#  Conditions for discreteness of the
spectrum to Schrödinger operator via
non-increasing rearrangement, Lagrangian
relaxation and perturbations

Leonid Zelenko  Department of Mathematics
University of Haifa
Haifa 31905
Israel<EMAIL_ADDRESS>
###### Abstract

This work is a continuation of our previos paper [22], where for the
Schrödinger operator $H=-\Delta+V({\mathbf{x}})\cdot$ $(V({\mathbf{x}})\geq
0)$, acting in the space $L_{2}({\mathbf{R}}^{d})\,(d\geq 3)$, some sufficient
conditions for discreteness of its spectrum have been obtained on the base of
well known Mazya -Shubin criterion and an optimization problem for a set
function, which is an infinite-dimensional generalization of a binary linear
programming problem. A sufficient condition for discreteness of the spectrum
is formulated in terms of the non-increasing rearrangement of the potential
$V({\mathbf{x}})$. Using the method of Lagrangian relaxation for this
optimization problem, we obtain a sufficient condition for discreteness of the
spectrum in terms of expectation and deviation of the potential. By means of
suitable perturbations of the potential we obtain conditions for discreteness
of the spectrum, covering potentials which tend to infinity only on subsets of
cubes, whose Lebesgue measures tend to zero when the cubes go to infinity.
Also the case where the operator $H$ is defined in the space $L_{2}(\Omega)$
is considered ($\Omega$ is an open domain in ${\mathbf{R}}^{d}$).

###### Key words and phrases:

Schrödinger operator, discreteness of the spectrum,
rearrangement of a function, optimization problem, Lagrangian
relaxation, perturbations.

###### 1991 Mathematics Subject Classification:

Primary 47F05, 47B25,
47D08, 35P05; Secondary 81Q10, 90C10, 90C27, 91A12

###### Contents

  1. 1 Introduction
  2. 2 Basic notations
  3. 3 Preliminaries
  4. 4 Some results from the previous work
  5. 5 Main results
    1. 5.1 Non-increasing rearrangement
    2. 5.2 Lagrangian relaxation
    3. 5.3 Perturbations of the potential
    4. 5.4 The case where the operator $H$ is defined in $L_{2}(\Omega)\;(\Omega\subset{\mathbf{R}}^{d})$
  6. 6 Proof of main results
    1. 6.1 Proof of Theorem 5.1
    2. 6.2 Proof of Theorem 5.3
    3. 6.3 Proof of Theorem 5.6
    4. 6.4 Proof of Theorem 5.7
    5. 6.5 Proof of Theorem 5.8
    6. 6.6 Proof of Theorem 5.11
    7. 6.7 Proof of Theorem 5.12
    8. 6.8 Proof of Theorem 5.13
    9. 6.9 Proof of Theorem 5.18
    10. 6.10 Proof of Corollary 5.19
    11. 6.11 Proof of Theorem 5.21
  7. 7 Some examples
  8. A Some classes of solutions of the divergence equation
    1. A.1 Reduction to ODE
    2. A.2 Periodic potential solutions
  9. B Some estimates

##  1\. Introduction

This work is a continuation of our previous paper [22].

We consider the Schrödinger operator $H=-\Delta+V({\mathbf{x}})\cdot$, acting
in the space $L_{2}({\mathbf{R}}^{d})$. In what follows we assume that $d\geq
3$, $V({\mathbf{x}})\geq 0$ and $V(\cdot)\in L_{1,loc}({\mathbf{R}}^{d})$.
Physically $V({\mathbf{x}})$ is the potential of an external electric field.
In [22] some sufficient conditions for discreteness of the spectrum of $H$
have been obtained on the base of well known Mazya -Shubin criterion [15] and
an optimization problem for a set function. Since also in the present paper we
shall use the Mazya -Shubin result, let us formulate it. Following to [15],
consider in ${\mathbf{R}}^{d}$ an open domain $\mathcal{G}$ satisfying the
conditions:

(a) $\mathcal{G}$ is bounded and star-shaped with respect to any point of an
open ball $B_{\rho}(0)\,(\rho>0)$ contained in $\mathcal{G}$;

(b) $\mathrm{diam}(\mathcal{G})=2$.

As it was noticed in [15], condition (a) implies that $\mathcal{G}$ can be
represented in the form

$\mathcal{G}=\\{{\mathbf{x}}\in{\mathbf{R}}^{d}:\,{\mathbf{x}}=r\omega,\,|\omega|=1,\,0\leq
r<r(\omega)\\},$ (1.1)

where $r(\omega)$ is a positive Lipschitz function on the standard unit sphere
$S^{d-1}\subset{\mathbf{R}}^{d}$. For $r>0$ and
${\mathbf{y}}\in{\mathbf{R}}^{d}$ denote

$\mathcal{G}_{r}({\mathbf{y}}):=\\{{\mathbf{x}}\in{\mathbf{R}}^{d}:\,r^{-1}{\mathbf{x}}\in\mathcal{G}\\}+\\{{\mathbf{y}}\\}.$
(1.2)

Denote by $\mathcal{N}_{\gamma}({\mathbf{y}},r)\;(\gamma\in(0,1))$ the
collection of all compact sets $F\subseteq\bar{\mathcal{G}}_{r}({\mathbf{y}})$
satisfying the condition

$\mathrm{cap}(F)\leq\gamma\,\mathrm{cap}(\bar{\mathcal{G}}_{r}({\mathbf{y}})),$
(1.3)

where $\mathrm{cap}(F)$ is the harmonic capacity, defined by (3).

######  Theorem 1.1.

[[15], Theorem 2,2] The spectrum of the operator $H$ is discrete, if for some
$r_{0}>0$ and for any $r\in(0,r_{0})$ the condition

$\lim_{{\mathbf{y}}\rightarrow\infty}\inf_{F\in\mathcal{N}_{\gamma(r)}({\mathbf{y}},r)}\int_{{\mathcal{G}}_{r}({\mathbf{y}})\setminus
F}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=\infty,$ (1.4)

is satisfied, where

$\forall\,r\in(0,\,r_{0}):\;\gamma(r)\in(0,1)\quad\mathrm{and}\quad\limsup_{r\downarrow
0}r^{-2}\gamma(r)=\infty,$ (1.5)

Let us notice that in [15] the above result was formulated for the more
general case, where the operator $H$ is defined in $L_{2}(\Omega)$ ($\Omega$
is an open domain in ${\mathbf{R}}^{d}$). Also a necessary condition for
discreteness of the spectrum was obtained there, which is close to sufficient
one. Notice that it was proved in [17] that the condition

$\forall\;r(0,\,r_{0}])\quad\lim_{|{\mathbf{y}}|\rightarrow\infty}\int_{Q_{r}({\mathbf{y}})}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=\infty$
(1.6)

is necessary for discreteness of the spectrum of $H$.

As we have noticed in [22], condition (1.4) of Theorem 1.1 is hardly
verifiable, because in order to test it, one needs to solve a difficult
optimization problem, whose cost functional is the set function
$\mathcal{I}(F)=\int_{{\mathcal{G}}_{r}({\mathbf{y}})\setminus
F}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}$ and the constrain
$F\in\mathcal{N}_{\gamma(r)}({\mathbf{y}},r)$ is submodular (because“cap” is a
submodular set function (definition (3.2))). The goal of the previous and
present papers is to obtain some easier verifiable (or at least better
intuitively explainable) sufficient conditions for discreteness of the
spectrum to $H$. Such kind of conditions was found in the papers [2], [20],
[12] and [5] without use of the Mazya -Shubin result. In [22] we have
estimated the cost functional $\mathcal{I}(F)$ in (1.4) from below using the
isocapacity inequality and replacing
$F\in\mathcal{N}_{\gamma(r)}({\mathbf{y}},r)$ by a weaker but additive
constrain. By this way on the base of Theorem 1.1 we have obtained in [22]
some easier verifiable sufficient conditions for discreteness of the spectrum
in terms of measures, which permit a reformulation in terms of non-increasing
rearrangement of the potential $V({\mathbf{x}})$. As we have shown, these
conditions are more general than ones obtained in the papers mentioned above.

In the present paper we have obtained some sufficient conditions for
discreteness of the spectrum of $H$, using along with the arguments mentioned
above also the method of Lagrangian relaxation for the optimization problem,
mentioned above. Notice that this problem is an infinite-dimensional
generalization a of binary problem from integer linear programming (Problem
4.1 and Remark 4.1 ). The method of Lagrangian relaxations for solving such
kind of problems was applied, for instance, in the paper [23]. We have used
some ideas from this paper. We have obtained also some result on perturbations
of the potential $V({\mathbf{x}})$ preserving discreteness of the spectrum of
$H$, which permit to obtain conditions for discreteness of the spectrum,
covering potentials that do not belong to the scope of other claims of the
paper. We consider also the case where operator $H$ is defined in the space
$L_{2}(\Omega)$, where $\Omega$ is an open domain in ${\mathbf{R}}^{d}$.

Let us describe briefly the main results of this paper.

Theorems 5.1, 5.3 and 5.6 were formulated and proved in the previous paper
[22], but in the proof of Theorem 5.1 a technical mistake was admitted (Remark
5.2). Therefore we give the proofs of these theorems in the present paper.

Theorem 5.1 is obtained by a direct use of the isocapacity inequality. It
yields a sufficient condition of discreteness of the spectrum for $H$ in terms
of an optimization problem involving Lebesgue measure instead of the capacity.
Let us notice that for the case $\gamma(r)\equiv const$ Theorem 5.1 was proved
in [11] (Theorem 6.1), where the operator $H$ on a Riemannian manifold with
bounded geometry was considered.

Theorem 5.3 yields a sufficient condition for discreteness of the spectrum of
$H$ in terms of the non-increasing rearrangement of $V({\mathbf{x}})$ with
respect to Lebesgue measure on cubes that are going to infinity. The sense of
this theorem is following: the spectrum of $H$ is discrete, if the normalized
Lebesgue measure of subsets of cubes, where $V({\mathbf{x}})$ tends to
infinity as the cubes go to infinity does not tend to zero too fast when the
sizes of cubes tend to zero.

Theorem 5.6 , based on Theorem 5.3, yields a condition for discreteness of the
spectrum for $H$ in terms of the non-increasing rearrangement of the potential
$V({\mathbf{x}})$ with respect to Lebesgue measure only on cubes from $m$-adic
partitions of unit cubes $Q_{1}(\vec{l})\,(\vec{l}\in{\mathbf{Z}}^{d})$. This
circumstance enables us to construct nontrivial examples of the Schrödinger
operator with discrete spectrum (Example 3). In the formulation of this
theorem we use our concept of $(\log_{m},\,\theta)$\- dense system of subsets
of a unit cube (Definition 3.2). Furthermore, in this formulation the lower
bound of the normalized Lebesgue measure of subsets of cubes from the $m$-adic
partition, where $V({\mathbf{x}})$ tends to infinity as the cubes go to
infinity, depends on sizes of these cubes. The property of
$(\log_{m},\,\theta)$-density ensures that on cubes with any placement and
arbitrarily small sizes a similar dependence on the size is preserved for the
lower bound of the normalized measure of the sets, mentioned above. This
circumstance enables us to use Theorem 5.3 in the proof of Theorem 5.6.

Theorem 5.7 yields a lower bound for optimal value of the functional in
optimization Problem 4.1 in terms of expectation and deviation of the function
$W(x)$ with respect to a probability measure. In the proof of this theorem the
method of Lagrangian relaxation is used.

Theorem 5.8 is based on Theorem 5.7. The sense of the sufficient condition for
discreteness of the spectrum given by this theorem is following: the spectrum
of $H$ is discrete, if the rate of growth to infinity of expectation of the
potential $V({\mathbf{x}})$ on a cube with respect to the normalized Lebesgue
measure is not less than the rate of growth of its deviation while the cube
goes to infinity. Let us notice that Theorem 5.8 covers more narrow class of
potentials than Theorem 5.3, since its condition limits the rate of growth of
the potential in the case where
$\lim_{{\mathbf{x}}\rightarrow\infty}V({\mathbf{x}})=+\infty$ (Example 6 and
Remarks 5.4, 5.9). Nevertheless, Example 4 shows that it covers potentials
that do not belong to the scope of criterion from [5], which is most resent
among conditions for discreteness of the spectrum of $H$ obtained by other
authors. Furthermore, in some cases the condition of Theorem 5.8 is easier
verifiable than the condition of Theorem 5.3 (Example 5).

Theorem 5.11 describes some perturbations $W({\mathbf{x}})$ of the potential
$V({\mathbf{x}})$ preserving discreteness of the spectrum of $H$. Its proof
uses upper estimate (B.2) for absolute value of the quadratic form
$\int_{\Omega}W({\mathbf{x}})|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}$,
obtained in Section B (claim (ii) of Proposition B.1). In the right hand side
of this estimate the $L_{d}(\Omega)$-norm of a vector field
$\vec{\Gamma}({\mathbf{x}})$ takes part. This field is a solution of the
divergence equation $\mathrm{div}\vec{\Gamma}=W$ in the domain $\Omega$, This
circumstance permits to make small the norm $\|\vec{\Gamma}\|_{d}$ via fast
sign-alternating oscillations of $W({\mathbf{x}})$ (Example 7). Let us notice
that a similar cancellation phenomenon was discovered and used in [16].

Theorem 5.12, based on Theorem 5.11, yields a sufficient condition for
discreteness of of the spectrum of $H$ in terms of the Fourier transform on
the torus ${\mathbf{T}}^{d}$ for the $1$-periodic continuation (by all the
variables) of the function $V|_{Q_{1}(\vec{l})}\,(\vec{l}\in{\mathbf{Z}}^{d})$
from the cube $Q_{1}(\vec{l})$ to the whole ${\mathbf{R}}^{d}$. In the proof
of this theorem we use Proposition A.2, where the existence of a periodic
potential solution of the divergence equation $\mathrm{div}\vec{\Gamma}=W$ is
established and its $L_{p}$-norm $(p>2)$ is estimated.

Theorem 5.13 is motivated by a counterexample, constructed by A.M. Molchanov
in [17], which shows that condition (1.6) is not sufficient for discreteness
of the spectrum of $H$ for $d>1$. In this example the support of
$V({\mathbf{x}})$ is a union of disjoint balls, whose centers form lattices on
cubes $Q_{1}(\vec{l})\,(\vec{l}\in{\mathbf{Z}}^{d})$ and densities of the
lattices tend to infinity as $|\vec{l}|\rightarrow\infty$ (Remark 5.16). The
following question appears: how to choose densities of lattices, radii of
balls and values of the potential on balls such that the spectrum of the
operator $H$ would be discrete ? The answer of Theorem 5.13 is following: this
occurs when condition (1.6) is fulfilled with $Q_{1}(\vec{l})$ instead of
$\mathcal{G}_{r}({\mathbf{y}})$ (Remark 5.14) and density of one-dimensional
sections of the lattices tends to infinity faster than values of the potential
$V({\mathbf{x}})$ on the balls as the cubes go to infinity. Let us notice that
in the counterexample of Molchanov the last condition is not fulfilled, since
it contradicts the“small capacity property” for the support of
$V({\mathbf{x}})$. The proof of Theorem 5.13 is based on Theorem 5.12. Also
notice that this theorem covers such potentials that Lebesgue measures of
their supports on unit cubes tend to zero as the cubes go to infinity, hence
they do not belong to the scope of all the claims, based on Theorem 5.1
(Remark 5.15).

Theorems 5.18 and 5.21 yield sufficient conditions for discreteness of the
spectrum of $H$ in the case where $H$ is defined in $L_{2}(\Omega)$ ($\Omega$
is an open domain in ${\mathbf{R}}^{d}$). They are generalizations of Theorems
5.1 and 5.8 to this case. Let us notice that if in Theorem 5.21 the Lebesgue
measure of ${\mathcal{G}}_{r}({\mathbf{y}})\cap\Omega$ approaches from above
some threshold for large $|{\mathbf{y}}|$, the condition of this theorem
becomes weaker (Remark 5.22). On the other hand, if it is less than this
threshold (i.e. the domain $\Omega$ becomes closer to bounded), then even the
potential $V({\mathbf{x}})\equiv 0$ satisfies this condition (Corollary 5.19).

Let us notice that in [21] Michael Taylor have found an alternative for the
Mazya-Shubin result. His necessary and sufficient conditions for discreteness
of the spectrum of $H$ are formulated in terms of the scattering length  of
the potential $V({\mathbf{x}})$ on boxes that are going to infinity. It would
be interesting to extract from this result an easier verifiable sufficient
condition which would be more general than results obtained in the present
paper.

The paper is organized as follows. After this Introduction, in Section 2, we
give basic notations, in Section 3 (Preliminaries) we introduce some concepts
used in the paper. In Section 4 we formulate some results from the previous
work [22], used in this paper. In Section 5 we formulate the main results of
the paper and in Section 6 we prove them. In Section 7 we recall briefly some
examples, constructed in [22] and construct some counterexamples for claims
obtained in the present paper. Sections A and B are appendices. In Section A
we prove some claims on solutions of the divergence equation and Section B
yields some estimates for
$|\int_{\Omega}W({\mathbf{x}})|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}|$,
mentioned above, by means of Sobolev Embedding Theorems.

##  2\. Basic notations

$\langle{\mathbf{x}},\,{\mathbf{y}}\rangle\;({\mathbf{x}},{\mathbf{y}}\in{\mathbf{R}}^{d})$
is the canonical inner product in the real vector space ${\mathbf{R}}^{d}$;
$|{\mathbf{x}}|=\sqrt{\langle{\mathbf{x}}\cdot{\mathbf{x}}\rangle}$ is the
Euclidean norm in ${\mathbf{R}}^{d}$;

If ${\mathbf{x}}=(x_{1},\,x_{2},\,\dots,\,x_{d})\in{\mathbf{R}}^{d}$, denote
$|{\mathbf{x}}|_{\infty}=\max_{1\leq j\leq d}|x_{j}|$;

$\mathrm{mes}_{d}$ is Lebesgue measure in ${\mathbf{R}}^{d}$;

$\Sigma_{L}(\Omega)$ is the $\sigma$-algebra of all Lebesgue measurable
subsets of of a domain $\Omega\subseteq{\mathbf{R}}^{d}$;

${\mathbf{T}}$ is the circle (one-dimensional torus) with the length $1$;

${\mathbf{T}}^{d}=\times_{k=1}^{d}{\mathbf{T}}$ is the $d$-dimensional
torus;${\mathbf{Z}}$ is the ring of integers; ${\mathbf{N}}$ is the set of
natural numbers;

${\mathbf{Z}}^{d}=\times_{k=1}^{d}{\mathbf{Z}}$;

$[x]$ is the integer part of a real number $x$;

$\mathcal{F}(f)(\vec{k})=\int_{{\mathbf{T}}^{d}}f({\mathbf{x}})\exp\big{(}-i\,2\pi\,\langle\vec{k},\,{\mathbf{x}}\rangle\big{)}\,\mathrm{d}{\mathbf{x}}\quad(\vec{k}\in{\mathbf{Z}}^{d})$
is Fourier transform of a complex-valued function $f({\mathbf{x}})$ defined on
${\mathbf{R}}^{d}$ and $1$-periodic by all the variables
$x_{1},\,x_{2},\,\dots,\,x_{d}$ (hence $f$ can be considered as defined on
${\mathbf{T}}^{d}$ and it is possible to say about Fourier transform of $f$ on
${\mathbf{T}}^{d}$);

$\mathcal{F}^{-1}(g)({\mathbf{x}})=\sum_{\vec{k}\in{\mathbf{Z}}^{d}}g(\vec{k})\exp\big{(}i\,2\pi\,\langle\vec{k},\,{\mathbf{x}}\rangle\big{)}\,(g:{\mathbf{Z}}^{d}\rightarrow{\mathbf{C}},\,{\mathbf{x}}\in{\mathbf{R}}^{d})$
is the transform inverse to $\mathcal{F}$;

$\hat{f}(\vec{\omega}):=\int_{{\mathbf{R}}^{d}}f({\mathbf{x}})\exp\big{(}-i2\pi\,\langle{\mathbf{x}},\,\vec{\omega}\rangle\big{)}\,\mathrm{d}{\mathbf{x}}\,(\vec{\omega}\in{\mathbf{R}}^{d})$
is Fourier transform on ${\mathbf{R}}^{d}$ of a function
$f:\,{\mathbf{R}}^{d}\rightarrow{\mathbf{C}}$;

For Fourier transform and inverse Fourier transform of vector functions we
shall use the same notations;

$\|f\|_{p}\,(p\in[1,\,\infty])$ is the $L_{p}$-norm of a function $f\in
L_{p}(\Omega)\,(\Omega\subseteq{\mathbf{R}}^{d}$ and the same notation we
shall use for the $L_{p}$-norm in $L_{p}({\mathbf{T}}^{d})$;

$l_{p}({\mathbf{Z}}^{d})\,(p\in[1,\,\infty])$ is the space of functions
$g:{\mathbf{Z}}^{d}\rightarrow{\mathbf{C}}$ such that

$\sum_{\vec{k}\in{\mathbf{Z}}^{d}}|g(\vec{k})|^{p}\\\ <\infty;$

$\|g\|_{l_{p}}$ is the $l_{p}$-norm of a function $g\in
l_{p}({\mathbf{Z}}^{d})$;

For the analogous spaces of vector-valued functions and the norms in them we
shall use the same notations;

$C_{0}^{\infty}(\Omega)$ is the collection of all functions from
$C^{\infty}(\Omega)$ having compact supports ($\Omega$ is an open domain in
${\mathbf{R}}^{d}$);

$\mathrm{supp}(f)$ is the support of a function
$f:\,\Omega\rightarrow{\mathbf{C}}$ ;

$W_{p}^{l}(\Omega)\,(p\geq 1)$ is Sobolev space of functions having
generalized partial derivatives up to order $l$, which belong to
$L_{p}(\Omega)$;

$\chi_{A}:\;X\rightarrow{\mathbf{R}}$ is the characteristic function of a
subset $A\subseteq X$;

$B_{r}({\mathbf{y}})$ is the open ball in ${\mathbf{R}}^{d}$ whose radius and
center are $r>0$ and ${\mathbf{y}}$;

The unit cube $Q=[0,\,1]^{d}\subset{\mathbf{R}}^{d}$ and dilation of it
$Q_{r}:=r\cdot Q\,(r>0)$ and translation of the latter
$Q_{r}({\mathbf{y}}):=Q_{r}+\\{{\mathbf{y}}\\}$.

Some specific notations will be introduced in what follows.

##  3\. Preliminaries

Let us come to agreement on some notations and terminology. Let $\Omega$ be an
open domain in $R^{d}$. We denote by $\Sigma_{B}(\bar{\Omega})$ the
$\sigma$-algebra of all Borel subsets of $\bar{\Omega}$. It is known that
$\Sigma_{B}(\bar{\Omega}\subseteq\Sigma_{L}(\bar{\Omega})$. Let us recall the
definition of the harmonic (or Newtonian) capacity111In the Russian literature
it is often called Wiener capacity. of a compact set
$E\subset{\mathbf{R}}^{d}$ ([15]):

$\displaystyle\mathrm{cap}(E):=\inf\Big{(}\big{\\{}\int_{{\mathbf{R}}^{d}}|\nabla
u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}\,:\;u\in
C^{\infty}({\mathbf{R}}^{d}),\;u\geq 1\;\mathrm{on}\;E,$ $\displaystyle
u({\mathbf{x}})\rightarrow
0\;\mathrm{as}\;|{\mathbf{x}}|\rightarrow\infty\big{\\}}\Big{)}.$ (3.1)

It is known ([4], [14], [13]) that the set function “cap” can be extended in a
suitable manner from the collection of all compact subsets of the space
${\mathbf{R}}^{d}$ to the collection of all Borel subsets of it and the set
function “cap” is monotonic and submodular (concave) in the sense that for any
pair of sets $A,\,B\in\Sigma_{B}(\bar{\Omega})$

$\mathrm{cap}(A\cup B)+\mathrm{cap}(A\cap
B)\leq\mathrm{cap}(A)+\mathrm{cap}(B)$ (3.2)

and the isocapacity inequality is valid:

$\forall\,F\in\Sigma_{B}(\bar{\Omega}):\quad\mathrm{mes}_{d}(F)\leq
c_{d}\,(\mathrm{cap}(F))^{d/(d-2)}.$ (3.3)

with
$c_{d}=\big{(}d(d-2)(\mathrm{mes}_{d}(B_{1}(0)))^{2/d}\big{)}^{-d/(d-2)}$,
which comes as identity if $F$ is a closed ball.

In [22] we have used the following concepts:

######  Definition 3.1.

We call a subset of ${\mathbf{R}}^{d}$ a regular parallelepiped, if it has the
form $\times_{k=1}^{d}[a_{k},\,b_{k}]$.

######  Definition 3.2.

Suppose that $m>1$ and $\theta\in(0,1)$. A sequence
$\\{D_{n}\\}_{n=1}^{\infty}$ of subsets of a cube $Q_{1}({\mathbf{y}})$ is
said to be a $(\log_{m},\,\theta)$\- dense system in $Q_{1}({\mathbf{y}})$, if

(a) each $D_{n}$ is a finite union of regular parallelepipeds;

(b) for any cube $Q_{r}({\mathbf{z}})\subseteq Q_{1}({\mathbf{y}})$ with
$r\in\big{(}0,\,\min\\{1,\,\frac{1}{\theta m^{2}}\\}\big{)}$ there is
$j\in\\{1,2,\dots,\big{[}\log_{m}\big{(}\frac{1}{\theta r}\big{)}\big{]}\\}$
such that for some regular parallelepiped $\Pi\subseteq D_{j}$ there is a cube
$Q_{\theta r}({\mathbf{s}})$, contained in $\Pi\cap Q_{r}({\mathbf{z}})$.

##  4\. Some results from the previous work

All the results of our previous paper [22] and partly of the present one are
based on the following optimization problem for set functions:

######  Problem 4.1.

Let $(X,\,\Sigma,\,\mu)$ be a measure space with a non-negative measure $\mu$
and $W(x)$ be a function defined on $X$ and belonging to $L_{1}(X,\,\mu)$. For
$t\in(0,\,\mu(X))$ consider the collection $\mathcal{E}(t,X,\,\mu)$ of all
$\mu$-measurable sets $E\subseteq X$ such that $\mu(E)\geq t$. The goal is to
find the quantity

$I_{W}(t,\,X,\,\mu)=\inf_{E\in\mathcal{E}(t,X,\,\mu)}\int_{E}W(x)\,\mu(\mathrm{d}x).$
(4.1)

######  Remark 4.1.

Problem 4.1 can be formulated in the following equivalent form: to find the
quantity

$I_{W}(t,\,X,\,\mu)=\inf_{E\in\Sigma}\int_{X}W(x)\chi_{E}(x)\,\mu(\mathrm{d}x)$

under the constrain $\int_{X}\chi_{E}(x)\,\mu(\mathrm{d}x\geq t$. We see that
this problem is an infinite-dimensional generalization of a binary problem
from integer linear programming [18].

######  Remark 4.2.

In [22] we have assumed in the formulation of Problem 4.1 that
$W({\mathbf{x}})\geq 0$. Here we omit this assumption, since the method of
Lagrangian relaxation, used below for lower estimation of
$I_{W}(t,\,X,\,\mu)$, does not need it.

In the formulation of next claim, proved in [22], we have used the following
notations. For the measure space and the function $W(x)$, introduced in
Problem 4.1, consider the quantity:

$J_{W}(t,\,X,\mu):=\int_{\mathcal{K}_{W}^{-}\big{(}t,\,X,\,\mu\big{)}}W(x)\,\mu(\mathrm{d}x)+\big{(}t-\kappa_{W}^{-}(t,\,X,\,\mu)\big{)}W_{\star}(t,\,X,\,\mu),$
(4.2)

where $W_{\star}(t,\,X,\,\mu)$ is the non-decreasing rearrangement of the
function $W(x)$, i.e,,

$W_{\star}(t,\,X,\,\mu):=\sup\\{s>0:\;\lambda_{\star}(s,\,W,\,X,\,\mu)<t\\}\quad(t>0)$
(4.3)

with

$\lambda_{\star}(s,\,W,\,X,\,\mu)=\mu({\mathcal{L}}_{\star}(s,W,X)),$ (4.4)
${\mathcal{L}}_{\star}(s,W,X)=\\{x\in X:\;W(x)\leq s\\}.$ (4.5)

Furthermore,

$\mathcal{K}_{W}^{-}\big{(}t,\,X,\,\mu\big{)}={\mathcal{L}}_{\star}(s^{-},W,X)|_{s=W_{\star}(t,\,X,\,\mu)},$
(4.6)

and

$\kappa_{W}^{-}(t,\,X,\,\mu)=\mu\big{(}\mathcal{K}_{W}^{-}(t,\,X,\,\mu)\big{)},$
(4.7)

where

${\mathcal{L}}_{\star}(s^{-},W,X)=\bigcup_{u<s}{\mathcal{L}}_{\star}(u,W,X)=\\{x\in
X:\;W(x)<s\\},$ (4.8)

The following claim from [22] solves Problem 4.1 for a non-atomic measure:

######  Proposition 4.3.

[[22], Theorem 3.3] Suppose that, in addition to conditions of Problem 4.1,
$W({\mathbf{x}})\geq 0$ and the measure $\mu$ is non-atomic. Then

(i) for any $t\in(0,\,\mu(X))$ there exists a $\mu$-measurable set
$\tilde{\mathcal{K}}\subseteq X$ such that

$\mu(\tilde{\mathcal{K}})=t,$ (4.9)

for the quantity $J_{W}(t,\,X,\,\mu)$, defined by (4.2)-(4.7), the
representation

$J_{W}\big{(}t,\,X,\,\mu\big{)}=\int_{\tilde{\mathcal{K}}}W(x)\,\mu(\mathrm{d}x)$
(4.10)

is valid and

$\displaystyle\forall\;x\in\tilde{\mathcal{K}}:\quad W(x)\leq
W_{\star}(t,\,X,\,\mu),$ $\displaystyle\forall\;x\in
X\setminus\tilde{\mathcal{K}}:\quad W(x)\geq W_{\star}(t,\,X,\,\mu);$ (4.11)

(ii) the equality

$I_{W}(t,\,X,\,\mu)=J_{W}(t,\,X,\,\mu)$ (4.12)

is valid.

In the next claim. proved in [22], we have obtained a two-sided estimate for
the solution $J_{W}(t,\,X,\,\mu)$ of Problem 4.1 via a non-increasing
rearrangement of the function $W(x)$ on $X$. This rearrangement is following:

$W^{\star}(t;\,X;\,\mu):=\sup\\{s>0\,:\;\lambda^{\star}(s;\,\,W;\,X;\,\mu)\geq
t\\}\quad(t>0),$ (4.13)

where

$\displaystyle\lambda^{\star}(s;\,W;\,X;\,\mu)=\mu(\mathcal{L}^{\star}(s;\,W;\,X)),$
$\displaystyle\mathcal{L}^{\star}(s;\,W;\,X)=\\{x\in X,:\;W(x)\geq s\\}.$
(4.14)

The promised claim is following:

######  Proposition 4.4.

[[22], Proposition 3.4] Suppose that, in addition to conditions of Problem 4.1
and Proposition 4.3, the measure $\mu$ is finite. Then for $\theta>1$ and
$t\in(0,\,\mu(X))$ the estimates

$J_{W}(\mu(X)-t/\theta,\,X,\,\mu))\geq\frac{(\theta-1)t}{\theta}W^{\star}(t;\,X;\,\mu),$
(4.15) $J_{W}(\mu(X)-t,\,X,\,\mu)\leq(\mu(X)-t)W^{\star}(t;\,X;\,\mu)$ (4.16)

are valid.

##  5\. Main results

###  5.1. Non-increasing rearrangement

In this subsection we shall assume that $V({\mathbf{x}})\geq 0$.

Denote by $\mathcal{M}_{\gamma}({\mathbf{y}},r)\;(\gamma\in(0,1))$ the
collection of all Borel sets $F\subseteq\mathcal{G}_{r}({\mathbf{y}})$
satisfying the condition
$\mathrm{mes}_{d}(F)\leq\gamma\,\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))$,
where the domain $\mathcal{G}_{r}({\mathbf{y}})$ is defined by (1.2), (1.1). A
direct use of isocapacity inequality (3.3) leads to the following claim:

######  Theorem 5.1.

If for some $r_{0}>0$ and a function $\gamma(r)$, satisfying the conditions

$\forall\,r\in(0,\,r_{0}):\;\gamma(r)\in(0,1)\quad\mathrm{and}\quad\limsup_{r\downarrow
0}\,r^{-2d/(d-2)}\,\gamma(r)=\infty,$ (5.1)

the condition

$\lim_{|{\mathbf{y}}|\rightarrow\infty}\inf_{F\in\mathcal{M}_{\gamma(r)}({\mathbf{y}},r)}\int_{\mathcal{G}_{r}({\mathbf{y}})\setminus
F}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=\infty$ (5.2)

is satisfied for any $r\in(0,r_{0}]$, then the spectrum of the operator
$H=-\Delta+V({\mathbf{x}})$ is discrete.

######  Remark 5.2.

In [22] in condition (5.1) the technical mistake was admitted: the exponent
$-2(d-2)/d$ of $r$ was there instead of $-2d/(d-2)$.

Let us introduce some notations. We shall omit $\mu$ in the notation (4.13) of
non-increasing rearrangement in the case where $X$ is an open domain
$\Omega\subseteq{\mathbf{R}}^{d}$ and $\mu=\mathrm{mes}_{d}$ , i.e., to write
$W^{\star}(t,\Omega)$. In the case where
$\Omega=\mathcal{G}_{r}({\mathbf{y}})$ we shall write
$W^{\star}(t,{\mathbf{y}},r)$. If $\mathcal{G}_{r}({\mathbf{y}})$ is a cube
$Q_{r}({\mathbf{y}})$, we shall use the same notation, if it could not lead to
a confusion.

On the base of Theorem 5.1 we get the following claim:

######  Theorem 5.3.

If for some $r_{0}>0$ and a function $\gamma(r)$, satisfying the conditions
(5.1), the condition

$\lim_{|{\mathbf{y}}|\rightarrow\infty}V^{\star}(\delta(r),{\mathbf{y}},r)=\infty$
(5.3)

is fulfilled for any $r\in(0,r_{0}]$ with
$\delta(r)=\gamma(r)\mathrm{mes}_{d}(\mathcal{G}_{r}(0)$, then the spectrum of
the operator $H=-\Delta+V({\mathbf{x}})$ is discrete.

######  Remark 5.4.

Notice that condition (5.3) is easier verifiable than condition (5.2). On the
other hand, Proposition 4.3 and estimates (4.15), (4.16) imply that these
conditions are equivalent in the following sense: for some function
$\tilde{\gamma}(r)$ satisfying conditions (5.1) the condition (5.2) is
fulfilled if and only if for some function $\gamma(r)$ satisfying conditions
(5.1) the condition (5.3) is fulfilled.

######  Remark 5.5.

From Theorem 6 of [5] the following criterion of discreteness of the spectrum
for operator $H=-\Delta+V({\mathbf{x}})$ follows (in our notations): if for
some numbers $\delta>0$, $c\in(0,1)$ and $r_{0}>0$ and for any
${\mathbf{y}}\in{\mathbf{R}}^{d}$, $r\in(0,r_{0})$ the condition

$\lambda^{\star}\Big{(}\frac{\delta}{\mathrm{mes}_{d}(Q_{r}(0))}\int_{Q_{r}({\mathbf{y}})}V({\mathbf{x}})\,d{\mathbf{x}},V,\,r,\,{\mathbf{y}}\Big{)}\geq
c\;\mathrm{mes}_{d}(Q_{r}(0))$ (5.4)

is fulfilled, then discreteness of the spectrum of the operator $H$ is
equivalent to the condition:
$\lim_{|{\mathbf{y}}|\rightarrow\infty}\int_{Q_{r}({\mathbf{y}})}V({\mathbf{x}})\,d{\mathbf{x}}=\infty$
for some (hence for every) $r>0$. It is easy to see that as a sufficient
condition this criterion follows from Theorem 5.3 with
$\mathcal{G}_{r}({\mathbf{y}})=Q_{r_{1}}({\mathbf{y}}-r\vec{a})$
($r_{1}=2r/\sqrt{d}$, $\vec{a}=(d^{-1/2},d^{-1/2},\dots,d^{-1/2})$), if one
takes $\gamma(r)\equiv c$.

Consider the covering of the space ${\mathbf{R}}^{d}$ by the cubes
$Q_{1}(\vec{l})\;(\vec{l}\in{\mathbf{Z}}^{d})$ and for any
$\vec{l}\in{\mathbf{Z}}^{d}$ consider a sequence
$\\{D_{j}(\vec{l})\\}_{j=1}^{\infty}$ of subsets of $Q_{1}(\vec{l})$.
Furthermore, for some integers $n>0$ and $m>1$ consider the $m$-adic partition
of each cube $Q_{1}(\vec{l})$:
$\\{Q(\vec{\xi},n)\\}_{\vec{\xi}\in\Xi(\vec{l},\,n)}$, where
$Q(\vec{\xi},n)=Q_{m^{-n}}(\vec{\xi})$ and $\Xi(\vec{l},\,n)=\\{\vec{\xi}\in
m^{-n}\cdot{\mathbf{Z}}^{d}\,:\,Q(\vec{\xi},n)\subset Q_{1}(\vec{l})\\}$.
Denote

$\Xi_{j}(\vec{l},\,n)=\\{\vec{\xi}\in
m^{-n}\cdot{\mathbf{Z}}^{d}\,:\,Q(\vec{\xi},n)\subseteq D_{j}(\vec{l})\\},$
(5.5)

The following claim is based on the previous claim and on the concept of a
$(\log_{m},\,\theta)$\- dense system (Definition 3.2):

######  Theorem 5.6.

[[22], Theorem 3.14] Suppose that $\theta\in(0,1)$ and for each
$\vec{l}\in{\mathbf{Z}}^{d}$ the sequence
$\\{D_{j}(\vec{l})\\}_{j=1}^{\infty}$ forms a $(\log_{m},\,\theta)$\- dense
system in $Q_{1}(\vec{l})$. Furthermore, suppose that

$\forall\;\vec{l}\in{\mathbf{Z}}^{d},\;j\in{\mathbf{N}},\;:\quad\Xi_{j}(\vec{l},j)\neq\emptyset.$
(5.6)

Let $\gamma(r)$ be a non-decreasing monotone function satisfying condition
(5.1). If for any natural $n$ the condition

$\lim_{|\vec{l}|\rightarrow\infty}\;\min_{\vec{\xi}\in\bigcup_{j=1}^{n}\Xi_{j}(\vec{l},\,n)}V^{\star}\big{(}\psi(n);\,Q(\vec{\xi},n)\big{)}=\infty$
(5.7)

is satisfied with $\psi(n)=\gamma(m^{-n})\mathrm{mes}_{d}(Q(\vec{0},n))$, then
the spectrum of the operator $H=-\Delta+V({\mathbf{x}})\cdot$ is discrete.

###  5.2. Lagrangian relaxation

Let us return to Problem 4.1 and assume that $(X,\,\Sigma,\,\mu)$ is a
probability space, i.e. $\mu(X)=1$. We shall estimate from below the value
$I_{W}(t,\,X,\,\mu)$, defined by (4.1), using the method of Lagrangian
relaxation ([23]). Before formulation of the results let us introduce some
notations. Consider the expectation $\mathrm{E}(W)$ of the function $W(x)$:

$\mathrm{E}(W)=\int_{X}W(x)\mu(dx)$ (5.8)

and its (standard) deviation

$\displaystyle\mathrm{Dev}(W)=\sqrt{E\big{(}(W-\mathrm{E}(W))^{2}\big{)}}=\sqrt{\int_{X}\big{(}W(x)-\mathrm{E}(W)\big{)}^{2}\mu(dx)}=$
$\displaystyle\sqrt{\mathrm{E}(W^{2})-\big{(}\mathrm{E}(W)\big{)}^{2}},$ (5.9)

assuming that $W\in L_{2}(X,\mu)$. The following claim is valid:

######  Theorem 5.7.

If $t\in(1/2,1)$ and $\sigma(t)=2t-1$, then

$I_{W}(t,\,X,\,\mu)\geq(1/2)(1+\sigma(t))\mathrm{E}(W)-(1/2)\sqrt{1-\sigma^{2}(t)}\cdot\mathrm{Dev}(W)$
(5.10)

Consider the probability space
$\big{(}\mathcal{G}_{r}({\mathbf{y}}),\Sigma_{L}(\mathcal{G}_{r}({\mathbf{y}})),m_{d,r}\big{)}$,
where $m_{d,r}$ is the normalized Lebesgue measure

$m_{d,r}(A):=\frac{\mathrm{mes}_{d}(A)}{\mathrm{mes}_{d}({\mathcal{G}}_{r}({\mathbf{y}}))}\quad(A\in\Sigma_{L}(\bar{\mathcal{G}}_{r}({\mathbf{y}}))).$
(5.11)

Denote by $\mathrm{E}_{{\mathbf{y}},r}(W)$ and
$\mathrm{Dev}_{{\mathbf{y}},r}(W)$ the expectation and deviation of a real-
valued function $W\in L_{2}(\mathcal{G}_{r}({\mathbf{y}}),\,m_{d,r})$.

The following claim is based on Theorem 5.7:

######  Theorem 5.8.

Suppose that $V({\mathbf{x}})\geq 0$, $V\in L_{2,loc}({\mathbf{R}}^{d})$ and
the condition

$\lim_{|{\mathbf{y}}|\rightarrow\infty}\Big{(}\mathrm{E}_{{\mathbf{y}},r}(V)-\sqrt{\gamma(r)}\cdot\mathrm{Dev}_{{\mathbf{y}},r}(V)\Big{)}=+\infty$
(5.12)

is satisfied for some $r_{0}>0$ and any $r\in(0,r_{0})$, where $\gamma(r)$
satisfies the condition (5.1). Then the spectrum of the operator
$H=-\Delta+V({\mathbf{x}})$ is discrete.

######  Remark 5.9.

We shall see from the proof of Theorem 5.8 that condition (5.12) implies
condition (5.2) of Theorem 5.1. Hence, in view of Remark 5.4, conditions of
Theorem 5.8 imply conditions of Theorem 5.3.

###  5.3. Perturbations of the potential

In this section we shall use a localization principle, which was established
in [8] and [10] for the case of the magnetic Schrödinger operator, but in
particular it is true also in absence of the magnetic field. Let $\Omega$ be
an open domain in ${\mathbf{R}}^{d}$ whose closure is compact. We denote by
$(\cdot,\cdot)_{\Omega}$ and $\|\cdot\|_{\Omega}$ the inner product and the
norm in the space $L_{2}(\Omega)$. Consider the quantities:

$\displaystyle\lambda_{0}(\Omega):=\inf_{u\in C_{0}^{\infty}(\Omega),\,u\neq
0}\frac{(Hu,u)_{\Omega}}{\|u\|_{\Omega}^{2}}=$ $\displaystyle\inf_{u\in
C_{0}^{\infty}(\Omega),\,u\neq 0}\frac{\int_{\Omega}(|\nabla
u({\mathbf{x}})|^{2}+V({\mathbf{x}})|u({\mathbf{x}})|^{2})\,\mathrm{d}{\mathbf{x}}}{\int_{\Omega}|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}},$
$\mu_{0}(\Omega):=\inf_{u\in C^{\infty}(\Omega),\,u\neq
0}\frac{(Hu,u)_{\Omega}}{\|u\|_{\Omega}^{2}}$

In essence $\lambda_{0}(\Omega)$ is the minimal eigenvalue of the generalized
Dirichlet boundary problem in the domain $\Omega$: $Hu=\lambda
u\;u|_{\partial\Omega}=0$ and $\mu_{0}(\Omega)$ is the minimal eigenvalue of
the generalized Neumann problem for the same equation, i.e., the corresponding
boundary condition is $\frac{\partial
u}{\partial\vec{n}}|_{\partial\Omega}=0$. As it is known, the spectra of these
problems are discrete. We denote briefly
$\lambda_{0}({\mathbf{y}},r)=\lambda_{0}(Q_{r}({\mathbf{y}}))$ and
$\mu_{0}({\mathbf{y}},r)=\mu_{0}(Q_{r}({\mathbf{y}}))$

######  Proposition 5.10.

[[10] Theorems 1.1, 1.2]

$(i)$ The operator $H$ is bounded below if and only if

$\exists\;r>0:\quad\liminf_{|{\mathbf{y}}|\rightarrow\infty}\lambda_{0}({\mathbf{y}},r)>-\infty;$
(5.13)

$(ii)$ The spectrum of $H$ is discrete and bounded below if and only if

$\exists\;r>0:\quad\lim_{|{\mathbf{y}}|\rightarrow\infty}\lambda_{0}({\mathbf{y}},r)=+\infty;$

$(iii)$ If $V({\mathbf{x}})\geq 0$, the spectrum of $H$ is discrete if and
only if

$\exists\;r>0:\quad\lim_{|{\mathbf{y}}|\rightarrow\infty}\mu_{0}({\mathbf{y}},r)=+\infty.$

Let $\Omega$ be an open and bounded domain in ${\mathbf{R}}^{d}.$ Following to
[16], for $W\in L_{1}(\Omega)$ and $s\geq 1$ consider the set
${\mathcal{D}}_{s}(W,\,\Omega)$ of all vector fields
$\vec{\Gamma}({\mathbf{x}})$ on $\Omega$ satisfying the divergence equation
$\mathrm{div}\vec{\Gamma}=W$ and belonging to the space $L_{s}(\Omega)$. We
assume that this equation is satisfied in the generalized sense (see (A)). By
Propositions A.1 and (A.2) ${\mathcal{D}}_{s}(W,\,\Omega)\neq\emptyset$ under
some additional conditions. Denote

$D_{s}(W,\Omega)=\inf_{\vec{\Gamma}\in{\mathcal{D}}_{s}(W,\,\Omega)}\|\vec{\Gamma}\|_{s},$
(5.14)

In the case where $\Omega=Q_{r}({\mathbf{y}})$ we shall write briefly
$D_{s}(W,{\mathbf{y}},r)$. Consider the constant:

$C(d)=\sqrt{\frac{1}{\pi
d(d-2)}}\Big{(}\frac{\Gamma(d)}{\Gamma(d/2)}\Big{)}^{1/d},$ (5.15)

The following claim on perturbation of Schrödinger operator, preserving
discreteness of its spectrum, is valid:

######  Theorem 5.11.

Suppose that the potential $V({\mathbf{x}})$ permit the representation
$V({\mathbf{x}})=V_{0}({\mathbf{x}})+W({\mathbf{x}})$ such that $V_{0}\in
L_{1,loc}({\mathbf{R}}^{d})$ and for some $r_{0}>0$ the condition

$\bar{D}(r_{0},W)=\limsup_{|{\mathbf{y}}|\rightarrow\infty}D_{d}(W,{\mathbf{y}},r_{0})<1/(2C(d)),$
(5.16)

is satisfied.

(i) If for some

$\sigma\in[]2C(d)\bar{D}(r_{0},W),\,1)$ (5.17)

the operator $H_{\sigma}=-(1-\sigma)\Delta+V_{0}({\mathbf{x}})\cdot$ is
bounded below, then so does operator $H=-\Delta+V({\mathbf{x}})\cdot$ ;

(ii) If additionally the spectrum of operator $H_{\sigma}$ is discrete, then
so does the spectrum of operator $H$.

In the next claim we shall use the Fourier transform $\mathcal{F}(f)$ on
${\mathbf{T}}^{d}$ of a function $f\in L_{2}({\mathbf{T}}^{d})$.

######  Theorem 5.12.

Suppose that $V\in L_{2,\,loc}({\mathbf{R}}^{d})$ and

$\limsup_{|\vec{l}|\rightarrow\infty}\Big{\|}\frac{\mathcal{F}\big{(}W_{\vec{l}}\big{)}(\vec{k})}{2\pi
i|\vec{k}|^{2}}\vec{k}\Big{\|}_{l_{d^{\prime}}}<\frac{1}{d2^{d+1}C(d)}\quad(\vec{l}\in{\mathbf{Z}}^{d}),$
(5.18)

where $d^{\prime}=d/(d-1)$ and
$W_{\vec{l}}({\mathbf{x}})\,(\vec{l}\in{\mathbf{Z}}^{d})$ is the $1$-periodic
continuation (by all the variables) of the function
$\big{(}V({\mathbf{x}})-\int_{Q_{1}(\vec{l})}V({\mathbf{s}})\,\mathrm{d}{\mathbf{s}}\big{)}|_{Q_{1}(\vec{l})}$
from the cube $Q_{1}(\vec{l})$ to the whole ${\mathbf{R}}^{d}$. Then the
spectrum of the operator $H$ is discrete and bounded below, if

$\lim_{|\vec{l}|\rightarrow\infty}\int_{Q_{1}(\vec{l})}V({\mathbf{s}})\,\mathrm{d}{\mathbf{s}}=+\infty.$
(5.19)

The next claim is motivated by a counterexample, constructed by A.M. Molchanov
in [17], and it answers the question, raised in Introduction. Let us construct
the potential $V({\mathbf{x}})$ in the following manner. Consider a function

$S\in L_{2}({\mathbf{R}}^{d})$ (5.20)

such that

$S({\mathbf{x}})\geq
0,\quad\mathrm{supp}(S)\subseteq\bar{B}_{1/2}(\vec{0})\quad\mathrm{and}\quad\int_{{\mathbf{R}}^{d}}S({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}>0,$
(5.21)

its dilation $S({\mathbf{x}}/r)\,(r\in(0,1/2))$, translation

$S\big{(}({\mathbf{x}}-\vec{c})/r\big{)}\quad(\vec{c}=(1/2,\,1/2,\,\dots,\,1/2))$

and the periodic summation of the last function

$V_{r}({\mathbf{x}})=\sum_{\vec{k}\in{\mathbf{Z}}^{d}}S\big{(}({\mathbf{x}}-\vec{c}-\vec{k})/r\big{)},$
(5.22)

which is $1$-periodic by all the variables $x_{1},\,x_{2},\,\dots,\,x_{d}$.
For a natural $m$ consider dilation $V_{r}(m{\mathbf{x}})$ of
$V_{r}({\mathbf{x}})$, which is $1/m$-periodic by all the variables. For some
functions $\mathcal{N}:\,{\mathbf{Z}}^{d}\rightarrow{\mathbf{R}}_{+}$,
$r:\,{\mathbf{Z}}^{d}\rightarrow(0,\,1/2)$ and
$m:\,{\mathbf{Z}}^{d}\rightarrow{\mathbf{N}}$ define the desired potential
$V({\mathbf{x}})$ in the following way:

$V({\mathbf{x}}):=\mathcal{N}(\vec{l})V_{r(\vec{l})}\big{(}m(\vec{l}){\mathbf{x}}\big{)}\quad\mathrm{for}\quad{\mathbf{x}}\in
Q_{1}(\vec{l}).$ (5.23)

We shall use Fourier transform $\hat{S}(\vec{\omega})$ on ${\mathbf{R}}^{d}$
of the function $S({\mathbf{x}})$. Denote by $\chi_{r}({\mathbf{x}})$ the
characteristic function of the ball $\bar{B}_{r}(\vec{0})$. Recall that the
constant $C(d)$ is defined by (5.15)

The promised claim is following::

######  Theorem 5.13.

Suppose that along with (5.20), (5.21) the condition

$\bar{H}=\sup_{r\in(0,\,1)}r^{d}\|H_{r}\|_{l_{d^{\prime}}({\mathbf{Z}}^{d})}<\infty$
(5.24)

is fulfilled with $d^{\prime}=d/(d-1)$, where

$\displaystyle
H_{r}(\vec{k})=\left\\{\begin{array}[]{ll}\frac{\hat{S}(r\vec{k})}{2\pi|\vec{k}|}&\mathrm{for}\quad\vec{k}\in{\mathbf{Z}}^{d}\setminus\\{\vec{0}\\},\\\
0&\mathrm{for}\quad\vec{k}=\vec{0}.\end{array}\right.$ (5.27)

Then

(i) the conditions

$\limsup_{|\vec{l}|\rightarrow\infty}\frac{\mathcal{N}(\vec{l})}{m(\vec{l})}<\big{(}d2^{d+1}C(d)\bar{H}\big{)}^{-1}$
(5.28)

and

$\lim_{|\vec{l}|\rightarrow\infty}\mathcal{N}(\vec{l})(r(\vec{l}))^{d}=\infty$
(5.29)

imply discreteness of the spectrum of $H=-\Delta+V({\mathbf{x}})\cdot$;

(ii) if $S({\mathbf{x}})=\chi_{1/2}({\mathbf{x}})$, condition (5.24)-(5.27) is
satisfied for $d\in\\{3,\,4\\}$.

######  Remark 5.14.

In view of (5.21)-(5.23), we have for $\vec{l}\in{\mathbf{Z}}^{d}:$

$\displaystyle\int_{Q_{1}(\vec{l})}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=\mathcal{N}(\vec{l})\int_{Q_{1}(\vec{0})}V_{r(\vec{l})}(m(\vec{l}){\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=$
$\displaystyle\frac{\mathcal{N}(\vec{l})}{(m(\vec{l}))^{d}}\int_{Q_{m(\vec{l})}(\vec{0})}V_{r(\vec{l})}({\mathbf{s}})\,\mathrm{d}{\mathbf{s}}=\mathcal{N}(\vec{l})\int_{Q_{1}(\vec{0})}V_{r(\vec{l})}({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=$
$\displaystyle\mathcal{N}(\vec{l})\int_{{\mathbf{R}}^{d}}S\big{(}({\mathbf{x}}-\vec{c})/r(\vec{l})\big{)}\,\mathrm{d}{\mathbf{x}}=\mathcal{N}(\vec{l})\big{(}r(\vec{l})\big{)}^{d}\int_{{\mathbf{R}}^{d}}S({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}.$

Hence condition (5.29) is equivalent to (5.19).

######  Remark 5.15.

If $S({\mathbf{x}})=\chi_{1/2}({\mathbf{x}})$, then using arguments from the
previous remark, we get:

$\displaystyle\mathrm{mes}_{d}\big{(}\\{{\mathbf{x}}\in
Q_{1}(\vec{l}):\;V({\mathbf{x}})>0\\}\big{)}=\int_{Q_{1}(\vec{0})}V_{r(\vec{l})}(m(\vec{l}){\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=$
$\displaystyle\big{(}r(\vec{l})\big{)}^{d}\mathrm{mes}_{d}(B_{1/2}(\vec{0})).$

Hence if $\lim_{|\vec{l}|\rightarrow\infty}r(\vec{l})=0$, the potential
$V({\mathbf{x}})$, constructed above, does not satisfy condition (5.3) of
Theorem 5.3 and condition (5.12) of Theorem 5.8. But in this case we can
choose the functions $\mathcal{N}(\vec{l})$ and $m(\vec{l})$ such that they
satisfy conditions (5.28) and (5.29) of Theorem 5.13, i.e., the spectrum of
$H=-\Delta+V({\mathbf{x}})\cdot$ will be discrete.

######  Remark 5.16.

The idea of a counterexamle from [17], mentioned above, is following. The
potential $V({\mathbf{x}})$ have been constructed as above with
$S({\mathbf{x}})=\chi_{1/2}({\mathbf{x}})$, such that

$\forall\,\vec{l}\in{\mathbf{Z}}^{d}:\quad r(\vec{l})\in(0,\,1/2)$ (5.30)

and condition (5.19) (i,e,. (5.29)) is fulfilled. We see from (5.21)-(5.23)
that on each cube $Q_{1}(\vec{l})\;(\vec{l}\in{\mathbf{Z}}^{d})$ the set
$A(\vec{l}):=\mathrm{supp}(V)\cap Q_{1}(\vec{l})$ is a union of
$\big{(}m(\vec{l})\big{)}^{d}$ balls having the radius
$a(\vec{l})=r(\vec{l})/m(\vec{l})$. Consider the set $B(\vec{l})$, which is
the union of all balls concentric with these balls and having the radius
$b(\vec{l})=2a(\vec{l})$. In [17] on each cube $Q_{1}(\vec{l})$ a nonnegative
“test function” $\phi_{\vec{l}}({\mathbf{x}})$ have been constructed such that
it belongs to $W_{2}^{1}(Q_{1}(\vec{l}))$ and has the properties:
$\phi_{\vec{l}}({\mathbf{x}})=0$ for ${\mathbf{x}}\in A(\vec{l})$,
$\phi_{\vec{l}}({\mathbf{x}})=1$ for ${\mathbf{x}}\in Q_{1}(\vec{l})\setminus
B(\vec{l})$ and it is defined in $B(\vec{l})\setminus A(\vec{l})$ such that

$\displaystyle\int_{Q_{1}(\vec{l})}|\nabla\phi_{\vec{l}}({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}=(d-2)\omega_{d}\big{(}m(\vec{l})\big{)}^{d}\frac{\big{(}a(\vec{l})\big{)}^{d-2}}{1-\big{(}a(\vec{l})/b(\vec{l})\big{)}^{d-2}}=$
(5.31) $\displaystyle
G_{d}\big{(}m(\vec{l})\big{)}^{2}r(\vec{l})\big{)}^{d-2}<1,$

where $\omega_{d}$ is the volume of unit sphere in ${\mathbf{R}}^{d}$ and
$G_{d}=(d-2)\omega_{d}\frac{2^{d-2}}{2^{d-2}-1}$. Condition (5.31) was called
in [17] “small capacity property” of the sets $A(\vec{l})$. On the other hand,
it is easy to show using (5.30) that for some $c>0$

$\forall\,\vec{l}\in{\mathbf{Z}}^{d}:\quad\int_{Q_{1}(\vec{l})}|\phi_{\vec{l}}({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}\geq\mathrm{mes}_{d}\big{(}Q_{1}(\vec{l})\setminus
B(\vec{l})\big{)}\geq c.$

Then, taking into account the density of $C^{\infty}(Q_{1}(\vec{l}))$ in
$W_{2}^{1}(Q_{1}(\vec{l}))$, we obtain:

$\displaystyle\mu_{0}(Q_{1}(\vec{l}))=\inf_{u\in
W_{2}^{1}(Q_{1}(\vec{l})),\,u\neq 0}\frac{\int_{Q_{1}(\vec{l})}(|\nabla
u({\mathbf{x}})|^{2}+V({\mathbf{x}})|u({\mathbf{x}})|^{2})\,\mathrm{d}{\mathbf{x}}}{\int_{Q_{1}(\vec{l})}|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}}\leq$
$\displaystyle\frac{\int_{Q_{1}(\vec{l})}|\nabla\phi_{\vec{l}}({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}}{\int_{Q_{1}(\vec{l})}|\phi_{\vec{l}}({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}}<1/c$

for all $\vec{l}\in{\mathbf{Z}}^{d}$. Hence by the localization principle
(Proposition 5.10), the spectrum of operator $H=-\Delta+V({\mathbf{x}})\cdot$
is not discrete, despite condition (1.6) is satisfied for the potential
$V({\mathbf{x}})$. Notice that in view of (5.31) and (5.30)

$\frac{\mathcal{N}(\vec{l})}{m(\vec{l})}\geq
2^{d/2+1}\sqrt{G_{d}}\mathcal{N}(\vec{l})(r(\vec{l}))^{d}.$

Hence (5.29) implies that condition (5.28) of Theorem 5.13 is not satisfied.
This fact is predictable, because otherwise by this theorem the spectrum of
$H$ would be discrete for $d\in\\{3,\,4\\}$.

###  5.4. The case where the operator $H$ is defined in
$L_{2}(\Omega)\;(\Omega\subset{\mathbf{R}}^{d})$

Let us notice that in [15] Theorem 1.1 have been formulated in a more general
form. Assume that a nonnegative potential $V({\mathbf{x}})$ is defined in a
open domain $\Omega\subset{\mathbf{R}}^{d}$ (which can be unbounded), $V\in
L_{1,\,loc}(\Omega)$ and the Schrödinger operator $H=-\Delta+V(\cdot)$ acts in
the space $L_{2}(\Omega)$. Denote by
$\mathcal{N}_{\gamma}({\mathbf{y}},r,\Omega)\;(\gamma\in(0,1))$ the collection
of all compact sets $F\subseteq\bar{\mathcal{G}}_{r}({\mathbf{y}})$ satisfying
the conditions

$\bar{\mathcal{G}}_{r}({\mathbf{y}})\setminus\Omega\subseteq F$ (5.32)

and (1.3), where $``\mathrm{cap}^{\prime\prime}$ is the harmonic capacity and
the domain $\mathcal{G}_{r}({\mathbf{y}})$ is defined by (1.2), (1.1). The
more general formulation of Theorem 1.1 is:

######  Theorem 5.17.

[[15], Theorem 2.2] The spectrum of the operator $H$ is discrete, if for some
$r_{0}>0$ and for any $r\in(0,r_{0})$ the condition

$\lim_{|{\mathbf{y}}|\rightarrow\infty}\inf_{F\in\mathcal{N}_{\gamma(r)}({\mathbf{y}},r,\Omega)}\int_{{\mathcal{G}}_{r}({\mathbf{y}})\setminus
F}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=+\infty,$ (5.33)

is satisfied, where $\gamma(r)$ satisfies the conditions (1.5).

Denote by

$\mathcal{P}_{\gamma}({\mathbf{y}},r,\Omega)\quad(\gamma\in(0,1))$

the collection of all $\mathrm{mes}_{d}$-measurable sets
$E\subseteq\mathcal{G}_{r}({\mathbf{y}})\cap\Omega$ satisfying the condition
$\mathrm{mes}_{d}(E)\geq(1-\gamma)\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))$.
For $W\in L_{1,\,loc}(\Omega)$ and $\gamma\in(0,1)$ consider the quantities:

$\displaystyle
R_{W}({\mathbf{y}},r,\Omega,\gamma)=\left\\{\begin{array}[]{ll}+\infty,&\mathrm{if}\\\
\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega)<(1-\gamma)\,\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})),\\\
\\\
\int_{\mathcal{G}_{r}({\mathbf{y}})\cap\Omega}W({\mathbf{x}})\,\mathrm{d}{\mathbf{x}},&\mathrm{if}\\\
\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega)=\\\
(1-\gamma)\,\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})),\\\ \\\
\inf_{E\in\mathcal{P}_{\gamma}({\mathbf{y}},r,\Omega)}\int_{E}W({\mathbf{x}})\,\mathrm{d}{\mathbf{x}},&\mathrm{if}\\\
\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega)>(1-\gamma)\,\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})),\end{array}\right.$
(5.43)

A direct use of isocapacity inequality (3.3) leads to the following claim:

######  Theorem 5.18.

Suppose that for some $r_{0}>0$ and any $r\in(0,r_{0})$ the condition

$\lim_{|{\mathbf{y}}|\rightarrow\infty}R_{V}({\mathbf{y}},r,\Omega,\gamma(r))=+\infty$
(5.44)

is fulfilled with a function $\gamma(r)$ satisfying condition (5.1). Then the
spectrum of the operator $H=-\Delta+V({\mathbf{x}})$ is discrete.

Taking $V({\mathbf{x}})\equiv 0$, we obtain the following consequence of
Theorem 5.18:

######  Corollary 5.19.

If for some $r_{0}>0$ and any $r\in(0,r_{0})$ the condition

$\limsup_{|{\mathbf{y}}|\rightarrow\infty}\frac{\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega)}{\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))}\leq
1-\gamma(r)$ (5.45)

is fulfilled with a function $\gamma(r)$ satisfying condition (5.1), then the
spectrum of the operator $-\Delta$ is discrete.

######  Remark 5.20.

Condition (5.45) can be written in the equivalent form:

$\liminf_{|{\mathbf{y}}|\rightarrow\infty}\frac{\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})\setminus\Omega)}{\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))}\geq\gamma(r)$

In this formulation for the case $\gamma(r)\equiv const$ the result of
Corollary 5.19 was obtained in [11] (Corollary 6.8).

Consider the probability space

$\big{(}\mathcal{G}_{r}({\mathbf{y}})\cap\Omega,\,\Sigma_{L}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega),\,m_{d,r,\Omega}\big{)},$

where $m_{d,r,\Omega}$ is the normalized Lebesgue measure on
$\mathcal{G}_{r}({\mathbf{y}}))\cap\Omega$:

$m_{d,r,\Omega}(A):=\frac{\mathrm{mes}_{d}(A)}{\mathrm{mes}_{d}({\mathcal{G}}_{r}({\mathbf{y}})\cap\Omega)}\quad(A\in\Sigma_{L}(\bar{\mathcal{G}}_{r}({\mathbf{y}})\cap\Omega)).$

Denote by $\mathrm{E}_{{\mathbf{y}},r,\Omega}(W)$ and
$\mathrm{Dev}_{{\mathbf{y}},r,\Omega}(W)$ the expectation and deviation of a
function $W\in
L_{2}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega,\,m_{d,r,\Omega})$. Consider the
quantity:

$\displaystyle
Y_{W}({\mathbf{y}},r,\Omega,\gamma)=\left\\{\begin{array}[]{ll}+\infty,&\mathrm{if}\\\
\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega)<(1-\gamma/2)\,\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})),\\\
\\\
\mathrm{E}_{{\mathbf{y}},r,\Omega}(W)-\sqrt{2\bar{\gamma}(\gamma,\Omega)}\cdot\mathrm{Dev}_{{\mathbf{y}},r,\Omega}(W),&\mathrm{if}\\\
\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega)\geq(1-\gamma/2)\,\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})),\end{array}\right.$
(5.51)

where

$\bar{\gamma}(\gamma,\Omega)=1-(1-\gamma/2)\frac{\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))}{\mathrm{mes}_{d}({\mathcal{G}}_{r}({\mathbf{y}})\cap\Omega)}.$
(5.52)

The following claim is based on Theorems 5.7 and 5.18:

######  Theorem 5.21.

Suppose that $V\in L_{2,loc}(\Omega)$ and the condition

$\lim_{|{\mathbf{y}}|\rightarrow\infty}Y_{V}({\mathbf{y}},r,\Omega,\gamma(r))=+\infty$
(5.53)

is satisfied for some $r_{0}>0$ and any $r\in(0,r_{0})$, where $\gamma(r)$
satisfies the condition (5.1). Then the spectrum of the operator
$H=-\Delta+V({\mathbf{x}})$ is discrete.

######  Remark 5.22.

We see from (5.51) and (5.52) that if in Theorem 5.21 the quantity
$\frac{(1-\gamma/2)\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))}{\mathrm{mes}_{d}({\mathcal{G}}_{r}({\mathbf{y}})\cap\Omega)}$
approaches $1$ from below for large $|{\mathbf{y}}|$, the condition (5.53)
becomes weaker because of a weaker influence of deviation of the potential
$V({\mathbf{x}})$ in ${\mathcal{G}}_{r}({\mathbf{y}})\cap\Omega$. On the other
hand, if it is bigger than $1$, then by Corollary 5.19 even
$V({\mathbf{x}})\equiv 0$ satisfies this condition.

##  6\. Proof of main results

###  6.1. Proof of Theorem 5.1

###### Proof.

Consider the following quantities connected with the domain $\mathcal{G}$
having the form (1.1):

$\bar{r}:=\max_{\omega\in S^{d-1}}r(\omega),\quad r_{m}:=\min_{\omega\in
S^{d-1}}r(\omega),$ (6.1) $G:=\Big{(}\frac{\bar{r}}{r_{m}}\Big{)}^{d}.$ (6.2)

we see from (1.1), (1.2), (6.1) and (6.2) that
$\mathcal{G}_{r}({\mathbf{y}})\subseteq B_{\bar{r}\cdot r}({\mathbf{y}})$ and

$\mathrm{mes}_{d}(B_{\bar{r}\cdot r}({\mathbf{y}}))\leq
G\cdot\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})).$

Let us define

$\tilde{\gamma}(r)=\big{(}\gamma(r)/G\big{)}^{(d-2)/d}.$ (6.3)

In view of (5.1). the function $\tilde{\gamma}(r)$ satisfies condition (1.5).
Suppose that $F\in\mathcal{N}_{\tilde{\gamma}(r)}({\mathbf{y}},r)$, where the
collection $\mathcal{N}_{\gamma}({\mathbf{y}},r)$ is defined by (1.3). In view
of the isocapacity inequality (3.3) and the fact that it comes as identity for
$F=\bar{B}_{\bar{r}\cdot r}({\mathbf{y}})$, we have, taking into account
(6.3):

$\displaystyle\frac{\mathrm{mes}_{d}(F)}{\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))}\leq
G\frac{\mathrm{mes}_{d}(F)}{\mathrm{mes}_{d}(B_{\bar{r}\cdot
r}({\mathbf{y}}))}\leq
G\Big{(}\frac{\mathrm{cap}(F)}{\mathrm{cap}(\bar{B}_{\bar{r}\cdot
r}({\mathbf{y}}))}\Big{)}^{d/(d-2)}\leq$ $\displaystyle
G\Big{(}\frac{\mathrm{cap}(F)}{\mathrm{cap}(\bar{\mathcal{G}}_{r}({\mathbf{y}}))}\Big{)}^{d/(d-2)}\leq
G(\tilde{\gamma}(r))^{d/(d-2)}=\gamma(r),$

i.e., $F\in\mathcal{M}_{\gamma(r)}({\mathbf{y}},r)$. Thus,
$\mathcal{N}_{\tilde{\gamma}(r)}({\mathbf{y}},r)\subseteq\mathcal{M}_{\gamma(r)}({\mathbf{y}},r)$.
Hence

$\inf_{F\in\mathcal{N}_{\tilde{\gamma}(r)}({\mathbf{y}},r)}\int_{\mathcal{G}_{r}({\mathbf{y}})\setminus
F}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}\geq\inf_{F\in\mathcal{M}_{\gamma(r)}({\mathbf{y}},r)}\int_{\mathcal{G}_{r}({\mathbf{y}})\setminus
F}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}.$

Therefore in view of condition (5.2), condition (1.4) is satisfied with
$\tilde{\gamma}(r)$ insyead of $\gamma(r)$. Hence by Theorem 1.1 the spectrum
of the operator $H$ is discrete. Theorem 5.1 is proven. ∎

###  6.2. Proof of Theorem 5.3

###### Proof.

Let us take $\theta>1$, $\tilde{\gamma}(r)=\gamma(r)/\theta$ and
$\sigma(r)=(1-\tilde{\gamma}(r))\mathrm{mes}_{d}(\mathcal{G}_{r}(0))$. Since
$\gamma(r)$ satisfies conditions (5.1), then $\tilde{\gamma}(r)$ satisfies
these conditions. Denote briefly for $t\in(0,1)$:
$\mathcal{E}(t,\,{\mathbf{y}}\,,r):=\mathcal{E}(t,\,\mathcal{G}_{r}({\mathbf{y}}),\,\mathrm{mes}_{d})$
(Problem 4.1). In view of Proposition 4.3 and estimate (4.15) (Proposition
4.4) with $W({\mathbf{x}})=V({\mathbf{x}})$, $X=\mathcal{G}_{r}({\mathbf{y}})$
and $t=\delta(r)$, we have:

$\displaystyle\hskip-14.22636pt\inf_{F\in\mathcal{M}_{\tilde{\gamma}(r)}({\mathbf{y}},r)}\int_{\mathcal{G}_{r}({\mathbf{y}})\setminus
F}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}\geq\inf_{G\in\mathcal{E}((1-\tilde{\gamma}(r))\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})),\,{\mathbf{y}},\,r)}\int_{G}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}\geq$
$\displaystyle\hskip-14.22636pt\frac{\theta-1}{\theta}\delta(r)V^{\star}(\delta(r),{\mathbf{y}},r).$

This estimate, condition (5.3) and Theorem 5.1 imply the desired claim.
Theorem 5.3 is proven. ∎

###  6.3. Proof of Theorem 5.6

###### Proof.

Let us take $r\in(0,1)$ and a cube $Q_{r}({\mathbf{y}})$. It is clear that
this cube intersects not more than $2^{d}$ adjacent cubes
$Q_{1}(\vec{l}_{k})\,(\vec{l}_{k}\in{\mathbf{Z}}^{d})$ and among them there is
a cube $Q_{1}(\vec{l}_{k})$ such that for some $\tilde{\mathbf{y}}\in
Q_{1}(\vec{l}_{k})$: $Q_{\tilde{r}}(\tilde{\mathbf{y}})\subseteq
Q_{1}(\vec{l}_{k})\cap Q_{r}({\mathbf{y}})$ with $\tilde{r}=r/2$. Thus, for
any ${\mathbf{y}}\in{\mathbf{R}}^{d}$ we can choose
$\vec{l}({\mathbf{y}},r)\in Z^{d}$ such that

$Q_{\tilde{r}}(\tilde{\mathbf{y}})\subseteq Q_{r}({\mathbf{y}})\cap
Q_{1}(\vec{l}({\mathbf{y}},r)).$ (6.4)

Denote

$\hat{\gamma}(\rho,K)=K\gamma(\theta\rho/m^{2})\theta^{d}\quad(K>0).$ (6.5)

By Lemma 6.2 and Lemma 6.1 with $W({\mathbf{x}})=V({\mathbf{x}})$, we get that
for some $K>0$ and $j\in\\{1,2,\dots,n\\}$ with
$n=\big{[}\log_{m}\big{(}\frac{1}{\theta\tilde{r}}\big{)}\big{]}+2$

$\displaystyle
V^{\star}\big{(}\hat{\gamma}(\tilde{r},K),\,Q_{r}({\mathbf{y}})\big{)}\geq$
$\displaystyle
V^{\star}\big{(}\hat{\gamma}(\tilde{r},K),\,Q_{\tilde{r}}(\tilde{\mathbf{y}})\big{)}\geq\min_{\vec{\xi}:\,Q(\vec{\xi},\,n)\subseteq
F_{j}}V^{\star}\big{(}\gamma(m^{-n}),\,\vec{\xi}\,,n),$ (6.6)

where $\bar{\gamma}(r)=\kappa\hat{\gamma}(r/\sqrt{d},\,K)$ and $F_{j}$ is a
non-empty union of cubes $Q(\vec{\xi},\,n)$ with $\vec{\xi}\in
m^{-n}\cdot{\mathbf{Z}}^{d}$, such that $F_{j}\subseteq
Q_{\bar{r}}(\tilde{\mathbf{y}})\cap D_{j}(\vec{l}({\mathbf{y}},r))$. Notice
that since the function $\gamma(r)$ satisfies conditions (5.1), the function
$\bar{\gamma}(r)$ satisfies this condition too for some $r_{0}>0$. Then in
view of estimate (6.3), the property (6.4) and condition (5.7), condition
(5.3) of Theorem 5.3 is satisfied with $\gamma(r)=\bar{\gamma}(r)$. Then the
spectrum of the operator $H=-\Delta+V({\mathbf{x}})\cdot$ is discrete. Theorem
5.6 is proven. ∎

In the proof of Theorem 5.6 we have used the following claims, proved in [22]:

######  Lemma 6.1.

[[22], Lemma 4.3] Suppose that in a cube
$Q_{1}(\vec{l})\,(\vec{l}\in{\mathbf{Z}}^{d})$ there is a sequence of subsets
$\\{D_{n}(\vec{l})\\}_{n=1}^{\infty}$ forming in it a
$(\log_{m},\,\theta)$-dense system. Let
$\gamma:\,(0,\,r_{0})\rightarrow{\mathbf{R}}$ be a monotone non-increasing
function with $r_{0}=\min\\{1,\,1/(m^{2}\theta)\\}$. Then for some $K>0$ and
for any cube $Q_{r}({\mathbf{y}})\subset Q_{1}(\vec{l})$ there are
$j\in\\{1,2,\dots,n\\}$ with

$n=\big{[}\log_{m}\big{(}\frac{1}{\theta r}\big{)}\big{]}+2$ (6.7)

and a non-empty set $F_{j}\subseteq Q_{r}({\mathbf{y}})\cap D_{j}(\vec{l})$,
which is a union of cubes $Q(\vec{\xi}\,,n)$ with $\vec{\xi}\in
m^{-n}\cdot{\mathbf{Z}}^{d}$, such that for any nonnegative function $W\in
L_{1}(Q_{r}({\mathbf{y}}))$ the inequality

$\displaystyle
W^{\star}(\hat{\gamma}(r,K),Q_{r}({\mathbf{y}}))\geq\min_{\vec{\xi}:\;Q(\vec{\xi},\,n)\subseteq
F_{j}}W^{\star}\big{(}\gamma(m^{-n}),\,\vec{\xi},\,n)\big{)}$ (6.8)

is valid, where the function $\hat{\gamma}(r,K)$ is defined by (6.5).

######  Lemma 6.2.

[[22], Lemma 4.5] Let $\Omega_{1}$ and $\Omega_{2}$ be measurable subsets of
${\mathbf{R}}^{d}$ such that $\Omega_{1}\subseteq\Omega_{2}$ and
$W({\mathbf{x}})$ be a non-negative measurable function defined on
$\Omega_{2}$. Then for any $t>0$ the inequality $W^{\star}(t,\Omega_{1})\leq
W^{\star}(t,\Omega_{2})$ is valid.

###  6.4. Proof of Theorem 5.7

###### Proof.

Let us use the reformulation of Problem 4.1 given in Remark 4.1. Denote
$u(x)=2\chi_{\mathrm{E}}(x)-1$. Then it is easy to see that

$I_{W}(t,\,X,\,\mu)\geq(1/2)\inf_{u\in\mathcal{F}(t,X,\,\mu)}\int_{X}W(x)(1+u(x))\,\mu(\mathrm{d}x),$

where

$\displaystyle\mathcal{F}(t,X,\,\mu):=$ $\displaystyle\\{u\in
L_{2}(X,\mu):\;u^{2}(x)\leq
1\;\mathrm{a.e.}\;\mathrm{in}\,X,\;\int_{X}u(x)\,\mu(dx)\geq\sigma(t)\\}.$
(6.9)

Consider the Lagrangian:

$\displaystyle
L(u,\lambda,\nu)=\int_{X}W(x)(1+u(x))\,\mu(\mathrm{d}x)-\int_{X}\lambda(x)(1-u^{2}(x))\,\mu(dx)-$
$\displaystyle\nu\big{(}\int_{X}u(x)\,\mu(dx)-\sigma(t)\big{)},$ (6.10)

where $\nu\geq 0$, $\lambda\in L_{2}^{+}(X,\,\mu)=\\{\lambda\in
L_{2}(X,\mu):\;\lambda(x)\geq 0\;\mathrm{a.e.}\;\mathrm{in}\,X\\}$. It is
clear that

$I_{W}(t,\,X,\,\mu)\geq(1/2)\sup_{\lambda\in
L_{1}^{+}(X,\mu),\,\nu>0}\;\inf_{u\in L_{2}(X,\mu)}L(u,\lambda,\nu).$

We shall use an easier version of the Lagrangian relaxation, assuming that
$\lambda(x)\equiv\lambda=const>0$ and $\nu>0$. Then

$I_{W}(t,\,X,\,\mu)\geq(1/2)\sup_{\lambda>0,\,\nu>0}m(\lambda,\nu),$ (6.11)

where

$m(\lambda,\nu)=\inf_{u\in L_{2}(X,\mu)}L(u,\lambda,\nu).$ (6.12)

Let us find $m(\lambda,\nu)$. To this end we shall find critical points of the
functional $L(u,\lambda,\nu)$ for fixed $\lambda>0,\,\nu>0$. We see that its
Gateaux differential by $u$ is:

$\displaystyle D_{u}L(u,\lambda,\nu)(\eta)=$
$\displaystyle\int_{X}W(x)\eta(x)\,\mu(dx)+2\lambda\int_{X}u(x)\eta(x)\,\mu(dx)-\nu\int_{x}\eta(x)\,\mu(dx),$

where $\eta\in L_{2}(X,\mu)$. Hence its gradient by $u$ is:

$\nabla_{u}L(u,\lambda,\nu)=W(x)-\nu+2\lambda u(x).$

Therefore for any fixed $\lambda>0,\,\nu>0$ the functional $L(u,\lambda,\nu)$
has only one critical point in $L_{2}(X,\mu)$:

$u_{\star}(x,\lambda,\nu)=\frac{\nu-W(x)}{2\lambda}.$

Consider the second Gateaux differential of $L(u,\lambda,\nu)$ by $u$:

$D_{u}^{2}L(u,\lambda,\nu)(\eta,\eta)=2\lambda\int_{X}(\eta(x))^{2}\mu(dx)>0,\quad\mathrm{if}\quad\|\eta\|_{L_{2}(X,\mu)}=1.$

This means that the functional $L(u,\lambda,\nu)$ is strictly convex on
$L_{2}(X,\,\mu)$ for any fixed $\lambda>0,\,\nu>0$, Hence the point
$u_{\star}(x,\lambda,\nu)$ yields the minimum for this functional. Therefore,
in view of (6.12), we have:

$\displaystyle\hskip-28.45274ptm(\lambda,\nu)=\int_{X}W(x)\Big{(}1+\frac{\nu-W(x)}{2\lambda}\Big{)}\,\mu(dx)+$
(6.13)
$\displaystyle\hskip-28.45274pt\lambda\int_{X}\Big{(}\frac{(\nu-W(x))^{2}}{4\lambda^{2}}-1\Big{)}\,\mu(dx)-\nu\Big{(}\int_{X}\frac{\nu-W(x)}{2\lambda}\,\mu(dx)-\sigma(t)\Big{)}=$
$\displaystyle\hskip-28.45274pt\big{(}1+\frac{\nu}{2\lambda}\big{)}\mathrm{E}(W)-\frac{1}{2\lambda}\mathrm{E}(W^{2})-\lambda+\frac{\nu^{2}}{4\lambda}-\frac{\nu}{2\lambda}\mathrm{E}(W)+\frac{1}{4\lambda}\mathrm{E}(W^{2})-$
$\displaystyle\hskip-28.45274pt\frac{\nu^{2}}{2\lambda}+\frac{\nu}{2\lambda}\mathrm{E}(W)+\nu\sigma(t)=\big{(}1+\frac{\nu}{2\lambda}\big{)}\mathrm{E}(W)-\frac{1}{4\lambda}\mathrm{E}(W^{2})-\lambda-\frac{\nu^{2}}{4\lambda}+$
$\displaystyle\hskip-28.45274pt\nu\sigma(t)=\mathrm{E}(W)-\frac{1}{4\lambda}\big{(}\nu-\mathrm{E}(W)\big{)}^{2}-\frac{1}{4\lambda}\Big{(}\mathrm{E}(W^{2})-(\mathrm{E}(W))^{2}\Big{)}-\lambda+$
$\displaystyle\hskip-28.45274pt\nu\sigma(t)$

Now let us find $\sup_{\lambda>0,\,\nu>0}m(\lambda,\nu)$. We see that

$\displaystyle\frac{\partial}{\partial\lambda}m(\lambda,\nu)=\frac{1}{4\lambda^{2}}\Big{[}\big{(}\nu-\mathrm{E}(W)\big{)}^{2}+\mathrm{E}(W^{2})-(\mathrm{E}(W))^{2}\Big{]}-1,$
$\displaystyle\frac{\partial}{\partial\nu}m(\lambda,\nu)=\frac{1}{2\lambda}\mathrm{E}(W)-\frac{\nu}{2\lambda}+\sigma(t).$
(6.14)

Hence we have the following system of equations for critical points of the
function $m(\lambda,\nu)$:

$\displaystyle\big{(}\nu-\mathrm{E}(W)\big{)}^{2}+\mathrm{E}(W^{2})-(\mathrm{E}(W))^{2}=4\lambda^{2},$
$\displaystyle\nu-\mathrm{E}(W)=2\lambda\sigma(t).$

Suppose that $\mathrm{E}(W^{2})-(\mathrm{E}(W))^{2}>0$. We see that in this
case the function $m(\lambda,\nu)$ has the unique critical point in the domain
$\lambda>0,\,\nu>0$:

$\lambda_{\star}=\frac{\sqrt{\mathrm{E}(W^{2})-(\mathrm{E}(W))^{2}}}{2\sqrt{1-\sigma^{2}(t)}},\quad\nu_{\star}=\mathrm{E}(W)+2\lambda_{\star}\sigma(t),$

Then in view of (6.13),

$\displaystyle\hskip-14.22636ptm(\lambda_{\star},\,\nu_{\star})=\mathrm{E}(W)-\lambda_{\star}(1+\sigma^{2}(t))-\frac{1}{4\lambda_{\star}}\Big{(}\mathrm{E}(W^{2})-(\mathrm{E}(W))^{2}\Big{)}+$
$\displaystyle\hskip-14.22636pt\mathrm{E}(W)\sigma(t)+2\lambda_{\star}\sigma^{2}(t)=\mathrm{E}(W)(1+\sigma(t))-\lambda_{\star}(1-\sigma^{2}(t))-$
$\displaystyle\hskip-14.22636pt\frac{1}{4\lambda_{\star}}\Big{(}\mathrm{E}(W^{2})-(\mathrm{E}(W))^{2}\Big{)}=(1+\sigma(t))\mathrm{E}(W)-$
(6.15)
$\displaystyle\hskip-14.22636pt\sqrt{1-\sigma^{2}(t)}\sqrt{\mathrm{E}(W^{2})-(\mathrm{E}(W))^{2}}=(1+\sigma(t))\mathrm{E}(W)-\sqrt{1-\sigma^{2}(t)}\cdot\mathrm{Dev}(W).$

Let us show that $m(\lambda_{\star},\,\nu_{\star})$ is the maximal value of
$m(\lambda,\nu)$ in the domain $\lambda>0,\,\nu>0$. To this end consider the
Hessian matrix $\mathrm{Hes}(m)(\lambda,\nu)$ of $m(\lambda,\nu)$ in this
domain. Using (6.4), we have:

$\displaystyle-\mathrm{Hes}(m)(\lambda,\nu)=\left(\begin{array}[]{ll}-\frac{\partial^{2}}{\partial\lambda^{2}}m(\lambda,\nu),&-\frac{\partial^{2}}{\partial\lambda\partial\nu}m(\lambda,\nu)\\\
-\frac{\partial^{2}}{\partial\lambda\partial\nu}m(\lambda,\nu),&-\frac{\partial^{2}}{\partial\nu^{2}}m(\lambda,\nu)\end{array}\right)=$
$\displaystyle\left(\begin{array}[]{ll}\frac{1}{2\lambda^{3}}\Big{[}\big{(}\nu-\mathrm{E}(W)\big{)}^{2}+\mathrm{E}(W^{2})-(\mathrm{E}(W))^{2}\Big{]},&\frac{1}{2\lambda^{2}}(\nu-\mathrm{E}(W))\\\
\frac{1}{2\lambda^{2}}(\mathrm{E}(W)-\nu),&\frac{1}{2\lambda}\end{array}\right)$

We see that $-\frac{\partial^{2}}{\partial\lambda^{2}}m(\lambda,\nu)>0$,
$-\frac{\partial^{2}}{\partial\nu^{2}}m(\lambda,\nu)>0$ and

$\det(-\mathrm{Hes}(m)(\lambda,\nu))>0,$

i.e., the matrix $-\mathrm{Hes}(m)(\lambda,\nu)$ is positive-definite in any
point of the convex domain $\lambda>0$, $\nu>0$. This means that the function
$m(\lambda,\nu)$ is strictly concave in this domain. Thus, in the case where
$\mathrm{Dev}(W)>0$ the equality holds:
$m(\lambda_{\star},\nu_{\star})=\max_{\lambda>0,\nu>0}m(\lambda,\nu)$. Hence
in view of (6.11) and (6.4) the inequality (5.10) is valid. It is clear that
if $\mathrm{Dev}(W)=0$, then $W(x)=\mathrm{E}(W)$ for $\mu$-almost all $x\in
X$. In view of definition (4.1), in this case $I_{W}(t,\,X,\,\mu)\geq
t\cdot\mathrm{E}(W)$. Since $\sigma(t)=2t-1$, the inequality (5.10) is valid
also in this case. Theorem 5.7 is proven. ∎

###  6.5. Proof of Theorem 5.8

###### Proof.

In view of estimate (5.10) (Theorem 5.7) with $t=1-\gamma(r)/2$ and
$\sigma(t)=2t-1=1-\gamma(r)$,

$\displaystyle\inf_{F\in\mathcal{M}_{\gamma(r)/2}({\mathbf{y}},r)}\int_{\mathcal{G}_{r}({\mathbf{y}})\setminus
F}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=\mathrm{mes}_{d}\big{(}\mathcal{G}_{r}({\mathbf{y}})\big{)}\times$
$\displaystyle\inf_{F\in\mathcal{M}_{\gamma(r)/2}({\mathbf{y}},r)}\int_{\mathcal{G}_{r}({\mathbf{y}})\setminus
F}V({\mathbf{x}})\,m_{{\mathbf{y}},r}(\mathrm{d}{\mathbf{x}})\geq$
$\displaystyle\mathrm{mes}_{d}\big{(}\mathcal{G}_{r}({\mathbf{y}})\big{)}\frac{1+\sigma(t)}{2}\Big{(}\mathrm{E}_{{\mathbf{y}},r}(V)-\sqrt{\frac{1-\sigma(t)}{1+\sigma(t)}}\cdot\mathrm{Dev}_{{\mathbf{y}},r}(V)\Big{)}\geq$
$\displaystyle\mathrm{mes}_{d}\big{(}\mathcal{G}_{r}({\mathbf{y}})\big{)}(1-\gamma(r)/2)\Big{(}\mathrm{E}_{{\mathbf{y}},r}(V)-\sqrt{\gamma(r)}\cdot\mathrm{Dev}_{{\mathbf{y}},r}(V)\Big{)}.$

This estimate, condition (5.12) and Theorem 5.1 (with $\gamma(r)/2$ instead of
$\gamma(r)$) imply the desired claim. Theorem 5.8 is proven. ∎

###  6.6. Proof of Theorem 5.11

###### Proof.

In view of definition (5.14) and condition (5.17) there exists $R>0$ such that
for any ${\mathbf{y}}\in{\mathbf{R}}^{d}\setminus B_{R}(0)$ there is a vector
field $\Gamma_{{\mathbf{y}},r}\in{\mathcal{D}}_{{\mathbf{y}},\,r}(W)$ such
that $\|\Gamma_{{\mathbf{y}},r}\|_{d}\leq\sigma/(2C(d))$. Taking into account
that
$W({\mathbf{x}})=\mathrm{div}\vec{\Gamma}_{{\mathbf{y}},r}({\mathbf{x}})\;({\mathbf{x}}\in
Q_{r}({\mathbf{y}}))$ and using claim (ii) of Proposition B.1, we obtain:

$\displaystyle\int_{Q_{r}({\mathbf{y}})}W({\mathbf{x}})|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}\leq\sigma\int_{Q_{r}({\mathbf{y}})}|\nabla
u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}},$

hence

$\displaystyle\hskip-14.22636pt(Hu,u)_{Q_{r}({\mathbf{y}})}=\int_{Q_{r}({\mathbf{y}})}\Big{(}|\nabla
u({\mathbf{x}})|^{2}+\big{(}V_{0}({\mathbf{x}})+W({\mathbf{x}})\big{)}|u({\mathbf{x}})|^{2}\Big{)}|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}\geq$
$\displaystyle\hskip-14.22636pt\big{(}1-\sigma\big{)}\int_{Q_{r}({\mathbf{y}})}|\nabla
u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}+\int_{Q_{r}({\mathbf{y}})}V_{0}({\mathbf{x}})|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}=(H_{\sigma}u,u)_{Q_{r}({\mathbf{y}})}.$

This inequality and Proposition 5.10 imply the desired claims. Theorem 5.11 is
proven. ∎

###  6.7. Proof of Theorem 5.12

###### Proof.

Consider the step function

$V_{0}({\mathbf{x}})=\int_{Q_{1}(\vec{l})}V({\mathbf{s}})\,\mathrm{d}{\mathbf{s}}\quad\mathrm{for}\quad{\mathbf{x}}\in
Q_{1}(\vec{l}).$ (6.18)

In view of condition (5.19),
$\lim_{|{\mathbf{x}}|\rightarrow\infty}V_{0}({\mathbf{x}})=\infty$. Hence the
spectrum of operator
$H_{\theta}=-\theta\Delta+V_{0}({\mathbf{x}})\cdot\;(\theta>0)$ is discrete
[6]. Denote $W({\mathbf{x}})=V({\mathbf{x}})-V_{0}({\mathbf{x}})$. We see from
(6.18) that for any $\vec{l}\in{\mathbf{Z}}^{d}$
$\int_{Q_{1}(\vec{l})}W({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=0$. Furthermore,
since $v\in L_{2,\,loc}({\mathbf{R}}^{d})$, then $W\in
L_{2,\,loc}({\mathbf{R}}^{d})$. Hence in view of condition (5.18) and
Proposition A.2, for any $\vec{l}\in{\mathbf{Z}}^{d}$ divergence equation
$\mathrm{div}\vec{\Gamma}=W$ has in the cube $Q_{1}(\vec{l})$ a potential
solution $\Gamma_{\vec{l}}({\mathbf{x}})$ belonging to the space
$L_{d}(Q_{1}(\vec{l}))$ and
$\limsup_{|\vec{l}|\rightarrow\infty}\|\vec{\Gamma}_{\vec{l}}\|_{d}<\frac{1}{2^{d+1}C(d)}$.
Notice that if $r_{0}\in(0,\,1)$, any cube $Q_{r_{0}}({\mathbf{y}})$
intersects at most $2^{d}$ cubes
$Q_{1}(\vec{l})\,(\vec{l}\in{\mathbf{Z}}^{d})$. Hence perturbation
$W({\mathbf{x}})$ of the potential $V_{0}({\mathbf{x}})$ satisfies condition
(5.16) of Theorem 5.11, therefore the spectrum of H is discrete and bounded
below. Theorem 5.12 is proven. ∎

###  6.8. Proof of Theorem 5.13

###### Proof.

(i) Consider the step functiom $V_{0}({\mathbf{x}})$, defined by (6.18) and
denote $W({\mathbf{x}})=V({\mathbf{x}})-V_{0}({\mathbf{x}})$. In view of
condition (5.29) and Remark 5.14, condition (5.19) of Theorem 5.12 is
fulfilled. Furthermore, conditions (5.20), (5.21), definitions (5.22), (5.23)
imply that $W\in L_{2,\,loc}({\mathbf{R}}^{d})$. Using dilation rule for the
Fourier transform $\mathcal{F}$ on ${\mathbf{T}}^{d}$ [9], we have that for
the function $V_{r}({\mathbf{x}})$, defined by (5.22), and a natural $m$ the
equality

$\displaystyle\mathcal{F}\big{(}V_{r}(m\cdot{\mathbf{x}})\big{)}(\vec{k})=\left\\{\begin{array}[]{ll}\mathcal{F}\big{(}V_{r}\big{)}(\vec{k}/m)&\mathrm{for}\quad\vec{k}\in
m\cdot{\mathbf{Z}}^{d},\\\ 0&\mathrm{otherwise}\end{array}\right.$

is valid. Furthermore, applying Poisson formula for Fourier transform on
${\mathbf{T}}^{d}$ of the periodic summation and using dilation and
translation rules for Fourier transform on ${\mathbf{R}}^{d}$ [9], we have:

$\mathcal{F}\big{(}V_{r}\big{)}(\vec{k})=\hat{S}\big{(}(\cdot-\vec{c})/r\big{)}(\vec{k})=r^{d}\exp\big{(}-i2\pi\langle\vec{c},\,\vec{k}\rangle\big{)}\hat{S}(r\vec{k}).$

Denote by $W_{\vec{l}}({\mathbf{x}})$ and $V_{\vec{l}}({\mathbf{x}})$ the
$1$-periodic continuations (by all the variables) of of the functions
$W|_{Q_{1}(\vec{l})}$ and $V|_{Q_{1}(\vec{l})}$ from the cube $Q_{1}(\vec{l})$
to the whole ${\mathbf{R}}^{d}$. Then in view of condition (5.24)-(5.27),

$\displaystyle\Big{\|}\frac{\mathcal{F}\big{(}W_{\vec{l}}\big{)}(\vec{k})}{2\pi
i|\vec{k}|^{2}}\vec{k}\Big{\|}_{l_{d^{\prime}}}=\Big{\|}\frac{\mathcal{F}\big{(}V_{\vec{l}}\big{)}(\vec{k})}{2\pi
i|\vec{k}|^{2}}\vec{k}\Big{\|}_{l_{d^{\prime}}({\mathbf{Z}}^{d}\setminus\\{\vec{0}\\})}=$
$\displaystyle\mathcal{N}(\vec{l})(r(\vec{l}))^{d}\Big{(}\sum_{\vec{k}\in
m(\vec{l})\cdot{\mathbf{Z}}^{d}\setminus\\{\vec{0}\\}}\Big{(}\frac{|\hat{S}\big{(}r(\vec{l})\vec{k}/m(\vec{l})\big{)}|}{2\pi|\vec{k}|}\Big{)}^{d^{\prime}}\Big{)}^{1/d^{\prime}}=$
$\displaystyle\frac{\mathcal{N}(\vec{l})(r(\vec{l}))^{d}}{m(\vec{l})}\Big{(}\sum_{\vec{k}\in{\mathbf{Z}}^{d}\setminus\\{\vec{0}\\}}\Big{(}\frac{|\hat{S}\big{(}r(\vec{l})\vec{k}\big{)}|}{2\pi|\vec{k}|}\Big{)}^{d^{\prime}}\Big{)}^{1/d^{\prime}}\leq\frac{\mathcal{N}(\vec{l})}{m(\vec{l})}\bar{H}<\infty.$

Then condition (5.28) implies that the functions
$W_{\vec{l}}({\mathbf{x}})\,(\vec{l}\in{\mathbf{Z}}^{d})$ satisfy condition
(5.18) of Theorem 5.12. Hence the spectrum of $H$ is discrete. Claim (i) is
proven.

(ii) Let us take $S({\mathbf{x}})=\chi_{1/2}({\mathbf{x}})$. It is known that
Fourier transform of the characteristic function of a ball is expressed via
Bessel function ([7], p. 578). In particular,

$\hat{\chi}_{1/2}(\vec{\omega})=A(d)\frac{1}{(\pi|\vec{\omega}|)^{d/2}}J_{d/2}(\pi|\vec{\omega}|).,$

where $A(d)=\frac{2\sqrt{\pi}v_{d-1}\Gamma\big{(}(d-1)/2\big{)}}{2^{d/2}}$ and
$v_{d-1}$ is the volume of $d-1$-dimensional unit ball. It is known that for
$\nu>0$ Bessel function $J_{\nu}(x)$ is continuous in $[]0,\,+\infty)$ and the
following asymptotic formula is valid for $x\rightarrow+\infty$:

$J_{\nu}(x)=\sqrt{\frac{2}{\pi
x}}\cos\big{(}x-\pi\nu/2-\pi/4\big{)}+O\big{(}\frac{1}{x^{3/2}}\big{)}$

([7]. pp.168, 580). Hence there is a constant $\bar{J}(\nu)>0$ such that
$|J_{\nu}(x)|\leq\frac{\hat{J}(\nu)}{\sqrt{x}}$ for any $x>0$. Therefore for
some constant $B(d)>0$ and any
$\vec{k}\in{\mathbf{Z}}^{d}\setminus\\{\vec{0}\\}$, $r\in(0,\,1)$

$r^{d}\frac{\hat{S}(r\vec{k})}{2\pi|\vec{k}|}\leq
B(d)\frac{r^{(d-1)/2}}{|\vec{k}|^{(d+3)/2}}\leq\frac{B(d)}{|\vec{k}|^{(d+3)/2}}.$

Since the series
$\sum_{\vec{k}\in{\mathbf{Z}}^{d}\setminus\\{\vec{0}\\}}\Big{(}\frac{1}{|\vec{k}|^{(d+3)/2}}\Big{)}^{d^{\prime}}$
converges for $d<5$, condition (5.24) is satisfied for $d\in\\{3,\,4\\}$.
Claim (ii) is proven. Hence Theorem 5.13 is proven too. ∎

###  6.9. Proof of Theorem 5.18

###### Proof.

Assume that $F\in\mathcal{N}_{\tilde{\gamma}(r)}({\mathbf{y}},r,\Omega)$,
where $\tilde{\gamma}(r)=\big{(}\gamma(r)/G\big{)}^{(d-2)/d}$ and
$G:=\Big{(}\frac{\bar{r}}{r_{m}}\Big{)}^{d}$, $\bar{r}:=\max_{\omega\in
S^{d-1}}r(\omega)$, $r_{m}:=\min_{\omega\in S^{d-1}}r(\omega)$. Notice that
inclusion (5.32), is equivalent to
$E\subseteq\bar{\mathcal{G}}_{r}({\mathbf{y}})\cap\Omega$, where
$E=\bar{\mathcal{G}}_{r}({\mathbf{y}})\setminus F$. Furthermore, in the proof
of Theorem 5.1 we have shown using the isocapacity inequality that condition
(1.3) implies
$\mathrm{mes}_{d}(F)\leq\gamma(r)\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))$,
i.e.,
$\mathrm{mes}_{d}(E)\geq(1-\gamma(r))\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))$.
In particular, these arguments imply that if
$\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega)<(1-\gamma(r))\,\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))$,
then $\mathcal{N}_{\tilde{\gamma}(r)}({\mathbf{y}},r,\Omega)=\emptyset$, i.e.,

$\inf_{F\in\mathcal{N}_{\tilde{\gamma}(r)}({\mathbf{y}},r,\Omega)}\int_{{\mathcal{G}}_{r}({\mathbf{y}})\setminus
F}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=+\infty.$

Then in view of definition (5.43), these circumstances mean that

$\inf_{F\in\mathcal{N}_{\tilde{\gamma}(r)}({\mathbf{y}},r,\Omega)}\int_{{\mathcal{G}}_{r}({\mathbf{y}})\setminus
F}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}\geq
R_{V}({\mathbf{y}},r,\Omega,\gamma(r)).$

This estimate, condition (5.44) and Theorem 5.17 imply the desired claim.
Theorem 5.18 is proven ∎

###  6.10. Proof of Corollary 5.19

###### Proof.

It is clear that the function $\tilde{\gamma}(r)=\gamma(r)/2$ satisfies
condition (5.1). Furthermore, we have from (5.45):

$\limsup_{|{\mathbf{y}}|\rightarrow\infty}\frac{\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega)}{\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))}<1-\tilde{\gamma}(r).$

This circumstance, definition (5.43) and Theorem 5.18 imply the desired claim.
Corollary 5.19 is proven. ∎

###  6.11. Proof of Theorem 5.21

###### Proof.

Assume that
$\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega)>(1-\gamma(r)/2)\,\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))$.
In view of estimate (5.10) (Theorem 5.7) with
$t=1-\bar{\gamma}(\gamma(r),\Omega)$ and
$\sigma(t)=2t-1=1-2\bar{\gamma}(\gamma(r),\Omega)$.

$\displaystyle\hskip-22.76219pt\inf_{E\in\mathcal{P}_{\gamma(r)/2}({\mathbf{y}},r,\Omega)}\int_{E}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=\mathrm{mes}_{d}\big{(}\mathcal{G}_{r}({\mathbf{y}})\cap\Omega\big{)}\times$
$\displaystyle\hskip-22.76219pt\inf_{E\in\mathcal{P}_{\gamma(r)/2}({\mathbf{y}},r,\Omega)}\int_{E}V({\mathbf{x}})\,m_{d,r,\Omega}(\mathrm{d}{\mathbf{x}})\geq$
$\displaystyle\hskip-22.76219pt\mathrm{mes}_{d}\big{(}\mathcal{G}_{r}({\mathbf{y}})\cap\Omega\big{)}\frac{1+\sigma(t)}{2}\Big{(}\mathrm{E}_{{\mathbf{y}},r,\Omega}(V)-\sqrt{\frac{1-\sigma(t)}{1+\sigma(t)}}\cdot\mathrm{Dev}_{{\mathbf{y}},r,\Omega}(V)\Big{)}\geq$
$\displaystyle\hskip-22.76219pt\mathrm{mes}_{d}\big{(}\mathcal{G}_{r}({\mathbf{y}})\big{)}(1-\gamma(r)/2)\Big{(}\mathrm{E}_{{\mathbf{y}},r,\Omega}(V)-\sqrt{2\bar{\gamma}(\gamma(r),\Omega)}\cdot\mathrm{Dev}_{{\mathbf{y}},r,\Omega}(V)\Big{)}.$

Notice that if
$\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}})\cap\Omega)=(1-\gamma(r)/2)\,\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))$
(i.e., $\bar{\gamma}(\gamma(r),\Omega)=0$), then

$\mathrm{E}_{{\mathbf{y}},r,\Omega}(V)=\frac{\int_{\mathcal{G}_{r}({\mathbf{y}})\cap\Omega}V({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}}{(1-\gamma(r)/2)\,\mathrm{mes}_{d}(\mathcal{G}_{r}({\mathbf{y}}))}$

These circumstances, condition (5.53) and Theorem 5.18 (with $\gamma(r)/2$)
instead of $\gamma(r)$) imply the desired claim. Theorem 5.21 is proven. ∎

##  7\. Some examples

First of all, let us recall some examples of the $(\log_{m},\,\theta)$-dense
system (Definition 3.2), constructed in [22].

######  Example 1.

Consider the classical middle third Cantor set $\mathcal{C}\subset[0,1]$, Let
$I_{n,k}\;(n=1,2,\dots),\,k=1,2,\dots,2^{n-1}$ be the closures of intervals
adjacent to $\mathcal{C}$. It is known that they are pairwise disjoint and for
any fixed $n$ and each $k\in\\{1,2,\dots,2^{n-1}\\}$
$\mathrm{mes}_{1}(I_{n,k})=3^{-n}$. For fixed $n$ we shall number the
intervals $I_{n,k}$ from the left to the right. Denote
$D_{n}=\bigcup_{k=1}^{2^{n-1}}I_{n,k}$. As we have shown in [22] (Example
5.1), the sequence $\\{D_{n}\\}_{n=1}^{\infty}$ forms in $[0,1]$ a
$(\log_{3},\,1/9)$-dense system.

######  Example 2.

Consider a cube $Q_{1}({\mathbf{y}})\subset{\mathbf{R}}^{d}$, represented in
the form $Q_{1}({\mathbf{y}})=Q_{1}({\mathbf{y}}_{1})\times
Q_{1}({\mathbf{y}}_{2})$, where
$Q_{1}({\mathbf{y}}_{1})\subset{\mathbf{R}}^{d_{1}}$ and
$Q_{1}({\mathbf{y}}_{2})\subset{\mathbf{R}}^{d_{2}}$. Let
$\\{D_{n}\\}_{n=1}^{\infty}$ be a sequence of subsets of the cube
$Q_{1}({\mathbf{y}}_{1})$, forming in it a $(\log_{m},\theta)$-dense system.
It is easy to see that the sequence $\\{D_{n}\times
Q_{1}({\mathbf{y}}_{2})\\}_{n=1}^{\infty}$ forms in $Q_{1}({\mathbf{y}})$ a
$(\log_{m},\theta)$-dense system too.

Now we shall consider counterexamples connected with conditions of
discreteness of the spectrum of the operator $H=-\Delta+V({\mathbf{x}})\cdot$,
obtained above.

######  Example 3.

In [22] we have constructed an example of the potential $V({\mathbf{x}})\geq
0$ which satisfies conditions of Theorem 5.6 (hence the spectrum of the
operator $H=-\Delta+V({\mathbf{x}})\cdot$ is discrete) , but the condition
(5.4) of the criterion from [5], formulated in Remark 5.5, is not satisfied
for it. In view of the technical mistake, admitted in [22] and pointed in
Remark 5.2, we shall recall and correct this construction. Let us return to
the sequence $\\{D_{n}\\}_{n=1}^{\infty}$ of subsets of the interval $[0,\,1]$
and considered in Example 1, and the following sequence of subsets of the cube
$Q_{1}(0)$:

$\mathcal{D}_{n}=D_{n}\times[0,\,1]^{d-1}.$ (7.1)

Consider also the translations of the cube $Q_{1}(0)$ and the sets
$\mathcal{D}_{n}$ by the vectors
$\vec{l}=(l_{1},l_{2},\dots,l_{d})\in{\mathbf{Z}}^{d}$:
$Q_{1}(\vec{l})=Q_{1}(0)+\\{\vec{l}\\}$,

$\mathcal{D}_{n}(\vec{l})=\mathcal{D}_{n}+\\{\vec{l}\\}.$ (7.2)

The arguments of Examples 1 and 2 imply that for any fixed
$\vec{l}\in{\mathbf{Z}}^{d}$ the sequence
$\\{\mathcal{D}_{n}(\vec{l})\\}_{n=1}^{\infty}$ forms in $Q_{1}(\vec{l})$ a
$(\log_{3},\,1/9)$-dense system. For $\beta\in(0,1)$ consider on
${\mathbf{R}}$ the $1$-periodic function $\theta_{\beta}(x)$, defined on the
interval $(0,1]$ in the following manner:

$\theta_{\beta}(x)=\left\\{\begin{array}[]{ll}1&\quad\mathrm{for}\quad
x\in(0,\beta],\\\ 0&\quad\mathrm{for}\quad x\in(\beta,1].\end{array}\right.$
(7.3)

Let us take

$\alpha\in\Big{(}0,\,2d/(d-2)\Big{)}$ (7.4)

(in [22] we have imposed the more restrictive condition $\alpha\in(0,\,2)$).
Consider the following function, defined on $(0,1]$:

$\displaystyle\Sigma_{N,\,p,\alpha}(x):=\left\\{\begin{array}[]{ll}0&\quad\mathrm{for}\quad
x\in(0,\,1]\setminus\bigcup_{n=1}^{\infty}D_{n},\\\
N\theta_{\beta}(3^{p}x)|_{\beta=3^{-\alpha n}}&\quad\mathrm{for}\quad x\in
D_{n}\;(n=1,2,\dots)\end{array}\right.$ (7.7)

$(N>0,\,p\in{\mathbf{N}})$. Consider a function
$\mathcal{N}:{\mathbf{Z}}^{d}\rightarrow{\mathbf{R}}_{+}$, satisfying the
condition

$\mathcal{N}(\vec{l})\geq
1,\quad\mathcal{N}(\vec{l})\rightarrow\infty\quad\mathrm{for}\quad|\vec{l}|_{\infty}\rightarrow\infty.$
(7.8)

Let us construct the desired potential in the following manner:

$\displaystyle
V_{\alpha}({\mathbf{x}}):=\Sigma_{N,\,p,\,\alpha}(P_{1}({\mathbf{x}}-\vec{l}))|_{N=\mathcal{N}(\vec{l}),\,p=|\vec{l}|_{\infty}+1}\quad\mathrm{for}\quad\vec{l}\in{\mathbf{Z}}^{d}$
(7.9) $\displaystyle\mathrm{and}\quad{\mathbf{x}}\in Q_{1}(\vec{l}),$

where the operator $P_{1}$ is defined in the following manner:

$\mathrm{for}\quad{\mathbf{x}}=(x_{1},x_{2},\dots,x_{d})\quad
P_{1}{\mathbf{x}}:=x_{1}.$ (7.10)

It is clear that $V_{\alpha}\in L_{\infty,\,loc}({\mathbf{R}}^{d})$. Let us
prove that the potential $V_{\alpha}({\mathbf{x}})$ satisfies all the
conditions of Theorem 5.6. Let us take a natural $n$,

$j\in\\{1,2,\dots,n\\}$ (7.11)

and a cube

$Q(\vec{\xi},\,n)\subseteq\mathcal{D}_{j}(\vec{l})$ (7.12)

of the $3$-adic partition of $Q_{1}(\vec{l})$. Let us notice that

$P_{1}\big{(}Q(\vec{\xi},\,n)\big{)}=[k\,3^{-n},\,(k+1)\,3^{-n}]$

for some $k\in{\mathbf{Z}}$. Then taking into account definitions (7.1)-(7.7),
(7.9) and the $1$-periodicity of $\theta_{\beta}(t)$, we get for
$|\vec{l}|_{\infty}>n$:

$\displaystyle\mathrm{mes}_{d}\Big{(}\big{\\{}{\mathbf{x}}\in
Q(\vec{\xi},n):\,V_{\alpha}({\mathbf{x}})>0\big{\\}}\Big{)}=$ (7.13)
$\displaystyle
3^{-(d-1)n}\mathrm{mes}_{1}\Big{(}\big{\\{}x\in[k\,3^{-n},\,(k+1)\,3^{-n}]:\,$
$\displaystyle\theta_{\beta}(3^{p}\,x)|_{\beta=3^{-\alpha
j},\,p=|\vec{l}|_{\infty}+1}>0\big{\\}}\Big{)}=$ $\displaystyle
3^{-(d-1)n}\mathrm{mes}_{1}\Big{(}\big{\\{}x\in[0,\,3^{-n}]:\,\theta_{\beta}(3^{p}\,x)|_{\beta=3^{-\alpha
j},\,p=|\vec{l}|_{\infty}+1}>0\big{\\}}\Big{)}=$
$\displaystyle\frac{3^{-(d-1)n}}{3^{|\vec{l}|_{\infty}+1}}\mathrm{mes}_{1}\Big{(}\big{\\{}t\in[0,\,3^{|\vec{l}|_{\infty}+1-n}]:\,\theta_{\beta}(t)|_{\beta=3^{-\alpha
j}}>0\big{\\}}\Big{)}=$ $\displaystyle
3^{-dn}\mathrm{mes}_{1}\Big{(}\big{\\{}t\in[0,\,1]:\,\theta_{\beta}(t)|_{\beta=3^{-\alpha
j}}>0\big{\\}}\Big{)}=$ $\displaystyle 3^{-\alpha
j}\mathrm{mes}_{d}\big{(}Q(\vec{\xi},\,n)\big{)}\geq 3^{-\alpha
n}\mathrm{mes}_{d}\big{(}Q(\vec{\xi},\,n)\big{)}.$

Therefore in view of definitions (4.13)-(4), (7.7) and (7.9),

$V_{\alpha}^{\star}\big{(}3^{-\alpha
n},\,\vec{\xi},\,n\big{)}\geq\mathcal{N}(\vec{l}),$

if conditions (7.11) and (7.12) are satisfied. This estimate and conditions
(7.4), (7.8)-b imply that condition (5.7) of Theorem 5.6 is satisfied for the
potential $V_{\alpha}({\mathbf{x}})$ with $\gamma(r)=r^{\alpha}$ satisfying
condition (5.1). Hence the spectrum of the operator
$H=-\Delta+V_{\alpha}({\mathbf{x}})\cdot$ is discrete.

######  Example 4.

We shall construct an example of the potential $V({\mathbf{x}})\geq 0$ which
satisfies conditions of Theorem 5.8 (hence the spectrum of the operator
$H=-\Delta+V({\mathbf{x}})\cdot$ is discrete) , but the condition (5.4) of the
criterion from [5], formulated in Remark 5.5, is not satisfied for it.
Consider the following partition of the interval $[0,1]$:
$[0,1]=\bigcup_{n=1}^{\infty}I_{n}$, where

$I_{n}=(2^{-n},\,2^{-n+1}]\quad(n=1,2,3,\dots)..$ (7.14)

Let us take

$\alpha\in(0,\;2d/(d-2))$ (7.15)

and consider the following function, defined on $(0,1]$:

$\displaystyle\hskip
28.45274pt\Psi_{N,\,p,\alpha}(x):=N\theta_{\beta}(px)|_{\beta=2^{-\alpha
n}}\quad\mathrm{for}\quad x\in I_{n}\;(n=1,2,\dots),$ (7.16)

where $\theta_{\beta}(x)$ is the $1$-periodic function, defined on $(0,1])$ by
(7.3). Let us construct the desired potential in the following manner:

$\displaystyle
V_{\alpha}({\mathbf{x}}):=\Psi_{N,\,p,\,\alpha}(P_{1}({\mathbf{x}}-\vec{l}))|_{N=\mathcal{N}(\vec{l}),\,p=|\vec{l}|_{\infty}+1}\quad\mathrm{for}\quad\vec{l}\in{\mathbf{Z}}^{d}$
(7.17) $\displaystyle\mathrm{and}\quad{\mathbf{x}}\in Q_{1}(\vec{l}),$

where the operator $P_{1}$ is defined by (7.10) and the function
$\mathcal{N}(\vec{l})$ has the properties (7.8). It is clear that
$V_{\alpha}\in L_{\infty,\,loc}({\mathbf{R}}^{d})$. Let us show that the
potential $V_{\alpha}({\mathbf{x}})$ satisfies condition (5.12) of Theorem
5.8. Consider a cube $Q_{r}({\mathbf{y}})\;(r\in(0,1])$. Assume initially that

$Q_{r}({\mathbf{y}})\subseteq
Q_{1}(\vec{l})\quad({\mathbf{y}}=(y_{1},y_{2},\dots,y_{d}),\;\vec{l}=(l_{1},l_{2},\dots,l_{d})).$

Denote

$F_{\delta}(x)=x-\sqrt{\delta}\cdot\sqrt{x-x^{2}}\quad(x\in(0,1],\;\delta>0)$
(7.18)

and

$\gamma(r)=Kr^{\alpha},$ (7.19)

where $K>0$ will be chosen in the sequel. Consider the probability measure on
the cube $Q_{r}({\mathbf{y}})$:

$m_{d,r}(A):=\frac{\mathrm{mes}_{d}(A)}{\mathrm{mes}_{d}(Q_{r}({\mathbf{y}}))}\quad(A\in\Sigma_{L}(Q_{r}({\mathbf{y}}))).$

Recall that we denote by $\mathrm{E}_{{\mathbf{y}},r}(W)$ and
$\mathrm{Dev}_{{\mathbf{y}},r}(W)$ the expectation and deviation of a real-
valued function $W\in L_{2}(Q_{r}({\mathbf{y}}),\,m_{d,r})$ (definitions
(5.8), (5.2)). In view of definitions (7.16), (7.17), (7.18) and Lemma 7.1, we
have:

$\displaystyle\hskip-42.67912pt\mathrm{E}_{{\mathbf{y}},r}(V_{\alpha})-\sqrt{\gamma(r)}\cdot\mathrm{Dev}_{{\mathbf{y}},r}(V_{\alpha})=$
$\displaystyle\hskip-42.67912pt\mathcal{N}(\vec{l})\cdot
F_{\gamma(r)}\Big{(}\sum_{n=1}^{\infty}\mu_{d,r}\big{(}\\{{\mathbf{x}}\in
Q_{r}({\mathbf{y}})\cap
D_{n}(\vec{l}):\;V_{\alpha}({\mathbf{x}})>0\\}\big{)}\Big{)}=$
$\displaystyle\hskip-42.67912pt\mathcal{N}(\vec{l})\cdot
F_{\gamma(r)}\Big{(}\sum_{n=1}^{\infty}2^{-\alpha
n}\frac{\mathrm{mes}_{1}\big{(}[y_{1}-l_{1},\,y_{1}-l_{1}+r]\cap
I_{n}\big{)}+\theta(n,\vec{l})}{r}\Big{)},$ (7.20)

where $[y_{1},\,y_{1}+r]\subseteq[l_{1},\,l_{1}+1]$ and the quantity
$\theta(n,\vec{l})$ satisfies the estimate:

$|\theta(n,\vec{l})|\leq\frac{2\cdot 2^{-\alpha n}}{|\vec{l}|_{\infty}+1}.$

Let $m_{g}$ be the measure on $[0,\,1]$, whose density $g(x)$ with respect to
$\mathrm{mes}_{1}$ is

$g(x)=2^{-\alpha n}\quad\mathrm{for}\quad x\in I_{n}.$ (7.21)

Then we have:

$\sum_{n=1}^{\infty}2^{-\alpha
n}\mathrm{mes}_{1}\big{(}[y_{1}-l_{1},\,y_{1}-l_{1}+r]\cap
I_{n}\big{)}=m_{g}\big{(}[y_{1}-l_{1},\,y_{1}-l_{1}+r]\big{)}.$ (7.22)

On the other hand, by claim (i) of Lemma 7.2,

$\displaystyle\hskip-17.07164ptF_{\gamma(r)}\Big{(}\frac{1}{r}m_{g}\big{(}[y_{1}-l_{1},\,y_{1}-l_{1}+r]\big{)}+2^{-\alpha
n}\frac{\theta(n,\vec{l})}{r}\Big{)}\geq$ (7.23)
$\displaystyle\hskip-17.07164ptF_{\gamma(r)}\Big{(}\frac{1}{r}m_{g}\big{(}[y_{1}-l_{1},\,y_{1}-l_{1}+r]\big{)}\Big{)}-\big{(}1+\sqrt{2\gamma(r)}\big{)}\sqrt{\frac{2}{r(1-2^{-\alpha})(|\vec{l}|_{\infty}+1)}}.$

Notice that in view of (7.21), (7.14), the function $g(x)$ is monotone
nondecreasing. Hence

$\displaystyle
m_{g}\big{(}[y_{1}-l_{1},\,y_{1}-l_{1}+r]\big{)}=\int_{y_{1}-l_{1}}^{y_{1}-l_{1}+r}g(x)\,\mathrm{d}x\geq$
$\displaystyle\int_{0}^{r}g(x)\,\mathrm{d}x=m_{g}\big{(}[0,r]\big{)}=\sum_{n=1}^{\infty}2^{-\alpha
n}\mathrm{mes}_{1}\big{(}[0,r]\cap I_{n}\big{)}.$

Let $n_{0}$ be the natural number such that that $2^{-n_{0}}\leq r<2\cdot
2^{-n_{0}}$. Then

$\displaystyle\sum_{n=1}^{\infty}2^{-\alpha
n}\frac{\mathrm{mes}_{1}\big{(}[0,r]\cap
I_{n}\big{)}}{r}\geq\sum_{n=n_{0}}^{\infty}2^{-\alpha
n}\frac{2^{-n+1}}{r}=\frac{2}{r}\cdot\frac{2^{-(\alpha+1)n_{0}}}{1-2^{-\alpha-1}}\geq$
$\displaystyle\frac{2}{r}\cdot\frac{(r/2)^{-(\alpha+1)}}{1-2^{-\alpha-1}}=\frac{2}{2^{\alpha+1}-1}r^{\alpha}.$

The last estimates, definition (7.19) and claims (ii), (iii) of Lemma 7.2
imply that if $0<K<\frac{2}{2^{\alpha+1}-1}$, then

$F_{\gamma(r)}\Big{(}\frac{2}{2^{\alpha+1}-1}r^{\alpha}\Big{)}>0$

and

$\displaystyle
F_{\gamma(r)}\Big{(}\frac{1}{r}m_{g}\big{(}[y_{1}-l_{1},\,y_{1}-l_{1}+r]\big{)}\Big{)}\geq
F_{\gamma(r)}\Big{(}\frac{2}{2^{\alpha+1}-1}r^{\alpha}\Big{)}.$

Hence in view of (4), (7.22) and (7.23),

$\displaystyle\mathrm{E}_{{\mathbf{y}},r}(V_{\alpha})-\sqrt{\gamma(r)}\cdot\mathrm{Dev}_{{\mathbf{y}},r}(V_{\alpha})\geq$
$\displaystyle\mathcal{N}(\vec{l})\cdot\Big{(}F_{\gamma(r)}\Big{(}\frac{2}{2^{\alpha+1}-1}r^{\alpha}\Big{)}-\big{(}1+\sqrt{2\gamma(r)}\big{)}\sqrt{\frac{2}{r(1-2^{-\alpha})(|\vec{l}|_{\infty}+1)}}\Big{)}.$

Now consider the case where the inclusion $Q_{r}({\mathbf{y}})\subseteq
Q_{1}(\vec{l})$ is not valid. It is clear that the cube $Q_{r}({\mathbf{y}})$
intersects not more than $2^{d}$ adjacent cubes
$Q_{1}(\vec{l}_{k})\,(\vec{l}_{k}\in{\mathbf{Z}}^{d},\,k=1,2,\dots,\nu)$ and
among them there is a cube $Q_{1}(\vec{l}_{k})$ such that for some
$\tilde{\mathbf{y}}\in Q_{1}(\vec{l}_{k})$:
$Q_{r/2}(\tilde{\mathbf{y}})\subseteq Q_{1}(\vec{l}_{k})\cap
Q_{r}({\mathbf{y}})$. Thus, for any ${\mathbf{y}}\in{\mathbf{R}}^{d}$ we can
choose $\tilde{\vec{l}}\in Z^{d}$ such that
$Q_{r/2}(\tilde{\mathbf{y}})\subseteq Q_{r}({\mathbf{y}})\cap
Q_{1}(\tilde{\vec{l}})$. Using again definitions (7.16), (7.17), (7.18) and
Lemma 7.1, we get:

$\displaystyle\mathrm{E}_{{\mathbf{y}},r}(V_{\alpha})-\sqrt{\gamma(r)}\cdot\mathrm{Dev}_{{\mathbf{y}},r}(V_{\alpha})=$
$\displaystyle
F_{\gamma(r)}\Big{(}\sum_{k=1}^{\nu}\mathcal{N}(\vec{l}_{k})\sum_{n=1}^{\infty}\mu_{d,r}\big{(}\\{{\mathbf{x}}\in
Q_{r}({\mathbf{y}})\cap
D_{n}(\vec{l}_{k}):\;V_{\alpha}({\mathbf{x}})>0\\}\big{)}\Big{)}$

and

$\displaystyle\sum_{k=1}^{\nu}\mathcal{N}(\vec{l}_{k})\sum_{n=1}^{\infty}\mu_{d,r}\big{(}\\{{\mathbf{x}}\in
Q_{r}({\mathbf{y}})\cap
D_{n}(\vec{l}_{k}):\;V_{\alpha}({\mathbf{x}})>0\\}\big{)}\geq$
$\displaystyle\mathcal{N}(\tilde{\vec{l}})\cdot\frac{1}{r}m_{g}\big{(}[\tilde{y}_{1}-\tilde{l}_{1},\,\tilde{y}_{1}-\tilde{l}_{1}+r/2]\big{)}-\max_{k\in\\{1,2,\dots,\nu\\}}\frac{2}{r(1-2^{-\alpha})(|\vec{l}_{k}|_{\infty}+1)},$

where $\tilde{y}_{1}=P_{1}(\tilde{\mathbf{y}})$ and
$\tilde{l}_{1}=P_{1}(\tilde{\vec{l}})$. Using the previous arguments, we can
choose $K>0$ in (7.19) small enough, such that

$\mathrm{E}_{{\mathbf{y}},r}(V_{\alpha})-\sqrt{\gamma(r)}\cdot\mathrm{Dev}_{{\mathbf{y}},r}(V_{\alpha})\geq\mathcal{N}(\tilde{\vec{l}})\cdot\Big{(}\Phi(r)-\max_{k\in\\{1,2,\dots,\nu\\}}\frac{\phi(r)}{\sqrt{|\vec{l}_{k}|_{\infty}+1}}\Big{)},$

where the quantities $\Phi(r)$ and $\phi(r)$ depend only on $r$ and they are
positive. This estimate and condition (7.8) imply that the potential
$V_{\alpha}({\mathbf{x}})$ satisfies condition (5.12). Furthermore, in view of
(7.19) and (7.4), the function $\gamma(r)$ satisfies conditions (5.1). Thus,
all the conditions of Theorem 5.8 are satisfied for the potential
$V_{\alpha}({\mathbf{x}})$, hence the spectrum of operator
$H=-\Delta+V_{\alpha}({\mathbf{x}})\cdot$ is discrete.

Let us show that the condition (5.4) of the criterion from [5] is not
satisfied for the potential $V_{\alpha}({\mathbf{x}})$. To this end for any
natural $n$ consider $\vec{l}_{n}\in{\mathbf{Z}}^{d}$ such that
$|\vec{l}_{n}|_{\infty}=n$ and take a cube

$Q(\vec{\xi}_{n},\,n+1)=Q_{2^{-n-1}}(\vec{\xi}_{n})\subset
I_{n}\times[0,1]^{d-1}+\\{\vec{l}_{n}\\}.$

In view of (7.3), (7.7), (7.9) and (7.8)-a,
$\int_{Q(\vec{\xi}_{n},\,n)}V_{\alpha}({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}>0$.
Then taking $\delta>0$, we have:

$\displaystyle\hskip-14.22636pt\big{\\{}{\mathbf{x}}\in
Q(\vec{\xi}_{n},\,n+1):\,V_{\alpha}({\mathbf{x}})\geq\frac{\delta}{\mathrm{mes}_{d}(Q(\vec{\xi}_{n},\,n+1))}\int_{Q(\vec{\xi}_{n},\,n+1)}V_{\alpha}({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}\big{\\}}\subseteq$
$\displaystyle\hskip-14.22636pt\big{\\{}{\mathbf{x}}\in
Q(\vec{\xi}_{n},\,n+1):\,V_{\alpha}({\mathbf{x}})>0\big{\\}}.$

On the other hand, in view of definitions (7.16), (7.17) and Lemma 7.1, we
get:

$\displaystyle\mathrm{mes}_{d}\Big{(}\big{\\{}{\mathbf{x}}\in
Q(\vec{\xi}_{n},\,n+1):\,V_{\alpha}({\mathbf{x}})>0\big{\\}}\Big{)}\leq$
$\displaystyle 2^{-\alpha
n}\Big{(}1+\frac{2}{n+1}\Big{)}\mathrm{mes}_{d}\big{(}Q(\vec{\xi}_{n},\,n+1)\big{)}.$

These circumstances mean that the sequence

$\frac{\mathrm{mes}_{d}\Big{(}\big{\\{}{\mathbf{x}}\in
Q(\vec{\xi}_{n},\,n+1):\,V_{\alpha}({\mathbf{x}})\geq\frac{\delta}{\mathrm{mes}_{d}(Q(\vec{\xi}_{n},\,n+1))}\int_{Q(\vec{\xi}_{n},\,n+1)}V_{\alpha}({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}\big{\\}}\Big{)}}{\mathrm{mes}_{d}\big{(}Q(\vec{\xi}_{n},\,n+1)\big{)}}$

tends to zero as $n\rightarrow\infty$. This means that condition (5.4) is not
satisfied for the potential $V_{\alpha}({\mathbf{x}})$.

In the construction of Example 4 we have used the following claims:

######  Lemma 7.1.

For an interval $[a,b]$ and $S>0$ consider the quantity

$M(S,\beta,a,b)=\mathrm{mes}_{1}\\{x\in[a,b]:\,\theta_{\beta}(Sx)>0\\},$

where $\theta_{\beta}(x)$ is defined by (7.3). It satisfies the inequalities

$M(S,\beta,a,b)\leq\beta(b-a)+2\beta/S,$ (7.24)
$M(S,\beta,a,b)\geq\beta(b-a)-2\beta/S;$ (7.25)

###### Proof.

Taking into account the $1$-periodicity of $\theta_{\beta}(x)$, we obtain:

$\displaystyle M(S,\beta,a,b)\leq M(S,\beta,[Sa]/S,([Sb]+1)/S)=$
$\displaystyle
S^{-1}M(1,\beta,[Sa],[Sb]+1)=\frac{\beta}{S}\big{(}[S]+1-[Sa]\big{)}\leq$
$\displaystyle\beta(b-a)+2\beta/S,$ $\displaystyle M(S,\beta,a,b)\geq
M(S,\beta,([Sa]+1)/S,[Sb]/S)=$ $\displaystyle
S^{-1}M(1,\beta,[Sa]+1,[Sb])=\frac{\beta}{S}\big{(}[Sb]-1-[Sa]\big{)}\geq$
$\displaystyle\beta(b-a)-2\beta/S.$

∎

######  Lemma 7.2.

The function $F_{\delta}(x)$, defined by (7.18) has the properties:

(i) for any $h\in(0,\,1)$ and $x\in(0,1-h)$

$|F_{\delta}(x+h)-F_{\delta}(x)|\leq\big{(}1+\sqrt{2\delta}\big{)}\sqrt{h};$

(ii) it is positive in $(\frac{\delta}{1+\delta},\,1]$; (iii) it is increasing
in $(\frac{\delta}{2(1+\delta+\sqrt{1+\delta})},\,1]$.

###### Proof.

(i) We have:

$F_{\delta}^{\prime}(x)=1-\sqrt{\delta}\frac{1-2x}{2\sqrt{x-x^{2}}}.$ (7.26)

Then

$|F_{\delta}(x+h)-F_{\delta}(x)|\leq\int_{x}^{x+h}|F_{\delta}^{\prime}(s)|\,\mathrm{d}s\leq
h+\frac{\sqrt{\delta}}{2}\int_{x}^{x+h}\frac{\mathrm{d}s}{\sqrt{s}\cdot\sqrt{1-s}}.$
(7.27)

Consider the case where $x\in[0,\,1/2]$. Then

$\displaystyle\int_{x}^{x+h}\frac{\mathrm{d}s}{\sqrt{s}\cdot\sqrt{1-s}}\leq\sqrt{2}\int_{x}^{x+h}\frac{\mathrm{d}s}{\sqrt{s}}=2\sqrt{2}\big{(}\sqrt{x+h}-\sqrt{x}\big{)}=$
$\displaystyle 2\sqrt{2}\frac{h}{\sqrt{x+h}+\sqrt{x}}\leq 2\sqrt{2h}.$

Now consider the case where $x\in(1/2,\,1-h]$. We have:

$\displaystyle\hskip-14.22636pt\int_{x}^{x+h}\frac{\mathrm{d}s}{\sqrt{s}\cdot\sqrt{1-s}}\leq\sqrt{2}\int_{x}^{x+h}\frac{\mathrm{d}s}{\sqrt{1-s}}=2\sqrt{2}\big{(}\sqrt{1-x}-\sqrt{1-h-x}\big{)}=$
$\displaystyle\hskip-14.22636pt2\sqrt{2}\frac{h}{\sqrt{1-x}+\sqrt{1-h-x}}\leq
2\sqrt{2h}.$

These estimates imply claim (i).

(ii) The representation
$F_{\delta}(x)=\frac{x^{2}-\delta(x-x^{2})}{x+\sqrt{\delta}\cdot\sqrt{x-x^{2}}}$
implies that $F_{\delta}(x)>0$ if and only if
$x\in(\frac{\delta}{1+\delta},\,1]$. Claim (ii) is proven.

(iii) We see from (7.26) that $F_{\delta}^{\prime}(x)>0$, if $x\in[1/2,\,1]$.
Consider the case where $x\in[0,\,1/2)$. The representation

$F_{\delta}^{\prime}(x)=\frac{2\sqrt{x-x^{2}}-\sqrt{\delta}(1-2x)}{2\sqrt{x-x^{2}}}=\frac{4(x-x^{2})-\delta(1-2x)^{2}}{2\sqrt{x-x^{2}}\big{(}2\sqrt{x-x^{2}}+\sqrt{\delta}(1-2x)\big{)}}$

implies that $F_{\delta}^{\prime}(x)>0$, if and only if
$x^{2}-x+\frac{\delta}{4(1+\delta)}<0$, i.e.,

$x\in\Big{(}\frac{\delta}{2(1+\delta+\sqrt{1+\delta})},\,1/2\Big{)}.$

Claim (iii) is proven. ∎

######  Example 5.

The following example shows that in some cases condition (5.12) of Theorem 5.8
is easier verifiable than condition (5.3) of Theorem 5.3. Consider the
potential

$V({\mathbf{x}})=\mathcal{N}(\vec{l})W_{\vec{l}}({\mathbf{x}})\quad\mathrm{for}\quad{\mathbf{x}}=(x_{1},x_{2},\dots.x_{d})\in
Q_{1}(\vec{l})\quad(\vec{l}\in{\mathbf{Z}}^{d}),$

where

$W_{\vec{l}}({\mathbf{x}})=\prod_{j=1}^{d}\big{(}1-\cos(m(\vec{l})x_{j})\big{)},$

$\mathcal{N}(\vec{l})>0$, $m(\vec{l})>0$ and

$\mathcal{N}(\vec{l})\rightarrow\infty,\quad
m(\vec{l})\rightarrow\infty\quad\mathrm{for}\quad|\vec{l}|\rightarrow\infty.$
(7.28)

Consider the case where $Q_{r}({\mathbf{y}})\subseteq
Q_{1}(\vec{l})\,(r\in(0,1/2),{\mathbf{y}}=(y_{1},y_{2},\dots,y_{d}),\,\vec{l}\in{\mathbf{Z}}^{d})$.
We have:

$\mathrm{E}_{{\mathbf{y}},r}(V)-\sqrt{\gamma}\cdot\mathrm{Dev}_{{\mathbf{y}},r}(V)=\mathcal{N}(\vec{l})\big{(}\mathrm{E}_{{\mathbf{y}},r}(W_{\vec{l}})-\sqrt{\gamma}\cdot\mathrm{Dev}_{{\mathbf{y}},r}(W_{\vec{l}})\big{)}$
(7.29)

$(\gamma>0)$,

$\displaystyle\mathrm{E}_{{\mathbf{y}},r}(W_{\vec{l}})=\frac{1}{r^{d}}\int_{Q_{r}({\mathbf{y}})}W_{\vec{l}}({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=\prod_{j=1}^{d}\frac{1}{r}\int_{y_{j}}^{y_{j}+r}\big{(}1-\cos(m(\vec{l})x_{j})\big{)}\,\mathrm{d}x_{j}=$
$\displaystyle\prod_{j=1}^{d}\Big{(}1+O\big{(}\frac{1}{m(\vec{l})}\big{)}\Big{)}$

and

$\displaystyle\hskip-14.22636pt\mathrm{E}_{{\mathbf{y}},r}(W_{\vec{l}}^{2})=\frac{1}{r^{d}}\int_{Q_{r}({\mathbf{y}})}W_{\vec{l}}^{2}({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=\prod_{j=1}^{d}\frac{1}{r}\int_{y_{j}}^{y_{j}+r}\big{(}1-\cos(m(\vec{l})x_{j})\big{)}^{2}\,\mathrm{d}x_{j}=$
$\displaystyle\hskip-14.22636pt\prod_{j=1}^{d}\Big{(}3/2+O\big{(}\frac{1}{m(\vec{l})}\big{)}\Big{)}$

for $|\vec{l}|\rightarrow\infty$. Hence

$\big{(}\mathrm{Dev}_{{\mathbf{y}},r}(W_{\vec{l}})\big{)}^{2}=\mathrm{E}_{{\mathbf{y}},r}(W_{\vec{l}}^{2})-\big{(}\mathrm{E}_{{\mathbf{y}},r}(W_{\vec{l}})\big{)}^{2}=\big{(}3/2\big{)}^{d}-1+O\big{(}\frac{1}{m(\vec{l})}\big{)}.$

and

$\displaystyle\mathrm{E}_{{\mathbf{y}},r}(W_{\vec{l}})-\sqrt{\gamma}\cdot\mathrm{Dev}_{{\mathbf{y}},r}(W_{\vec{l}})=1-\sqrt{\gamma}\cdot\sqrt{\big{(}3/2\big{)}^{d}-1}+O\big{(}\frac{1}{m(\vec{l})}\big{)}.$
(7.30)

In the case, where the inclusion $Q_{r}({\mathbf{y}})\subseteq Q_{1}(\vec{l})$
is not valid for any $\vec{l}\in{\mathbf{Z}}^{d}$, the cube
$Q_{r}({\mathbf{y}})$ intersects not more than $2^{d}$ adjacent cubes
$Q_{1}(\vec{l}_{k})\,(\vec{l}_{k}\in{\mathbf{Z}}^{d})$. Making analogous
asymptotic estimates for integrals $\int_{Q_{r}({\mathbf{y}}\cap
Q_{1}(\vec{l}_{k})}W_{\vec{l}_{k}}({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}$ and
$\int_{Q_{r}({\mathbf{y}}\cap
Q_{1}(\vec{l}_{k})}W_{\vec{l}_{k}}^{2}({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}$
and summing each of them by $k$, we obtain the asymptotic estimate with the
same leading term as in (7.30). From (7.28)-(7.30) we obtain that if
$\gamma\in\Big{(}0,\,\frac{1}{\big{(}3/2\big{)}^{d}-1}\Big{)}$, the potential
$V({\mathbf{x}})$ satisfies condition (5.12) of Theorem 5.8 with
$Q_{r}({\mathbf{y}})$ instead of $\mathcal{G}_{r}({\mathbf{y}})$ and
$\gamma(r)\equiv\gamma$. Hence the spectrum of operator
$H=-\Delta+V({\mathbf{x}})\cdot$ is discrete. Notice that in view of Remark
5.9, conditions of Theorem 5.8 imply conditions of Theorem 5.3. But the
immediate application of Theorem 5.3 to this example needs technically
complicated geometrical considerations for calculation of the non-increasing
rearrangement of the potential $V({\mathbf{x}})$.

######  Example 6.

The potential $V({\mathbf{x}})$, considered below, satisfies the conditions of
Theorem 5.3, but condition (5.12) of Theorem 5.8 is not satisfied for it. We
put

$V({\mathbf{x}})=e^{|{\mathbf{x}}|^{2}}.$ (7.31)

Since $\lim_{|{\mathbf{x}}|\rightarrow\infty}V({\mathbf{x}})=+\infty$, it is
clear that $V({\mathbf{x}})$ satisfies the conditions of Theorem 5.3. On the
other hand, the representation

$\displaystyle\mathrm{E}_{{\mathbf{y}},r}(V)-\sqrt{\gamma(r)}\cdot\mathrm{Dev}_{{\mathbf{y}},r}(V)=$
$\displaystyle-\sqrt{\mathrm{E}_{{\mathbf{y}},r}(V^{2})}\Big{(}\sqrt{\gamma(r)}\sqrt{1-\frac{(\mathrm{E}_{{\mathbf{y}},r}(V))^{2}}{\mathrm{E}_{{\mathbf{y}},r}(V^{2})}}-\frac{\mathrm{E}_{{\mathbf{y}},r}(V)}{\sqrt{\mathrm{E}_{{\mathbf{y}},r}(V^{2})}}\Big{)}$

implies that condition (5.12) is not satisfied, if

$\lim_{|{\mathbf{y}}|\rightarrow\infty}\frac{(\mathrm{E}_{{\mathbf{y}},r}(V))^{2}}{\mathrm{E}_{{\mathbf{y}},r}(V^{2})}=0.$
(7.32)

Let us take $\mathcal{G}_{r}({\mathbf{y}})=Q_{r_{1}}({\mathbf{y}}-r\vec{a})$
($r_{1}=2r/\sqrt{d}$, $\vec{a}=(d^{-1/2},d^{-1/2},\dots,d^{-1/2})$). Using
twice L’Hospital’s rule, we have:

$\displaystyle\lim_{|y|\rightarrow\infty}\frac{\big{(}\int_{y}^{y+r_{1}}e^{x^{2}}\,\mathrm{d}x\big{)}^{2}}{\int_{y}^{y+r_{1}}e^{2x^{2}}\,\mathrm{d}x}=2\lim_{|y|\rightarrow\infty}\frac{\big{(}\int_{y}^{y+r_{1}}e^{x^{2}}\,\mathrm{d}x\big{)}\big{(}e^{(y+r_{1})^{2}}-e^{y^{2}}\big{)}}{e^{2(y+r_{1})^{2}}-e^{2y^{2}}}=$
$\displaystyle\lim_{|y|\rightarrow\infty}\frac{e^{(y+r_{1})^{2}}-e^{y^{2}}}{(y+r_{1})e^{(y+r_{1})^{2}}+ye^{y^{2}}}=0.$

This result and definition (7.31) imply equality (7.32). Thus, the considered
potential does not satisfy condition (5.12).

######  Example 7.

We shall construct an example of the potential $V({\mathbf{x}})$, for which
the conditions of claim (ii) of Theorem 5.11 are satisfied (hence the spectrum
of the operator $H=-\Delta+V({\mathbf{x}})\cdot$ is discrete), but condition
(5.3) of Theorem 5.3 is not satisfied for it. Suppose that the
$(\log_{3},\,1/9)$-dense systems $\mathcal{D}_{n}(\vec{l})$ in
$Q_{1}(\vec{l})$ $(\vec{l}\in{\mathbf{Z}}^{d})$ and the potential
$V_{\alpha}({\mathbf{x}})$ are the same as in Example 3. Assume that

$\log_{3}2<\alpha<2d/(d-2).$ (7.33)

Consider a monotone non-decreasing function $\psi:\,(0,\,1)\rightarrow(0,\,1)$
satisfying the condition

$\lim_{r\downarrow 0}r^{-2d/(d-2)}\psi(r)<\infty$ (7.34)

and the function

$\displaystyle\hskip-14.22636pt\tilde{\Sigma}_{N,\,p,\,\psi}(x):=$
$\displaystyle\hskip-14.22636pt\left\\{\begin{array}[]{ll}0&\quad\mathrm{for}\quad
x\in[0,\,1]\setminus\bigcup_{n=1}^{\infty}D_{n},\\\ N\frac{3^{-\alpha
n}}{\beta}\theta_{\beta}(3^{p}x)|_{\beta=\psi(3^{-(n+1)})}&\quad\mathrm{for}\quad
x\in D_{n}\;(n=1,2,\dots)\end{array}\right.$ (7.37)

$(N>0,\,p\in{\mathbf{N}})$, where $\theta_{\beta}(x)$ is defined by (7.3).
Consider the potential

$\displaystyle
V_{\psi}({\mathbf{x}}):=\tilde{\Sigma}_{N,\,p,\,\psi}(P_{1}({\mathbf{x}}-\vec{l}))|_{N=\mathcal{N}(\vec{l}),\,p=|\vec{l}|_{\infty}+1}\quad\mathrm{for}\quad\vec{l}\in{\mathbf{Z}}^{d}$
$\displaystyle\mathrm{and}\quad{\mathbf{x}}\in Q_{1}(\vec{l}),$ (7.38)

where the function $\mathcal{N}(\vec{l})$ satisfies conditions (7.8). Recall
that the operator $P_{1}$ is defined by (7.10). Let $I_{n,k}\,(1\leq k\leq
2^{n-1})$ be the intervals forming the set $D_{n}$, defined in Example 3.
Taking into account the left inequality of (7.33) and using Lemma 7.1, we
have:

$\displaystyle\hskip-14.22636pt\int_{Q_{1}(\vec{l})}V_{\psi}({\mathbf{x}})\,d{\mathbf{x}}=\mathcal{N}(\vec{l})\sum_{n=1}^{\infty}\sum_{k=1}^{2^{n-1}}\int_{I_{n,k}}\frac{3^{-\alpha
n}}{\beta}\theta_{\beta}(3^{|\vec{l}|_{\infty}+1}x)|_{\beta=\psi(3^{-(n+1)})}\,dx\leq$
$\displaystyle\hskip-14.22636pt\mathcal{N}(\vec{l})\sum_{n=1}^{\infty}2^{n-1}\Big{(}3^{-(\alpha+1)n}+\frac{2}{3^{|\vec{l}|_{\infty}+1}}3^{-\alpha
n}\Big{)}<\infty.$

Hence $V_{\psi}\in L_{1,\,loc}({\mathbf{R}}^{d})$. Let us prove that the
potential $V_{\psi}({\mathbf{x}})$ satisfies the conditions of claim (ii) of
Theorem 5.11. Denote
$W_{\psi,\alpha}({\mathbf{x}})=V_{\psi}({\mathbf{x}})-V_{\alpha}({\mathbf{x}})$.
Then, in view of (7.9),

$\displaystyle
W_{\psi,\alpha}({\mathbf{x}}):=\hat{\Sigma}_{N,\,p,\,\psi,\,\alpha}(P_{1}({\mathbf{x}}-\vec{l}))|_{N=\mathcal{N}(\vec{l}),\,p=|\vec{l}|_{\infty}+1}\quad\mathrm{for}\quad\vec{l}\in{\mathbf{Z}}^{d}$
$\displaystyle\mathrm{and}\quad{\mathbf{x}}\in Q_{1}(\vec{l}),$ (7.39)

where

$\displaystyle\hat{\Sigma}_{N,\,p,\,\psi,\,\alpha}(x):=$
$\displaystyle\left\\{\begin{array}[]{ll}0&\quad\mathrm{for}\quad
x\in[0,\,1]\setminus\bigcup_{n=1}^{\infty}D_{n},\\\
N\tilde{\theta}_{\psi,\alpha,n}(3^{p}x)&\quad\mathrm{for}\quad x\in
D_{n}\;(n=1,2,\dots),\end{array}\right.$ (7.42)

and

$\tilde{\theta}_{\psi,\alpha,n}(t)=\frac{3^{-\alpha
n}}{\beta}\theta_{\beta}(t)|_{\beta=\psi(3^{-(n+1)})}-\theta_{\delta}(t)|_{\delta=3^{-\alpha
n}}.$ (7.43)

Then, in view of (7.3),

$\int_{0}^{1}\tilde{\theta}_{\psi,\alpha,n}(t)\,dt=0.$ (7.44)

Consider the following vector field in the cube
$Q_{1}({\mathbf{y}})\,({\mathbf{y}}\in{\mathbf{R}}^{d})$:

$\vec{\Gamma}_{{\mathbf{y}}}({\mathbf{x}})=\Big{(}\int_{P_{1}{\mathbf{y}}}^{P_{1}{\mathbf{x}}}W_{\psi,\alpha}(\xi,\,(I-P_{1}){\mathbf{x}})\,d\xi,\,0,\dots,0\Big{)}.$
(7.45)

By Proposition A.1, it is a generalized solution of the equation
$\mathrm{div}(\vec{\Gamma}_{{\mathbf{y}}}({\mathbf{x}}))=W_{\psi,\alpha}({\mathbf{x}})$
in the cube $Q_{1}({\mathbf{y}})$. Consider the cubes
$\\{Q_{1}(\vec{l}_{i})\\}_{i=1}^{K}\;(\vec{l}_{i}\in{\mathbf{Z}}^{d})$
intersecting the cube $Q_{1}({\mathbf{y}})$. It is clear that $K\leq 2^{d}$.
Denote
$J({\mathbf{y}},i)=[P_{1}{\mathbf{y}},\,P_{1}{\mathbf{y}}+1]\cap[P_{1}\vec{l}_{i},\,P_{1}\vec{l}_{i}+1]$,
$J({\mathbf{y}},i,n,k)=J({\mathbf{y}},i)\cap(I_{n,k}+P_{1}\vec{l}_{i})$. Then,
in view of definitions (7), (7) and (7.43), property (7.44) and Lemma 7.3, we
have for ${\mathbf{x}}\in Q_{1}({\mathbf{y}})\cap Q_{1}(\vec{l}_{i})$

$\displaystyle|\vec{\Gamma}_{{\mathbf{y}}}({\mathbf{x}})|=\Big{|}\int_{J({\mathbf{y}},i)}W_{\psi,\alpha}(\xi,\,(I-P_{1}){\mathbf{x}})\,d\xi\Big{|}\leq$
$\displaystyle\sum_{n=1}^{\infty}\sum_{k=1}^{2^{n-1}}\Big{|}\int_{J({\mathbf{y}},i.n,k)}W_{\psi,\alpha}(\xi,\,(I-P_{1}){\mathbf{x}})\,d\xi\Big{|}=$
$\displaystyle\mathcal{N}(\vec{l}_{i})\sum_{n=1}^{\infty}\sum_{k=1}^{2^{n-1}}\Big{|}\int_{J({\mathbf{y}},i.n,k)}\tilde{\theta}_{\psi,\alpha,n}(3^{p}\xi)|_{p=|\vec{l}_{i}|_{\infty}+1}\,d\xi\Big{|}\leq$
$\displaystyle\frac{2\mathcal{N}(\vec{l}_{i})}{3^{|\vec{l}_{i}|_{\infty}+1}}\sum_{n=1}^{\infty}\sum_{k=1}^{2^{n-1}}\int_{0}^{1}|\tilde{\theta}_{\psi,\alpha,n}(t)|\,dt\leq\frac{2\mathcal{N}(\vec{l}_{i})}{3^{|\vec{l}_{i}|_{\infty}+1}}\sum_{n=1}^{\infty}\Big{(}\frac{2}{3^{\alpha}}\Big{)}^{n}.$
(7.46)

The left inequality of (7.33) implies that the last series in (7) converges.
Assume that along with conditions (7.8) the function $\mathcal{N}(\vec{l})$
satisfies the condition

$\frac{\mathcal{N}(\vec{l})}{3^{|\vec{l}|_{\infty}}}\rightarrow
0\quad\mathrm{as}\quad|\vec{l}|_{\infty}\rightarrow\infty,$ (7.47)

Then estimate (7) implies that

$\|\vec{\Gamma}\|_{L_{d}(Q_{r}({\mathbf{y}}))}\leq\sup_{{\mathbf{x}}\in
Q_{1}({\mathbf{y}})}|\vec{\Gamma}_{{\mathbf{y}}}({\mathbf{x}})|\rightarrow
0\quad\mathrm{as}\quad|{\mathbf{y}}|\rightarrow\infty,$

hence condition (5.16) is satisfied with $W=W_{\psi,\alpha}$. On the other
hand, we have shown in Example 3 that the right inequality of (7.33) and
condition (7.8) imply discreteness of the spectrum of operator
$-\Delta+V_{\alpha}({\mathbf{x}})\cdot$. These circumstances imply that the
conditions of claim (ii) of Theorem 5.11 are satisfied for the potential
$V_{\psi}({\mathbf{x}})$. Thus, under conditions (7.8), (7.33) and (7.47) the
spectrum of operator $-\Delta+V_{\psi}({\mathbf{x}})\cdot$ is discrete.

Let us show that the potential $V_{\psi}({\mathbf{x}})$ does not satisfy
condition (5.3) of Theorem 5.3. Let us take a function $\hat{\gamma}(r)$
satisfying condition (5.1) for some $r_{0}>0$. Then by (7.34),
$\limsup_{r\downarrow 0}\frac{\hat{\gamma}(r)}{\psi(r)}=\infty$. Hence for
some positive decreasing sequence $\\{r_{j}\\}_{j=1}^{\infty}$ tending to zero

$\lim_{j\rightarrow\infty}\frac{\hat{\gamma}(r_{j})}{\psi(r_{j})}=\infty.$
(7.48)

Let us choose an increasing sequence of natural numbers
$\\{n_{j}\\}_{j=1}^{\infty}$ such that

$3^{-(n_{j}+1)}\leq r_{j}<3^{-n_{j}}.$ (7.49)

Consider the vectors $\vec{l}\in{\mathbf{Z}}^{d}$ of the form
$\vec{l}=(l,0,\dots,0)$, where $l\in{\mathbf{Z}}$ will be chosen below.
Consider the intervals $I_{n_{j},1}=[a_{n_{j},1},\,b_{n_{j},1}]\subset
D_{n_{j}}$ and the cubes $Q_{j,l}=Q_{3^{-n_{j}}}(\tilde{\mathbf{y}}_{j,l})$,
where $\tilde{\mathbf{y}}_{j,l}=\big{(}a_{n_{j},1}+l,\,0,\dots,0\big{)}$. Then
$P_{1}(Q_{j,l})=I_{n_{j},1}+l$. In view of the right inequality of (7.49),
$Q_{r_{j}}(\tilde{\mathbf{y}}_{j,l})\subset Q_{j,l}$. Then using definitions
(7.43) and (7), Lemma 7.1, the left inequality of (7.49) and taking $l=n_{j}$,
we have:

$\displaystyle\mathrm{mes}_{d}\big{(}\\{{\mathbf{x}}\in
Q_{r_{j}}(\tilde{\mathbf{y}}_{j,l}):\,V_{\psi}({\mathbf{x}})>0\\}\big{)}\leq\mathrm{mes}_{d}\big{(}\\{{\mathbf{x}}\in
Q_{j,l}:\,V_{\psi}({\mathbf{x}})>0\\}\big{)}=$ $\displaystyle
3^{-n_{j}(d-1)}\mathrm{mes}_{1}\big{(}\\{x\in
I_{n_{j},1}:\,\theta_{\beta}(3^{l+1}x)|_{\beta=\psi(3^{-(n_{j}+1)})}\\}\big{)}\leq$
$\displaystyle\psi(3^{-(n_{j}+1)})(3^{-n_{j}d}+3\cdot
3^{-n_{j}(d-1)}3^{-l-1})=$ $\displaystyle
2\psi(3^{-(n_{j}+1)})3^{-(n_{j}+1)d}3^{d}\leq 2\cdot
3^{d}\psi(r_{j})\mathrm{mes}_{d}(Q_{r_{j}}(\tilde{\mathbf{y}}_{j,l})).$ (7.50)

On the other hand, in view of (7.48),

$\exists\,J>0\quad\forall\,j\geq J:\quad 2\cdot
3^{d}\psi(r_{j})<\hat{\gamma}(r_{j}).$

This circumstance, estimate (7) and definition (4.13)-(4) imply that

$\forall\,j\geq
J:\quad\bar{V}_{\psi}^{\star}\big{(}\hat{\gamma}(r_{j})\mathrm{mes}_{d}(Q_{r_{j}}({\mathbf{y}}_{j})),\,Q_{r_{j}}({\mathbf{y}}_{j})\big{)}=0,$

where ${\mathbf{y}}_{j}=\tilde{\mathbf{y}}_{j,n_{j}}$. This means that the
potential $V_{\psi}({\mathbf{x}})$ does not satisfy condition (5.3).

In the construction of Example 7 we have used the following claim:

######  Lemma 7.3.

Let $\tilde{\theta}:\,{\mathbf{R}}\rightarrow{\mathbf{R}}$ be $1$-periodic
locally integrable function such that $\int_{0}^{1}\tilde{\theta}(t)\,dt=0$.
Then for any interval $[a,b]$ and $S>0$ the inequality

$|\int_{a}^{b}\tilde{\theta}(Sx)\,dx|\leq\frac{2}{S}\int_{0}^{1}|\tilde{\theta}(t)|\,dt$

is valid.

###### Proof.

The representation

$\displaystyle\int_{a}^{b}\tilde{\theta}(Sx)\,dx=\frac{1}{S}\int_{Sa}^{Sb}\tilde{\theta}(t)\,dt=\frac{1}{S}\Big{(}\int_{[Sa]}^{[Sb]}\tilde{\theta}(t)\,dt-\int_{[Sa]}^{Sa}\tilde{\theta}(t)\,dt+$
$\displaystyle\int_{[Sb]}^{Sb}\tilde{\theta}(t)\,dt\Big{)}=\frac{1}{S}\Big{(}-\int_{[Sa]}^{Sa}\tilde{\theta}(t)\,dt+\int_{[Sb]}^{Sb}\tilde{\theta}(t)\,dt\Big{)}$

implies the desired estimate. ∎

##  Appendix A Some classes of solutions of the divergence equation

In this section we consider a class of generalized solutions of the divergence
equation

$\mathrm{div}\vec{\Gamma}=W$ (A.1)

in the cube $Q_{r}({\mathbf{y}})$ Here $W$ is a distribution on
$Q_{r}({\mathbf{y}})$. Recall that a vector distribution $\vec{\Gamma}$ is
called a generalized solution of the above equation, if it satisfies (A.1) in
the distributional sense. In particular, this means, that if $W\in
L_{1}(Q_{r}({\mathbf{y}}))$ and $\vec{\Gamma}\in L_{1}(Q_{r}({\mathbf{y}})$,
then

$\displaystyle\forall\,\phi\in C_{0}^{\infty}(Q_{r}({\mathbf{y}})):$
$\displaystyle\int_{Q_{r}({\mathbf{y}})}W({\mathbf{x}})\phi({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=-\int_{Q_{r}({\mathbf{y}})}\langle\vec{\Gamma}({\mathbf{x}}),\,\nabla\phi({\mathbf{x}})\rangle\,\mathrm{d}{\mathbf{x}},$
(A.2)

([3], [19]).

###  A.1. Reduction to ODE

Suppose that $W\in L_{1}(Q_{r}({\mathbf{y}}))$. We shall look for a solution
of equation (A.1) of the form:

$\vec{\Gamma}({\mathbf{x}})=u({\mathbf{x}})\,\vec{g}({\mathbf{x}})\quad({\mathbf{x}}=(x_{1},\,x_{2},\,\dots\,x_{d})),$
(A.3)

where $\vec{g}({\mathbf{x}})$ is a given vector field on
$Q_{r}({\mathbf{y}})\;({\mathbf{y}}=(y_{1},\,y_{2},\,\dots\,y_{d}))$ and
$u({\mathbf{x}})$ is a unknown scalar function. Let us take the constant
vector field

$\vec{g}({\mathbf{x}})=(1,\,0,\dots,\,0).$ (A.4)

Then (A.1) is reduced to $\frac{\partial u}{\partial x_{1}}=W({\mathbf{x}})$.
Consider the solution of last equation in $Q_{r}({\mathbf{y}})$, satisfying
the initial condition $u({\mathbf{x}})|_{x_{1}=y_{1}}=0$. It has the form

$u({\mathbf{x}})=\int_{y_{1}}^{x_{1}}W(s,{\mathbf{x}}^{\prime})\,\mathrm{d}s\quad({\mathbf{x}}^{\prime}=(x_{2},\,x_{3},\,\dots,\,x_{d})).$
(A.5)

The following claim is valid:

######  Proposition A.1.

Let $\vec{\Gamma}({\mathbf{x}})$ be the vector field, defined by (A.3), (A.4)
and (A.5). (i) If $W\in C^{1}(Q_{r}({\mathbf{y}}))$, then
$\vec{\Gamma}({\mathbf{x}})$ is a classical solution of divergence equation
(A.1) in $Q_{r}({\mathbf{y}})$;

(ii) The linear operator
$U(W)({\mathbf{x}}):=\int_{y_{1}}^{x_{1}}W(s,{\mathbf{x}}^{\prime})\,\mathrm{d}s$,
taking part in the right hand side of (A.5) acts in the space
$L_{p}(Q_{r}({\mathbf{y}}))\,(p\geq 1)$ and it is bounded;

(iii) If $W\in L_{p}(Q_{r}({\mathbf{y}}))$, then $\vec{\Gamma}({\mathbf{x}})$
is a generalized solution of (A.1), belonging to $L_{p}(Q_{r}({\mathbf{y}}))$.

###### Proof.

(i) We see from (A.5) that the function $u({\mathbf{x}})$ belongs to
$C^{1}(Q_{r}({\mathbf{y}}))$ and it satisfies the equation $\frac{\partial
u}{\partial x_{1}}=W({\mathbf{x}})$ in the cube $Q_{r}({\mathbf{y}})$. Hence
$\vec{\Gamma}({\mathbf{x}})$ is a classical solution of (A.1) in
$Q_{r}({\mathbf{y}})$. Claim (i) is proven

(ii) We have for $W\in L_{p}(Q_{r}({\mathbf{y}}))$, using Hölder inequality:

$\displaystyle\|U(W)\|_{p}^{p}=\int_{Q_{r}({\mathbf{y}})}\Big{|}\int_{y_{1}}^{x_{1}}W(s,{\mathbf{x}}^{\prime})\,\mathrm{d}s\Big{|}^{p}\,\mathrm{d}{\mathbf{x}}\leq$
$\displaystyle\int_{Q_{r}({\mathbf{y}})}(x_{1}-y_{1})^{p-1}\int_{y_{1}}^{x_{1}}|W(s,{\mathbf{x}}^{\prime})|^{p}\,\mathrm{d}s\,\mathrm{d}x_{1}\,\mathrm{d}{\mathbf{x}}^{\prime}\leq\frac{r^{p}}{p}\|W\|_{p}^{p}.$

Claim (ii) is proven.

(iii) By claim (i), for any $W\in C^{1}(Q_{r}({\mathbf{y}}))$ the vector field
$\vec{\Gamma}({\mathbf{x}})$ is a classical solution of equation (A.1), hence
it is a generalized solution of it, i.e. the relation (A) is valid. On the
other hand, in view of (A.3), (A.4), (A.5) and claim (ii) , the correspondence
$W\rightarrow\vec{\Gamma}$ is continuous with respect to $L_{p}$-norm. Using
the density of $C^{1}(Q_{r}({\mathbf{y}}))$ in $L_{p}(Q_{r}({\mathbf{y}}))$,
we can extend the relation (A) by continuity from $W\in
C^{1}(Q_{r}({\mathbf{y}}))$ to $W\in L_{p}(Q_{r}({\mathbf{y}}))$. Claim (iii)
is proven. ∎

###  A.2. Periodic potential solutions

Let us look for solutions of the divergence equation (A.1) in the cube
$Q_{1}(\vec{0})$, which satisfy the periodic boundary condition ([3]). This
means that we consider these solutions $\vec{\Gamma}({\mathbf{x}})$ defined on
the whole ${\mathbf{R}}^{d}$, which are $1$-periodic by all the variables
$x_{1},\,x_{2},\dots,\,x_{d}$, i.e., they can be considered as defined on the
torus ${\mathbf{T}}^{d}$. We shall consider potential solutions of (A.1),
i.e., those $\vec{\Gamma}({\mathbf{x}})$, for which there is a scalar function
$\phi({\mathbf{x}})$ such that
$\vec{\Gamma}({\mathbf{x}})=\nabla\phi({\mathbf{x}})$. Then this function
should satisfy the Poisson equation

$\Delta\phi=W.$ (A.6)

Recall that Fourier operator $\mathcal{F}$ on ${\mathbf{T}}^{d}$ maps
isometrically the space $L_{2}({\mathbf{T}}^{d})$ onto the space
$l_{2}({\mathbf{Z}}^{d})$ [9]. Denote by $L_{p}^{\\#}({\mathbf{T}}^{d})$ the
set of all functions $f\in L_{p}({\mathbf{T}}^{d})$, for which
$\int_{{\mathbf{T}}^{d}}f({\mathbf{x}})\,\mathrm{d}{\mathbf{x}}=0$, i.e.,
$\mathcal{F}(f)(\vec{0})=0$. Also denote $l_{p}^{0}({\mathbf{Z}}^{d}):=\\{g\in
l_{p}({\mathbf{Z}}^{d}):\;g(\vec{0})=0\\}$. We have the following claim:

######  Proposition A.2.

Suppose that $p>2$, $1/p+1/q=1$, $W\in L_{2}^{\\#}({\mathbf{T}}^{d})$ and

$\vec{G}\in l_{q}\big{(}{\mathbf{Z}}^{d}\big{)},$ (A.7)

where

$\vec{G}(\vec{k})=\frac{\mathcal{F}(W)(\vec{k})}{2\pi i|\vec{k}|^{2}}\vec{k}.$
(A.8)

Then the vector function

$\vec{\Gamma}({\mathbf{x}})=\mathcal{F}^{-1}(\vec{G})({\mathbf{x}})$ (A.9)

belongs to $L_{p}^{\\#}({\mathbf{T}}^{d})$ and it is a potential solution of
divergence equation (A.1). Furthermore,

$|\|\vec{\Gamma}\|_{p}\leq d\|\vec{G}\|_{l_{q}}.$ (A.10)

###### Proof.

For brevity we shall omit $({\mathbf{T}}^{d})$ and $({\mathbf{Z}}^{d})$ in the
notations $L_{p}({\mathbf{T}}^{d})$, $L_{p}^{\\#}({\mathbf{T}}^{d})$,
$l_{q}({\mathbf{Z}}^{d})$ and $l_{q}^{0}({\mathbf{Z}}^{d})$. Notice that since
the operator $\mathcal{F}^{-1}:\,l_{2}\rightarrow L_{2}$ is isometric, for any
$g\in l_{2}\;$ $\|\mathcal{F}^{-1}(g)_{2}=\|g\|_{l_{2}}$. On the other hand,
in view of the formula for $\mathcal{F}^{-1}$, given in Section 2,
$\mathcal{F}^{-1}$ maps $l_{1}$ into $L_{\infty}$ and for any $g\in l_{1}$
$\|\mathcal{F}^{-1}(g)\|_{\infty}\leq\|g\|_{l_{1}}$. Then by the Riesz-Thorin
Interpolation Theorem ([7], Theorem 1.3.4) , $\mathcal{F}^{-1}$ maps $l_{q}$
into $L_{p}$ and for any $g\in l_{q}$
$\|\mathcal{F}^{-1}(g)\|_{p}\leq\|g\|_{l_{q}}$. Notice that since $p>2$,
$l_{q}\subset l_{2}$ and $L_{p}\subset L_{2}$. Then since
$\mathcal{F}^{-1}:\,l_{2}\rightarrow L_{2}$ is injective,
$\mathcal{F}^{-1}:\,l_{q}\rightarrow L_{p}$ is injective too. It is clear that
$\mathcal{F}^{-1}$ maps injectively $l_{q}^{0}$ into $L_{p}^{\\#}$, It is easy
to see that the analogous property is valid for vector functions:
$\mathcal{F}^{-1}$ maps injectively $l_{q}^{0}$ into $L_{p}^{\\#}$ and for any
$\vec{H}\in l_{q}^{0}$

$\|\mathcal{F}^{-1}(\vec{H})\|_{p}\leq d\|\vec{H}\|_{l_{q}}.$ (A.11)

Applying the Fourier transform to equation (A.6) and denoting
$u(\vec{k})=\mathcal{F}(\phi)(\vec{k})$, we obtain:

$(2\pi i)^{2}|\vec{k}|^{2}u(\vec{k})=\mathcal{F}(W)(\vec{k}).$

Since $W\in L_{2}^{\\#}$ (hence $\mathcal{F}(W)\in l_{2}$ and
$\mathcal{F}(W)(\vec{0})=0$ ), the last equation has the solution

$\displaystyle u(\vec{k})=\frac{\mathcal{F}(W)(\vec{k})}{(2\pi
i)^{2}|\vec{k}|^{2}}.$

this solution belongs to the subspace $l_{2}^{0}$ and it is unique there.
Hence the function $\phi=\mathcal{F}^{-1}(u)$ is a unique solution of (A.6) in
the subspace $L_{2}^{\\#}$. Moreover, since $|\vec{k}|^{2}\ u\in l_{2}$, this
solution belongs to the Sobolev space $W_{2}^{2}({\mathbf{T}}^{d})$. Then
$\vec{\Gamma}=\nabla\phi$ is a solution of divergence equation (A.1) and it is
expressed by (A.9). Furthermore, in view of (A.8), (A.7) and (A.11), inclusion
$\vec{\Gamma}\in L_{p}^{\\#}$ and estimate (A.10) are valid. ∎

##  Appendix B Some estimates

Let $\Omega$ be a bounded open domain in ${\mathbf{R}}^{d}$.

######  Proposition B.1.

Suppose that $d>2$.

(i) If $W\in L_{d/2}(\Omega)$, then for any $u\in C_{0}^{\infty}(\Omega)$

$|\int_{\Omega}W({\mathbf{x}})|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}|\leq
C^{2}(d)\|W\|_{d/2}\int_{\Omega}|\nabla
u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}},$ (B.1)

where $C(d)$ is expressed by (5.15),

(ii) Suppose that there is a vector field $\vec{\Gamma}({\mathbf{x}})$ on
$\Omega$ belonging to $L_{d}(\Omega)$ and satisfying the equation
$\mathrm{div}\vec{\Gamma}({\mathbf{x}})=W({\mathbf{x}})$ in the generalized
sense, where $W\in L_{1}(\Omega)$. Then for any $u\in C_{0}^{\infty}(\Omega)$

$|\int_{\Omega}W({\mathbf{x}})|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}|\leq
2\,C(d)\,\|\vec{\Gamma}\|_{d}\int_{\Omega}|\nabla
u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}};$ (B.2)

###### Proof.

(i) Using Hölder inequality, we get:

$|\int_{\Omega}W({\mathbf{x}})|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}|\leq\|W\|_{d/2}\|u\|_{q}^{2},$

where $q=\frac{2d}{d-2}$. On the other hand, by the Sobolev’s theorem, the
space $W_{2}^{1}({\mathbf{R}}^{d})$ is embedded continuously into
$L_{q}({\mathbf{R}}^{d})$. Hence continuing each function $u\in
C_{0}^{\infty}(\Omega)$ by zero from $\Omega$ to the whole ${\mathbf{R}}^{d}$,
we get:

$\|u\|_{q}^{2}\leq C^{2}(d)\int_{\Omega}|\nabla
u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}},$ (B.3)

([1], [24]). These circumstances imply the desired claim.

(ii) Making integration by parts, we get for $u\in C_{0}^{\infty}(\Omega)$:

$\int_{\Omega}W({\mathbf{x}})|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}=-2\int_{\Omega}\Re\big{(}\langle\vec{\Gamma}{\mathbf{x}}),\,\overline{u({\mathbf{x}})}\,\nabla
u({\mathbf{x}})\rangle\big{)}\,\mathrm{d}{\mathbf{x}}.$

Using Schwartz’s inequality, we obtain:

$\int_{\Omega}W({\mathbf{x}})|u({\mathbf{x}})|^{2}\,\mathrm{d}{\mathbf{x}}\leq\\\
2\,\|\vec{\Gamma}u\|_{2}\|\nabla u|\|_{2}.$

On the other hand, by claim (i):

$\|\vec{\Gamma}u\|_{2}\leq
C(d)\Big{(}\||\vec{\Gamma}|^{2}\|_{d/2}\Big{)}^{1/2}\|\nabla
u|\|_{2}.=C(d)\|\vec{\Gamma}\|_{d}\|\nabla u|\|_{2}.$

Te last two estimates imply the desired inequality (B.2). Claim (ii) is
proven. ∎

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