n\in{\mathbb{Z}}^{d}}$ are independent random variables
uniformly distributed on the interval $[-a,0]$. Assume that that the quantity
$d(n)$ defined by (2.2) obeys the condition

$\lim_{|n|\to\infty,\,n\in\Omega}\frac{d(n)}{|n|^{\delta}}=\infty$

for some $\delta>0$. Then the operator $H_{\xi}=-\Delta+V_{\xi}$ almost surely
has a dense pure point spectrum in the interval $[\lambda_{0},0)$.

According to Theorems 2.1 and 2.2, there are discrete Schrödinger operators
whose absolutely continuous spectrum ”fills” the interval $[0,4d]$ while their
pure point spectrum is dense in $[\lambda_{0},0)$.

##  3\.  Proof of Theorem 2.1.

Standard notations. For a closed linear operator $T$, the symbols
${\mathcal{D}}(T)$, $\sigma(T)$ and $\sigma_{p}(T)$ denote its domain, its
spectrum and the set of eigenvalues (which does not have to be closed). If $T$
is self-adjoint, then $E_{T}(\cdot)$ denotes the operator-valued spectral
measure of the operator $T$.

Besides the standard wave operators $W_{\pm}$, we will consider the modified
operators $W_{\pm}(J)$ defined by

(3.1) $W_{\pm}(J)=\text{s -}\lim_{t\to\pm\infty}e^{-itH}Je^{itH_{0}},$

where $J:\ell^{2}({\mathbb{Z}}^{d})\to\ell^{2}({\mathbb{Z}}^{d})$ is assumed
to be bounded. We will employ one of the statements of the scattering theory,
using the notion of a trace class operator. One says that an operator $T$ on a
separable Hilbert space belongs to the class ${\mathfrak{S}}_{p}$, if the
sequence $\\{s_{j}(T)\\}$ of singular values of this operator is an
$\ell^{p}$-sequence. In this case, the norm of $T$ in ${\mathfrak{S}}_{p}$ is
defined as the norm of the sequence $\\{s_{j}(T)\\}$ in $\ell^{p}$. The
classes ${\mathfrak{S}}_{1}$ and ${\mathfrak{S}}_{2}$ are called the trace
class and the Hilbert-Schmidt class, correspondingly.

One of theorems by Pearson (see [14]) leads to the following result:

######  Theorem 3.1.

Let $E_{0}(\cdot)$ and $E(\cdot)$ be the operator-valued spectral measures of
$H_{0}$ and $H$ correspondingly. Assume that the operator

(3.2) $E(a,b)\bigl{(}HJ-JH_{0}\bigr{)}E_{0}(a,b)\in{\mathfrak{S}}_{1}$

is a trace class operator for any bounded interval $(a,b)$. Then the limits
$W_{\pm}(J)$ in (3.1) exist.

We will apply this theorem in the case where $J$ commutes with $H_{0}$. In
this case, the condition (3.2) turns into the relation

(3.3) $E(a,b)VJE_{0}(a,b)\in{\mathfrak{S}}_{1},$

while the limit (3.1) coincides with the limit

$\text{s -}\lim_{t\to\pm\infty}e^{-itH}e^{itH_{0}}J.$

The latter observation implies the following lemma.

######  Lemma 3.1.

Let $\\{J_{l}\\}_{l\in{\mathbb{N}}}$ be a family of bounded operators
commuting with $H_{0}$ and such that

(3.4) $\text{s-}\lim_{l\to\infty}J_{l}=I.$

Assume that

$VJ_{l}E_{0}(a,b)\in{\mathfrak{S}}_{1}$

for any bounded interval $(a,b)$ and for any $l$. Then the limits $W_{\pm}$ in
(2.1) exist.

Remark. Replacement of (3.4) by the assumption

$\text{s-}\lim_{l\to\infty}J_{l}=J.$

leads to the existence of the limits (3.1).

In order to construct operators $J$ needed for our purposes, one has to
introduce the operator
$\Phi:L^{2}({\mathbb{T}}^{d})\to\ell^{2}({\mathbb{Z}}^{d})$ setting

$\bigl{[}\Phi
u\bigr{]}(n)=\frac{1}{(2\pi)^{d/2}}\int_{{\mathbb{T}}^{d}}e^{-i\xi
n}u(\xi)d\xi,\qquad{\mathbb{T}}^{d}=[0,2\pi)^{d}.$

It is very well known that

(3.5) $H_{0}=\Phi\bigl{[}a\bigr{]}\Phi^{-1},$

where $[a]$ denotes the operator of multiplication by the function

$a(\xi)=\sum_{j=1}^{d}(2-2\cos(\xi_{j})),\qquad\xi=(\xi_{1},\xi_{2},\dots,\xi_{d}),$

called the symbol of the operator $H_{0}$.

Definition of a regular set. An open set $\Omega\subset{\mathbb{T}}^{d}$ is
called regular, if

(1) $\nabla a(\xi)\neq 0,\qquad\forall\xi\in\Omega$.

(2) The curvatures of the level surfaces $a(\xi)=const$ are different from
zero at all points $\xi\in\Omega$.

(3) $\Omega$ is diffeomorphic to a set $[\alpha,\beta]\times U$, where $U$ is
an open set in ${\mathbb{R}}^{d-1}$. Moreover, the corresponding
diffeomorphism

$\tau:\Omega\to[\alpha,\beta]\times U$

is a function $\tau=(\tau_{1},\tau_{2},\dots,\tau_{d})$ having the property
$\tau_{1}(\xi)=a(\xi),\quad\forall\xi\in\Omega$. Without loss of generality,
we will assume that the diffeomorphism $\tau$ can be extended into a larger
domain $\tilde{\Omega}$ containing $\Omega$, so that the corresponding
Jacobian is a bounded separated from zero function on $\Omega$.

Note that for any $\varepsilon>0$, there is a disjoint collection of sets
$\\{\Omega_{j}\\}_{j=1}^{N}$ having the properties (1)-(3) such that the
Lebesgue measure of the set

${\mathbb{T}}^{d}\setminus\bigcup_{j=1}^{N}\Omega_{j}$

is smaller than $\varepsilon$. Therefore, the collection of orthogonal
projections $P_{\Omega_{j}}$ in $L^{2}({\mathbb{T}}^{d})$ onto the space of
functions vanishing outside of $\Omega_{j}$ is a family of operators with the
property

$\sum_{j}P_{\Omega_{j}}f\to f,\qquad\text{as}\quad\varepsilon\to
0,\quad\forall f\in L^{2}({\mathbb{T}}^{d}).$

Consequently, we obtain the following statement:

######  Corollary 3.1.

Let $J_{\Omega}:\ell^{2}({\mathbb{Z}}^{d})\mapsto\ell^{2}({\mathbb{Z}}^{d})$
be the operator

$J_{\Omega}=\Phi\chi_{\Omega}\Phi^{-1}$

where $\chi_{\Omega}$ is the characteristic function of a set
$\Omega\subset{\mathbb{T}}^{d}$. If the limits $W_{\pm}(J_{\Omega})$ defined
by (3.1) with $J=J_{\Omega}$ exist for all sets $\Omega$ having the properties
(1)-(3), then the wave operators $W_{\pm}$ defined in (2.1) also exist.

The next step in our arguments will be an approximation of the operators
$J_{\Omega}$ by operators $J_{\Omega,n}$ commuting with $H_{0}$ and having the
property that

$J_{\Omega,n}\,f\to J_{\Omega}\,f,\qquad\text{as }\quad
n\to\infty,\quad\forall f\in\ell^{2}({\mathbb{Z}}^{d}).$

Obviously, existence of the limits (3.1) with $J=J_{\Omega,n}$ would imply
existence of operators $W_{\pm}(J_{\Omega})$. Therefore, according to
Corollary 3.1, the wave operators $W_{\pm}$ would also exist.

Note that, for any $f\in\ell^{2}({\mathbb{Z}}^{d})$, the element
$J_{\Omega}f\in\ell^{2}({\mathbb{Z}}^{d})$ is the sequence

(3.6)
$\bigl{[}J_{\Omega}f\bigr{]}(j)=\frac{1}{(2\pi)^{d/2}}\int_{[\alpha,\beta]\times
U}e^{i\,j\cdot\xi(\tau)}\omega(\tau)\bigl{[}\Phi^{-1}f\bigr{]}(\xi(\tau))d\tau,\quad
j\in{\mathbb{Z}}^{d},$

where $\xi(\cdot)$ is the inverse of the mapping $\tau(\cdot)$ and
$\omega(\tau)=\bigl{|}\frac{\partial\xi}{\partial\tau}\bigr{|}$ is its
Jacobian.

Let $\\{\phi_{k}\\}_{k=1}^{\infty}$ be an orthonormal basis in $L^{2}(U)$.
Without loss of generality, one can assume that $\phi_{k}\in
C_{0}^{\infty}(U)$. We define the orthogonal projections
$P_{n}:L^{2}([\alpha,\beta]\times U)\to L^{2}([\alpha,\beta]\times U)$ setting

$\bigl{[}P_{n}u\bigr{]}(\tau)=\sum_{k=1}^{n}\phi_{k}(\tau^{\prime})\int_{U}u(\tau_{1},\eta)\bar{\phi}_{k}(\eta)d\eta,\qquad\tau=(\tau_{1},\tau^{\prime})\in[\alpha,\beta]\times
U.$

After that, we define the operators $J_{\Omega,n}$ by

(3.7)
$\bigl{[}J_{\Omega,n}f\bigr{]}(j)=\frac{1}{(2\pi)^{d/2}}\int_{[\alpha,\beta]\times
U}e^{i\,j\cdot\xi(\tau)}\omega(\tau)\Bigl{[}P_{n}\bigl{[}[\Phi^{-1}f](\xi(\cdot))\bigr{]}\Bigr{]}(\tau)\,d\tau,\quad
j\in{\mathbb{Z}}^{d}.$

Taking into account the fact that $a(\xi(\tau))=\tau_{1}$ for all
$\tau\in[\alpha,\beta]\times U$, that is, the symbol of $H_{0}$ coincides with
the first of the $\tau$-coordinates, we obtain from (3.5) and (3.7) that
$\bigl{[}P_{n}([\Phi^{-1}H_{0}f]\circ\xi)\bigr{]}(\tau)=\tau_{1}\bigl{[}P_{n}([\Phi^{-1}f]\circ\xi)\bigr{]}(\tau)$.
Hence, the operators $J_{\Omega,n}$ commute with $H_{0}$. Furthermore, since
the sequence $P_{n}$ converges to $I$ strongly in $L^{2}([\alpha,\beta]\times
U)$, it converges to the same limit in the weighted space
$L^{2}([\alpha,\beta]\times U,\omega)$, because the norms in these two spaces
are equivalent. Comparing (3.6) and (3.7), we conclude that $J_{\Omega,n}\to
J_{\Omega}$ strongly as $n\to\infty$.

Thus, it remains to establish existence of limits (3.1) for $J=J_{\Omega,n}$.
The latter follows from the statement below.

######  Proposition 3.1.

Let $\chi_{j}$ be the characteristic function of the one-point set
$\\{j\\}\subset{\mathbb{Z}}^{d}$. Let $J_{\Omega,n}$ be the operators defined
above. Then there is a positive constant $C=C(d,\Omega,n)$ depending only on
$d,\Omega$ and the choice of the collection $\\{\phi_{k}\\}_{k=1}^{n}$ such
that

(3.8) $\|V\chi_{j}J_{\Omega,n}\|\leq
C\frac{|V(j)|}{1+|j|^{(d-1)/2}},\qquad\forall j\in{\mathbb{Z}}^{d}.$

.

Proof. The relation (3.8) follows from the estimate

(3.9)
$Q_{k}(\tau_{1},j):=\Bigl{|}\int_{U}e^{ij\cdot\xi(\tau_{1},\tau^{\prime})}\phi_{k}(\tau^{\prime})\omega(\tau_{1},\tau^{\prime})d\tau^{\prime}\Bigr{|}\leq\frac{C_{\tau_{1}}}{1+|j|^{(d-1)/2}},\qquad\forall
j\in{\mathbb{Z}}^{d},$

which holds for each fixed $\tau_{1}\in[\alpha,\beta]$. Such inequalities were
studied systematically in the paper [12] by Shaban and Vainberg. In
particular, it was shown that $C_{\tau_{1}}$ is a bounded function of
$\tau_{1}$ on $[\alpha,\beta]$. To prove (3.8), one needs to observe that, for
any $u\in L^{2}({\mathbb{T}}^{d})$,

(3.10) $\Bigl{|}\bigl{[}VJ_{\Omega,n}\Phi
u\bigr{]}(j)\Bigr{|}\leq(2\pi)^{-d/2}|V(j)|\sum_{k=1}^{n}\int_{\alpha}^{\beta}\,Q_{k}(\tau_{1},j)g_{k}(\tau_{1})d\tau_{1},$

where

(3.11)
$g_{k}(\tau_{1})=\Bigl{|}\int_{U}u\bigl{(}\xi(\tau_{1},\tau^{\prime})\bigr{)}\bar{\phi}_{k}(\tau^{\prime})d\tau^{\prime}\Bigr{|}\leq\Bigl{(}\int_{U}|u\bigl{(}\xi(\tau_{1},\tau^{\prime})\bigr{)}|^{2}d\tau^{\prime}\Bigr{)}^{1/2}.$

The inequality (3.11) implies that
$\int_{\alpha}^{\beta}g_{k}(\tau_{1})d\tau_{1}\leq
C\|u\|_{L^{2}({\mathbb{T}}^{d})}$. Combining this estimate with the relation
(3.9), we infer (3.8) from (3.10). $\,\,\,\,\,\Box$

Since $V\chi_{j}J_{\Omega,n}$ is a rank one operator, its trace class norm
coincides with the usual norm. Consequently, we obtain:

######  Corollary 3.2.

Assume that

$\sum_{j\in{\mathbb{Z}}^{d}\setminus\\{0\\}}\frac{|V(j)|}{|j|^{(d-1)/2}}<\infty.$

Then the operator $VJ_{\Omega,n}$ is a trace class operator.

Theorem 2.1 follows from Lemma 3.1 and Corollary 3.2. $\,\,\,\,\Box$

##  4\. Proof of Theorem 2.2

Our study of the pure point spectrum is based on the following celebrated
result of Simon and Wolff [13]:

######  Theorem 4.1.

Let $A$ be a self-adjoint operator in a Hilbert space ${\mathfrak{H}}$ such
that $\phi$ is a cyclic vector for $A$. Let $\mu_{\phi}$ be the spectral
measure of $A$ corresponding to the vector $\phi$. Let $P$ be the orthogonal
projection onto the space of scalar multiples of $\phi$. The spectrum of
$A+tP$ is pure point on a Borel set $\Omega\subset{\mathbb{R}}$ for almost
every $t\in{\mathbb{R}}$, if and only if

(4.1)
$\int_{{\mathbb{R}}}\frac{d\mu_{\phi}(t)}{(t-\lambda)^{2}}<\infty,\qquad\text{for
almost every}\quad\lambda\in\Omega.$

In applications to discrete Schrödinger operators, it is convenient to
interpret (4.1) as the condition that $(A-\lambda)^{-1}\phi$ is also an
element of the Hilbert space ${\mathfrak{H}}$. Indeed, let us define
$U:{\mathfrak{H}}\to L^{2}({\mathbb{R}},\mu_{\phi})$ as a bounded operator
mapping an element of the form $f(A)\phi$ to the function $f(t)$ (Here, one
needs to consider only such functions that $\phi\in{\mathcal{D}}(f(A))$). This
operator $U$ is a unitary operator from ${\mathfrak{H}}$ onto
$L^{2}({\mathbb{R}},\mu_{\phi})$. Moreover, $U$ diagonalizes the operator$A$
in the sense that $UAU^{-1}$ is the operator of multiplication by the
independent variable $t$. Note now that $U\phi$ is the function that is
identically equal to 1. Consequently, using the fact that the operator
$(A-\lambda)^{-1}$ is well defined for any $\lambda$ that is not an eigenvalue
of $A$, we obtain:

$\frac{1}{t-\lambda}\in
L^{2}({\mathbb{R}},\mu_{\phi})\iff\lambda\notin\sigma_{p}(A)\quad\text{and}\quad\phi\in{\mathcal{D}}\bigl{(}(A-\lambda)^{-1}\bigr{)}.$

Thus, the Simon-Wolff theorem can be reformulated as follows.

######  Theorem 4.2.

Let $A$ be a self- adjoint operator in a Hilbert space ${\mathfrak{H}}$ such
that $\phi$ is a cyclic vector for $A$. Let $P$ be the orthogonal projection
onto the space of scalar multiples of $\phi$. The spectrum of $A+tP$ is pure
point on a Borel set $\Omega$ for almost every $t\in{\mathbb{R}}$, if and only
if for almost every $\lambda\in\Omega\setminus\sigma_{p}(A)$,

(4.2) $\phi\in{\mathcal{D}}\bigl{(}(A-\lambda)^{-1}\bigr{)}.$

One is tempted to say that this theorem could be applied to Schrödinger
operators on the lattice ${\mathbb{Z}}^{d}$, because one can decompose
$H=-\Delta+V$ into the orthogonal sum of operators whose spectra have
multiplicities equal to 1. However, the arguments that allow us to use Theorem
4.2 are more complicated.

######  Proposition 4.1.

Let ${\mathfrak{H}}_{0}$ be an invariant subspace for the operator
$H=-\Delta+V$ that is orthogonal to vectors $\chi_{j}$ for all $j\in$ supp$V$.
Then the spectrum of the restriction of $H$ to the subspace
${\mathfrak{H}}_{0}$ is contained in $[0,4d]$.

Proof. The statement is obvious and its proof is left as an exercise.
$\,\,\,\Box$

######  Proposition 4.2.

Let $j\in{\mathbb{Z}}^{d}$ be a fixed point of the lattice. Assume that

$\chi_{j}\in{\mathcal{D}}\bigl{(}(H-\lambda)^{-1}\bigr{)}$

for almost every $\lambda\in(a,b)\setminus\sigma_{p}(H)$. Let
${\mathfrak{H}}_{0}(t)$ be an invariant subspace for the operator
$H+t\chi_{j}(\cdot,\chi_{j})$ on which the restriction of this operator has
coninuous spectrum that is contained in $[a,b]$. Then $\chi_{j}$ is orthogonal
to ${\mathfrak{H}}_{0}(t)$ for almost every $t\in{\mathbb{R}}$.

Proof. Let ${\mathfrak{H}}_{1}$ be the span of vectors
$\\{\chi_{j},H\chi_{j},H^{2}\chi_{j},\dots\\}$. The subspace
${\mathfrak{H}}_{1}$ is invariant for the operator
$H+t\chi_{j}(\cdot,\chi_{j})$ whose restriction to ${\mathfrak{H}}_{1}$ is an
operator having pure point spectrum in $[a,b]$ for almost every $t$.
Consequently, ${\mathfrak{H}}_{0}(t)$ is orthogonal to ${\mathfrak{H}}_{1}$.
$\,\,\,\Box$

######  Corollary 4.1.

Let $\\{\xi_{n}\\}_{n\in{\mathbb{Z}}^{d}}$ be independent random variables
uniformly distributed on some interval $[-\alpha,0]$ where $\alpha>0$. Let
$\Omega\subset{\mathbb{Z}}^{d}$ be a fixed subset of the lattice, such that

$V_{\xi}(n)=\begin{cases}\xi_{n},\qquad\text{if}\qquad n\in\Omega,\\\
0,\qquad\text{if}\qquad n\notin\Omega.\end{cases}$

Let $H_{\xi}=-\Delta+V_{\xi}$. Let also $a<b<0$. Assume that, for all $\xi$
and all $j\in\Omega$,

$\chi_{j}\in{\mathcal{D}}\bigl{(}(H_{\xi}-\lambda)^{-1}\bigr{)}$

for almost every $\lambda\in(a,b)\setminus\sigma_{p}(H_{\xi})$. Then the
operator $H_{\xi}$ almost surely has pure point spectrum in $(a,b)$.

Proof. Let ${\mathfrak{H}}_{c}(\xi)$ be the invariant subspace on which the
restriction of $H_{\xi}$ has continuous spectrum contained in $[a,b]$. By
Proposition 4.2, $\chi_{j}$ is orthogonal to ${\mathfrak{H}}_{c}(\xi)$ for
almost every $\xi_{j}\in[-\alpha,0]$ provided the values of all other random
variables are fixed. By Fubini’s theorem, $\chi_{j}$ is orthogonal to
${\mathfrak{H}}_{c}(\xi)$ almost surely. By Proposition 4.1, the spectrum of
the restriction of $H_{\xi}$ to ${\mathfrak{H}}_{c}(\xi)$ does not intersect
the interval $[a,b]$, because $b<0$. The obtained contradiction implies that a
nontrivial subspace ${\mathfrak{H}}_{c}(\xi)$ does not exist. $\,\,\,\,\,\Box$

Let $H=-\Delta+V$ be a Schrödinger operator on the lattice ${\mathbb{Z}}^{d}$.
Our goal is to find conditions guaranteeing that the sequence

$\psi(n)=\bigl{[}(H-\lambda)^{-1}\chi_{j}\bigr{]}(n)$

is square summable for any fixed $j\in{\mathbb{Z}}^{d}$. This litteraly means
that $\chi_{j}\in{\mathcal{D}}\bigl{(}(H-\lambda)^{-1}\bigr{)}$. We remind the
reader that, for $\lambda\notin[0,4d]$, the sequence $\psi$ is a solution of
the equation

(4.3) $\psi=\psi_{0}-(H_{0}-\lambda)^{-1}V\psi,$

where $\psi_{0}$ is the sequence

$\psi_{0}(n)=\bigl{[}(H_{0}-\lambda)^{-1}\chi_{j}\bigr{]}(n)$

One the other hand, once we know the values $\psi(n)$ for $n\in$ supp$(V)$, we
immediately obtain $\psi$ for all $n$, from (4.3).

(4.4)
$\psi(n)=\psi_{0}(n)-\Bigl{(}(H_{0}-\lambda)^{-1}\chi_{n},\chi_{n}\Bigr{)}V(n)\psi(n)-\sum_{l\neq
n}\Bigl{(}(H_{0}-\lambda)^{-1}\chi_{l},\chi_{n}\Bigr{)}V(l)\psi(l),$

First, observe that, for any $\varepsilon>0$, there are constants
$\gamma_{\varepsilon}>0$ and $C_{\varepsilon}>0$ depending on $\varepsilon$
such that

(4.5)
$\Bigl{|}\bigl{(}(H_{0}-\lambda)^{-1}\chi_{l},\chi_{n}\bigr{)}\Bigr{|}\leq
C_{\varepsilon}e^{-\gamma_{\varepsilon}|n-l|},\qquad\forall
n,l\in{\mathbb{Z}}^{d},\qquad\forall\lambda\in{\mathbb{R}}\setminus[-\varepsilon,4d+\varepsilon].$

######  Proposition 4.3.

Let $V$ be a bounded potential and let $\varepsilon>0.$ Then for almost every
$\lambda\notin[-\varepsilon,4d+\varepsilon]$, there is a number $k(\lambda)$
such that

(4.6)
$\Bigl{|}1+\bigl{(}(H_{0}-\lambda)^{-1}\chi_{n},\chi_{n}\bigr{)}V(n)\Bigr{|}\geq\frac{1}{|n|^{-d-\varepsilon}}$

for $|n|>k(\lambda)$.

Proof. Observe first, that if
$\bigl{|}\bigl{(}(H_{0}-\lambda)^{-1}\chi_{n},\chi_{n}\bigr{)}V(n)\bigr{|}<1/2$
and $|n|>2$, then (4.6) holds. Threfore, we only need to consider the case
$\bigl{|}\bigl{(}(H_{0}-\lambda)^{-1}\chi_{n},\chi_{n}\bigr{)}V(n)\bigr{|}\geq
1/2$. The latter implies that we only need to consider the points
$\lambda\in[-2\|V\|_{\infty},-\varepsilon]\cup[4d+\varepsilon,4d+2\|V\|_{\infty}]$
and points $n\in{\mathbb{Z}}^{d}$ for which $|V(n)|\geq\varepsilon/2$. On that
set of $\lambda$-s, the absolute value of the derivative of the monotone
function

$f(\lambda)=1+\bigl{(}(H_{0}-\lambda)^{-1}\chi_{n},\chi_{n}\bigr{)}V(n)$

is bounded below by $\varepsilon/\|V\|^{2}_{\infty}$. Therefore, the Lebesgue
measure of the set

$S_{\varepsilon}(n):=\Bigl{\\{}\lambda\in{\mathbb{R}}\setminus[-\varepsilon,4d+\varepsilon]:\qquad\Bigl{|}1+\bigl{(}(H_{0}-\lambda)^{-1}\chi_{n},\chi_{n}\bigr{)}V(n)\Bigr{|}<\frac{1}{|n|^{d+\varepsilon}}\Bigr{\\}}$

is bounded by

$|S_{\varepsilon}(n)|\leq\varepsilon^{-1}\|V\|^{2}_{\infty}|n|^{-d-\varepsilon}.$

Consequently,

$\sum_{n\in{\mathbb{Z}}^{d}}|S_{\varepsilon}(n)|<\infty.$

Using the Borel-Cantelli lemma, we conclude that, for almost every
$\lambda\in[-\|V\|_{\infty},-\varepsilon]\cup[4d+\varepsilon,4d+\|V\|_{\infty}]$,
there is a number $k(\lambda)$ such that

$\Bigl{|}1+\bigl{(}(H_{0}-\lambda)^{-1}\chi_{n},\chi_{n}\bigr{)}V(n)\Bigr{|}\geq\frac{1}{|n|^{-d-\varepsilon}}$

for $|n|>k(\lambda)$. $\,\,\,\,\Box$

The equation (4.4) can be written in the form

(4.7) $\psi(n)=\alpha(n)\psi_{0}(n)-\alpha(n)\sum_{l\neq
n}\Bigl{(}(H_{0}-\lambda)^{-1}\chi_{l},\chi_{n}\Bigr{)}V(l)\psi(l),$

where the sequence

$\alpha(n)=\bigl{(}1+\bigl{(}(H_{0}-\lambda)^{-1}\chi_{n},\chi_{n}\bigr{)}V(n)\bigr{)}^{-1}$

obeys the bound

(4.8) $|\alpha(n)|\leq|n|^{d+\varepsilon}\quad\text{ for
}\quad|n|>k(\lambda).$

Consequently, (4.7) can be written in the form

(4.9) $\psi=\tilde{\psi}_{0}-T\psi,$

where $\tilde{\psi}(n)=\alpha(n)\psi(n)$ and $T$ is the operator defined by

(4.10) $[Tu](n)=\alpha(n)\sum_{l\neq
n}\Bigl{(}(H_{0}-\lambda)^{-1}\chi_{l},\chi_{n}\Bigr{)}V(l)u(l),$

Combining (4.5) with (4.8), we conclude that $T$ is a compact operator on
$\ell^{2}({\rm supp}(V))$ provided the quantity

(4.11)
$d(n)=\text{dist}\,\bigl{(}n,\,\,\text{supp}(V)\setminus\\{n\\}\bigr{)}$

obeys the sparseness condition

(4.12)
$\lim_{|n|\to\infty,\,n\in\text{supp}(V)}\frac{d(n)}{|n|^{\delta}}=\infty,\qquad\text{for
some}\,\,\delta>0.$

######  Proposition 4.4.

Let $V$ be a bounded potential such that (4.12) holds. Let $\chi_{{\rm
supp}(V)}$ be the characteristic function of the support of $V$. Let $T$ be
defined by (4.10). Then the operator $\chi_{{\rm supp}(V)}T$ is compact.

Proof. For any $R>0$, let $\chi_{B_{R}}$ be the characteristic function of the
set $B_{R}=\\{n\in{\mathbb{Z}}^{d}:\,\,|n|\leq R\\}$. Consider the
decomposition

$T=\chi_{B_{R}}T+(1-\chi_{B_{R}})T.$

Since $\chi_{B_{R}}T$ is a finite rank operator, it is sufficient to prove
that

$\|(1-\chi_{B_{R}})\chi_{{\rm supp}(V)}T\|\to 0,\qquad{\rm as}\quad
R\to\infty.$

The latter property follows from the estimate

(4.13) $\|(1-\chi_{B_{R}})\chi_{{\rm supp}(V)}T\|\leq
C_{\varepsilon}\tilde{C}\sup_{n\in{\rm
supp}(V):\,|n|>R}\bigl{(}|n|^{d+\varepsilon}(\sum_{l:\,|l-n|\geq
d(n)}e^{-\gamma_{\varepsilon}|n-l|})^{1/2}\bigr{)},$

which follows from the Schur estimate for the norm of an operator $K$ whose
matrix consists of the elements $k(n,m)=\rho_{1}(n,m)\rho_{2}(n,m)$. This
estimate says that

$\|K\|\leq\sup_{n}\bigl{(}\sum_{m}|\rho_{1}(n,m)|^{2}\bigr{)}^{1/2}\sup_{m}\bigl{(}\sum_{n}|\rho_{2}(n,m)|^{2}\bigr{)}^{1/2}.$

The constant $\tilde{C}$ in (4.13) is equal to the quantity
$(\sum_{n\in{\mathbb{Z}}^{d}}e^{-\gamma_{\varepsilon}|n|})^{1/2}$.
$\,\,\,\,\Box$

Proposition 4.4 implies that either (4.9) is uniquely solvable, or the
equation

(4.14) $\psi=-T\psi$

has a non-trivial solution. On the other hand, since (4.14) is equivalent to
the equation

$\psi=-(H_{0}-\lambda)^{-1}V\psi,$

which can be written in the form $H\psi=\lambda\psi$, the equation (4.14) has
a non-zero solution if and only if $\lambda$ is an eigenvalue of the operator
$H$. Consequently, (4.9) has a unique square summable solution $\psi$ for
almost every $\lambda\notin[0,4d]$. Thus, we obtain the following result.

######  Theorem 4.3.

Let $V$ be bounded. Assume that the quantity (4.11) obeys

$\lim_{|n|\to\infty,\,n\in\text{supp}(V)}\frac{d(n)}{|n|^{\delta}}=\infty$

for some $\delta>0$. Then the sequence

$\psi(n)=\bigl{[}(H-\lambda)^{-1}\chi_{j}\bigr{]}(n)$

is square summable for almost every $\lambda\notin[0,4d]$.

As a consequence, we obtain the following result.

######  Theorem 4.4.

Let the conditions of Theorem 2.2 be fulfilled. Then the operator $H_{\xi}$
almost surely has pure point spectrum on ${\mathbb{R}}\setminus[0,4d]$.

Let us now explain why the spectrum of the operator $H_{\xi}$ in Theorem 2.2
fills the interval $[\lambda_{0},0]$.

######  Theorem 4.5.

Let $V$ be a bounded potential for which the quantity (4.11) obeys the
condition

$d(n)\to\infty,\qquad\text{as}\quad|n|\to\infty.$

Assume that there is a sequence of ponts $n_{j}\in{\mathbb{Z}}^{d}$ such that

$V(n_{j})\to\beta<0,\qquad\text{as}\quad j\to\infty.$

Let $\lambda<0$ satisfy the equation

$1+\beta\bigl{(}(H_{0}-\lambda)^{-1}\chi_{0},\chi_{0}\bigr{)}=0.$

Then $\lambda\in\sigma(-\Delta+V)$.

Proof. To prove Theorem 4.5, we use the three statements below.

######  Proposition 4.5.

Let $H=-\Delta+V$ and $H_{n}=-\Delta+V_{n}$ be Schrodinger operators on the
lattice ${\mathbb{Z}}^{d}$ with bounded potentials $V$ and $V_{n}$
correspondingly. Assume that the sequence $V_{n}$ converges to $V$ pointwise
so that

$\sup_{n\in{\mathbb{N}}}\|V_{n}\|_{\infty}<\infty.$

Then the sequence of measures

$\mu_{n}(\cdot)=\bigl{(}(E_{H_{n}}(\cdot)\chi_{0},\chi_{0})\bigr{)}$

converges weakly to the measure

$\mu(\cdot)=\bigl{(}(E_{H}(\cdot)\chi_{0},\chi_{0})\bigr{)},\qquad\text{as}\quad|n|\to\infty.$

Proof. Observe that

$\int_{\mathbb{R}}\frac{d\mu(t)}{t-z}=\bigl{(}(H-z)^{-1}\chi_{0},\chi_{0}\bigr{)},\qquad
z\in{\mathbb{C}}_{+}=\\{z\in{\mathbb{C}}:\quad{\rm Im}\,z>0\\},$

and a similar representation can be written for the measure $\mu_{n}$. Since
the span of functions of the form

$f(t)={\rm Im}\,\frac{1}{t-z},\qquad z\in{\mathbb{C}}_{+},$

is dense in the space of continuous decaying at infinity functions
$C_{0}({\mathbb{R}})$, it is sufficient to show that

$\bigl{(}(H_{n}-z)^{-1}\chi_{0},\chi_{0}\Bigr{)}\to\bigl{(}(H-z)^{-1}\chi_{0},\chi_{0}\bigr{)},\qquad\text{as}\quad
n\to\infty,$

uniformly on compact subsets of ${\mathbb{C}}_{+}$. The latter follows from
the Hilbert identity:

$\bigl{(}(H_{n}-z)^{-1}\chi_{0},\chi_{0}\bigr{)}-\bigl{(}(H-z)^{-1}\chi_{0},\chi_{0}\bigr{)}=\bigl{(}(H_{n}-z)^{-1}(V-V_{n})(H-z)^{-1}\chi_{0},\chi_{0}\bigr{)}.$

$\,\,\,\,\,\Box$

######  Corollary 4.2.

Let $H=-\Delta+V$ be the Schrödinger operator with a bounded potential $V$ for
which the quantity (4.11) obeys the condition

$d(n)\to\infty,\qquad\text{as}\quad|n|\to\infty.$

Let $n_{j}\in{\mathbb{Z}}^{d}$ be a sequence of distinct points such that

$V(n_{j})\to\beta<0,\qquad\text{as}\quad j\to\infty.$

Then the sequence of measures

$\mu_{j}(\cdot)=\bigl{(}E_{H}(\cdot)\chi_{n_{j}},\chi_{n_{j}}\bigr{)}$

converges weakly to the measure

$\mu(\cdot)=\bigl{(}E_{-\Delta+\beta P_{0}}(\cdot)\chi_{0},\chi_{0}\bigr{)},$

where $P_{0}=(\cdot,\chi_{0})\chi_{0}$ is the operator of multiplication by
the function $\chi_{0}$.

Proof. It is sufficient to note that

$\bigl{(}E_{H}(\cdot)\chi_{n},\chi_{n}\bigr{)}=\bigl{(}E_{H_{n}}(\cdot)\chi_{0},\chi_{0}\bigr{)}$

where $H_{n}=-\Delta+V_{n}$ with $V_{n}(x)=V(x-n)$. $\,\,\,\,\Box$

######  Corollary 4.3.

Let the conditions of Corollary 4.2 be fulfilled. Let $\lambda<0$ satisfy the
equation

$1+\beta\bigl{(}(H_{0}-\lambda)^{-1}\chi_{0},\chi_{0}\bigr{)}=0.$

Then $\lambda\in\sigma(H)$.

Proof. Indeed, since the spectrum of $H$ contains the support of the measure
$\mu_{n}$ for all $n\in{\mathbb{Z}}^{d}$, we conclude that

$\sigma(H)\supset{\rm supp}\bigl{(}E_{-\Delta+\beta
P_{0}}(\cdot)\chi_{0},\chi_{0}\bigr{)}\ni{\lambda}.$

$\,\,\,\,\,\Box$

Now Theorem 4.5 follows from Corollary 4.3. $\,\,\,\,\Box$

Finally, we use the following obvious assertion.

######  Proposition 4.6.

Let $\Omega$ be an unbounded subset of the lattice ${\mathbb{Z}}^{d}$. Let
$\\{\xi_{n}\\}_{n\in\Omega}$ be independent random variables uniformly
distributed on $[-a,0]$ with some $a>0$. Then, for any $\beta\in[-a,0]$ and
almost surely, there is a sequence of distinct points $n_{j}\in\Omega$ such
that

$\xi_{n_{j}}\to\beta,\qquad\text{as}\quad j\to\infty.$

Proof. Let $B_{R}=\\{n\in{\mathbb{Z}}^{d}:\,\,|n|\leq R\\}$, where $R>0$. It
is enough to observe that for every $\varepsilon>0$ and every $R>0$, the
probability of the event

$\\{\xi_{n}\notin(\beta-\varepsilon,\beta+\varepsilon),\qquad\forall
n\in\Omega\setminus B_{R}\\}$

is zero. Consequently, there is at least one $n_{1}\in\Omega\setminus B_{R}$
for which $\xi_{n_{1}}\in(\beta-\varepsilon,\beta+\varepsilon)$. Continuing
inductively, we construct a sequence $n_{j}$ such that
$\xi_{n_{j}}\in(\beta-2^{-j+1}\varepsilon,\beta+2^{-j+1}\varepsilon)$.
$\,\,\,\,\Box$

Theorem 4.4, Theorem 4.5 and Proposition 4.6 imply Theorem 2.2.

Acknowledgments: The work of S. Molchanov was supported by the Russian Science
Foundation, project ${\rm N}^{o}$ 20-11-20119. The work of B. Vainberg was
supported by the Simons Foundation grant 527180.

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