# Spectral properties of the resolvent difference for singularly perturbed
operators

Grigori Rozenblum  Chalmers Univ. of Technology
<EMAIL_ADDRESS>
###### Abstract.

We obtain order sharp spectral estimates for the difference of resolvents of
singularly perturbed elliptic operators ${\mathbf{A}}+{\mathbf{V}}_{1}$ and
${\mathbf{A}}+{\mathbf{V}}_{2}$ in a domain
$\Omega\subseteq{\mathbb{R}}^{\mathbf{N}}$ with perturbations
${\mathbf{V}}_{1},{\mathbf{V}}_{2}$ generated by $V_{1}\mu,V_{2}\mu,$ where
$\mu$ is a measure singular with respect to the Lebesgue measure and
satisfying two-sided or one-sided conditions of Ahlfors type, while
$V_{1},V_{2}$ are weight functions subject to some integral conditions. As an
important special case, spectral estimates for the difference of resolvents of
two Robin realizations of the operator ${\mathbf{A}}$ with different weight
functions are obtained. For the case when the support of the measure is a
compact Lipschitz hypersurface in $\Omega$ or, more generally, a rectifiable
set of Haußdorff dimension $d={\mathbf{N}}-1$, the Weyl type asymptotics for
eigenvalues is justified.

Dedicated to Gerd Grubb on her Jubilee.

##  1\. Introduction

###  1.1. Resolvent perturbations

Let ${\mathbf{A}}$ be a self-adjoint second order elliptic differential
operator in an open (possibly, unbounded) connected set
$\Omega\subseteq{\mathbb{R}}^{{\mathbf{N}}}$. Suppose that ${\mathbf{V}}_{1}$
and ${\mathbf{V}}_{2}$ are some perturbations of ${\mathbf{A}}$ (one of them
may be the zero one), so, at least symbolically,

${\mathbf{A}}_{1}={\mathbf{A}}+{\mathbf{V}}_{1},\,{\mathbf{A}}_{2}={\mathbf{A}}+{\mathbf{V}}_{2}.$
(1.1)

Generally, the perturbations ${\mathbf{V}}_{1},{\mathbf{V}}_{2}$ can be rather
singular, and the expressions for ${\mathbf{A}}_{1},{\mathbf{A}}_{2}$ in (1.1)
can be understood in a quite generalized and often highly formal way. In
particular, such a vague setting may embrace the change in the boundary
conditions for an elliptic operator. The question about the spectral
properties of the difference of resolvents,

${\mathbf{R}}^{(1,2)}=({\mathbf{A}}_{1}+t)^{-1}-({\mathbf{A}}_{2}+t)^{-1},$
(1.2)

where the real number $t$ is chosen in such way that both operators in (1.2)
are invertible, is one of the classical ones in Spectral Theory, with long
history. In particular, an efficient method was elaborated and fundamental
results were obtained in [21], [22], where one can find referrals to earlier
papers and a motivation for studying this kind of problems, the most important
being the needs of the scattering theory. Further on, quite general abstract
operator perturbation methods have been developed, see, in particular, [2],
[13], [18], [15], [33], [34], [35], [43], [61], [53] and many other sources.
Being applied to concrete differential operators, these abstract perturbation
schemes have produced a number of impressive results on the spectral
properties, semigroup properties, regularity etc.

One of important directions of study is the investigation of the spectrum of
the _difference of resolvents_ of operators generated by the same elliptic
differential expression with different boundary conditions. For the difference
of resolvents of the Dirichlet and Neumann operators, eigenvalue estimates we
obtained in the already cited paper [22] by M.Birman in 1962, while later, in
[24], general methods were developed producing eigenvalue _asymptotics_ for
many problems, including the above one.

Even this, rather particular, topic attracted great attention in the latest
two or even three decennia, with scores of papers published. We mention here
the papers [2], [43], [42], [15], [12], and a vast amount of references
therein. For the study of these problems, the extensive machinery of abstract
Krein-von-Neumann perturbation formulas, quasi-boundary triples, and the
generalized H.Weyl $M$-function have been elaborated. This approach, in
various modifications, was used to the study of many kinds of spectral
problems, a systematic description and an impressive bibliography can be found
in [13].

One of special singular problems under consideration has been the study of the
spectral properties of the difference of resolvents of self-adjoint operators
${\mathbf{A}}_{\iota},$ $\iota=1,2,$ in
$\Omega\subseteq{\mathbb{R}}^{\mathbf{N}}$, corresponding to the Laplacian (or
a general symmetric second order elliptic operator ${\mathcal{L}}$) with
weighted Robin boundary conditions containing two different weight functions,

$\partial_{\nu({\mathcal{L}})}u(X)=V_{\iota}(X)u(X),\,\iota=1,2,X\in\Gamma=\partial\Omega,$

where $\partial_{\nu({\mathcal{L}})}u$ is the (co)normal derivative at the
boundary, associated with ${\mathcal{L}}$. For the singular values $s_{j}$ of
this difference, in [22], the estimate
$s_{j}=O(j^{-\frac{2}{{\mathbf{N}}-1}})$ was obtained, it was improved to
$s_{j}=O(j^{-\frac{3}{{\mathbf{N}}-1}})$ in [15]. Further on, in [41], the
latter estimate was elevated to the asymptotic formula

$s_{j}\sim({\mathcal{A}}j)^{-\frac{3}{{\mathbf{N}}-1}},\,{\mathcal{A}}=C\int_{\Sigma}|V_{1}(X)-V_{2}(X)|^{\frac{{\mathbf{N}}-1}{3}}\omega(X)\sigma(dX),$
(1.3)

where the density $\omega(X)$ is determined (in some complicated way, see
Theorem 3.5 in [41]) by the coefficients of the operator ${\mathcal{L}}$ at
the boundary. We do not need the particular expression for this weight
function.

Formula (1.3) was established in [41] under rather restrictive regularity
conditions for the functions $V_{\iota}$, namely, that they and their
difference should be piecewise Hölder continuous, with allowed discontinuities
only along some smooth hypersurfaces in $\Gamma$, common for $V_{1},V_{2}$ (in
fact, a little less restrictive regularity is required, expressed in terms of
spaces of Bessel potentials.) These restrictions were caused by the
pseudodifferential technique used in [41]; further on, the regularity
conditions were somewhat relaxed in [42]. Important results were also obtained
for various singular perturbations in [18], [33], [53], and many more.

Another perturbation problem, also subject to an extensive study, is the
analysis of spectral propertied of the difference of resolvents of
’Schrödinger-like operators,’

$({\mathbf{A}}+V_{1}(X)+t)^{-1}-({\mathbf{A}}+V_{2}(X)+t)^{-1},$ (1.4)

where $V_{1},V_{2}$ are _functions_ defined in $\Omega.$ Such problems were
being studied systematically, starting, probably, with [9] and [73]. For
functions $V_{\iota}$ with compact support and some summability conditions,
eigenvalue estimates and eigenvalue asymptotics were known for the operator
(1.4) since long ago. More generally, operators of the type $-\Delta+V$, with
the _singular_ perturbation $V$ being the $\delta$-distribution supported on a
hyper-surface, with some weight, were being considered, see, especially, [29],
[30] and references therein; such mathematical models were treated as ’leaky
quantum graphs’. More recently, the studies were concerned with the spectrum
of resolvent differences for such singularly perturbed operators, see, e.g.,
[12], [11], [31]. The coefficients in front of the distributional potential
were supposed to be smooth functions on the hypersurface. More generally, one
can consider singular perturbations problems for Schrödinger-like operators,
as above, but with the roles of potentials played by singular measures
supported on some sets of zero Lebesgue measure, possibly, with fractional
dimension, sometimes called ’transmission across a fractal set,’ see, e.g.,
[5], [6], [25], [26], [37], [45], [48], [49], [50], etc.

###  1.2. Singular perturbations

In the present paper, we show that the study of spectral properties of the
problems discussed above can be performed in a uniform way, as versions of one
and the same construction. Namely, for a Radon measure $\mu$ supported in
$\overline{\Omega}$, possibly, singular with respect to the Lebesgue measure,
and a $\mu$-measurable function $V$ on the support of $\mu$ and subject to
some summability conditions, the operator ${\mathbf{A}}+V\mu$ in
$L_{2}(\Omega)$ can be effectively defined by means of quadratic forms, and
the eigenvalues of the difference of its resolvent and the resolvent of
${\mathbf{A}}$ (as well as of the difference of powers of resolvents) can be
estimated. The order of such estimates is determined by the class of Ahlfors
regularity of the measure $\mu$ (roughly speaking, the local Haußdorff
dimension of the support of $\mu$), while the coefficient in the estimates
depends on a certain integral norm of the ’weight’ function $V$ with respect
to the measure $\mu$. In a similar way, the spectrum of the difference of
(powers of) resolvents of operators with two different weight functions
$V_{1},V_{2}$ can be studied, producing eigenvalue estimates and, in certain
cases, asymptotics, generalizing (1.3).

In this setting, the Robin problem corresponds to the measure $\mu$ supported
on some set in the boundary of $\Omega,$ while the latter type of problems,
which we call ’the Schrödinger type’ ones, corresponds to the measure $\mu$
supported inside $\Omega.$ It is possible, of course, to consider a
combination of these perturbations.

Eigenvalue estimates obtained in the present paper involve integral
characteristics of the weight functions. This circumstance provides us quite a
freedom in extending the existing results on the _eigenvalue asymptotics_ to
considerably less regular weight functions $V_{\iota}$ for the difference of
Robin resolvents, compared, e.g., with the results in [41]. Note that in
addition to the asymptotics of singular numbers, studied previously, we are
able to separate positive and negative parts of $V_{1}-V_{2}$, thus finding
the asymptotics of negative and positive eigenvalues separately. Similarly,
for problems with singular potentials supported inside $\Omega$, the
eigenvalue asymptotics is found for the measure $\mu$ being the Haußdorff
measure on a Lipschitz hypersurface in $\Omega$ or on some unions of such
surfaces (uniformly rectifiable sets). When the set $\Sigma=\hbox{{\rm
supp}}\,\mu$ is Ahlfors regular of a fractional dimension, asymptotic formulas
for eigenvalues are, probably, not accessible in a general case, however, our
estimates are order sharp: the general upper estimates are accompanied by
lower estimates of the same order.

There is a huge literature on the operator perturbation theory, with extensive
machinery, involving boundary triples, generalized boundary pairs, Weyl
$M$-functions etc.; this activity has produced quite a lot of impressive
results. In our case, where we deal with perturbation of operators defined by
means of quadratic forms, using the approach originating in [21] and developed
further in [25] and in [61]; the latter is the closest to ours, being much
more general. We, however, do not need this approach in its full generality;
our perturbation scheme is rather elementary and does not use the above
general constructions, giving at the same time convenient explicit formulas.

In dealing with spectral estimates and asymptotics for singular measures, we
follow the approach developed recently in the papers [46], [65], [62], [67],
[64] concerning spectral properties of Birman-Schwinger type operators with
singular weights and use some important results of these papers, especially,
[67]. When needed, we reproduce some formulations from these papers; we
explain the strategy of proving asymptotic formulas as well.

In this paper, we restrict ourselves to perturbations and boundary problems
for second order uniformly elliptic operator ${\mathcal{L}}$ with smooth
coefficients. The regularity of the boundary, if it is present, depends on the
particular problem under consideration. More general results involving
elliptic operators with less regular coefficients and boundaries (like in
[63]), as well as higher order operators, would require essentially more
complicated considerations which would dim simple ideas in our approach to
spectral problems. We hope to return to these general topics on some other
occasion.

There are numerous papers devoted to various aspects of spectral theory of
singular perturbations. Our initial attempts to include a complete
bibliography led to the reference list filling almost one third of the paper.
Therefore, we decided to make the reference list (more or less) reasonably
short, so that the interested Reader may find an additional information using
reference lists of the papers cited here and the citation lists in the entries
of these papers in Math.Sci.Net.

##  2\. The abstract perturbation formula and related spectral properties

We consider perturbations of a given operator by quadratic forms. The formulas
for the resolvent of the perturbed operator are somewhat similar to the ones
established in [25] and, later, in [61], however this particular form enables
us to apply previously obtained results concerning spectral estimates.

###  2.1. Form perturbations

Let ${\mathbf{A}}>0$ be a self-adjoint operator in the Hilbert space
$\mathfrak{H}.$ We suppose that ${\mathbf{A}}\geq c_{0}>0$ is defined by the
positive quadratic form
${\mathbf{a}}[u]=({\mathbf{B}}u,{\mathbf{B}}u)_{\mathfrak{H}}$ with domain
$\mathfrak{H}^{1}={\mathcal{D}}({\mathbf{A}}^{\frac{1}{2}}).$ Here
${\mathbf{B}}$ is a bounded operator acting from the Hilbert space
$\mathfrak{H}^{1}$ to $\mathfrak{H}$. Of course, there is a considerable
freedom in choosing the operator ${\mathbf{B}}$ generating one and the same
operator ${\mathbf{A}}.$ The choice, always possible, is
${\mathbf{B}}={\mathbf{A}}^{\frac{1}{2}}$.

The perturbation is supposed to be defined by means of the quadratic form
${\mathbf{v}}[u]$ with domain
$\mathfrak{d}[{\mathbf{v}}]\supset{\mathcal{D}}({\mathbf{A}}^{\frac{1}{2}})\equiv\mathfrak{H}^{1}$
and bounded there,

$|{\mathbf{v}}[u]|\leq C{\mathbf{a}}[u],u\in\mathfrak{H}^{1}.$ (2.1)

Therefore, the quadratic form ${\mathbf{v}}[u]$ defines a bounded self-adjoint
operator ${\mathbf{V}}$ in $\mathfrak{H}^{1}$ satisfying

$({\mathbf{V}}u,u)_{\mathfrak{H}^{1}}={\mathbf{v}}[u],u\in\mathfrak{H}^{1}.$

Under these conditions, we define the perturbed operator
${\mathbf{A}}_{{\mathbf{V}}}$ in $\mathfrak{H}$ by means of the quadratic form

${\mathbf{a}}_{\mathbf{V}}[u]={\mathbf{a}}[u]+{\mathbf{v}}[u],u\in\mathfrak{H}^{1},$
(2.2)

In this paper, we suppose that the quadratic form (2.2) is positive in
$\mathfrak{H}$. Then, by the KLMN theorem, this quadratic form is closed in
$\mathfrak{H}$ and, in fact, defines a lower semi-bounded self-adjoint
operator ${\mathbf{A}}_{{\mathbf{V}}}$. Having the quadratic form definition
of the operator ${\mathbf{A}}_{{\mathbf{V}}}$, we are going now to describe
conveniently the operator itself.

Set $u={\mathbf{A}}^{-\frac{1}{2}}v,$ $v\in\mathfrak{H}$, in (2.2). We obtain

${\mathbf{v}}[u]={\mathbf{v}}[{\mathbf{A}}^{-\frac{1}{2}}v].$

We suppose that the perturbing quadratic form has the following structure. Let
$\mathfrak{G}$ be a Hilbert space and $\gamma$ be a (boundary) operator,
$\gamma:\mathfrak{H}^{1}\to\mathfrak{G}$ such that

$\|\gamma u\|^{2}_{\mathfrak{G}}\leq
C\|u\|^{2}_{\mathfrak{H}^{1}},\,u\in\mathfrak{H}^{1}.$

Further on, let ${\mathbf{F}}$ be an operator in the space $\mathfrak{G}$ such
that its domain ${\mathcal{D}}({\mathbf{F}})$ contains
$\gamma\mathfrak{H}^{1}$ and

$\|{\mathbf{F}}\gamma u\|_{\mathfrak{G}}\leq C\|u\|_{\mathfrak{H}^{1}}$

(note that we do not expect that ${\mathbf{F}}$ is a bounded operator in
$\mathfrak{G}$.) Finally, let ${\mathbf{U}}$ be a bounded self-adjoint
operator in $\mathfrak{G}$.

Under the above conditions, the quadratic form ${\mathbf{v}}$ is supposed to
be defined by

${\mathbf{v}}[u]=({\mathbf{U}}{\mathbf{F}}\gamma u,{\mathbf{F}}\gamma
u)_{\mathfrak{G}},u\in\mathfrak{H}^{1}.$ (2.3)

The expression (2.3) is defined, at least, on $\mathfrak{H}^{1}$ and it is
bounded with respect to the $\mathfrak{H}^{1}$ norm, see (2.1). Since
$u={\mathbf{A}}^{-\frac{1}{2}}v$, $v\in\mathfrak{H}$, (2.3) transforms to

${\mathbf{v}}[{\mathbf{A}}^{-\frac{1}{2}}v]=({\mathbf{U}}{\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}v,{\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}v)_{\mathfrak{G}}.$
(2.4)

By the conditions imposed above,
${\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}$ is a bounded operator from
$\mathfrak{H}$ to $\mathfrak{G}.$ Therefore, the expression in (2.4) can be
represented as

${\mathbf{v}}[{\mathbf{A}}^{-\frac{1}{2}}v]=(({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})v,v)_{\mathfrak{H}}.$
(2.5)

Here, $({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}$ acts from
$\mathfrak{G}$ to $\mathfrak{H}$; it is the adjoint operator to
${\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}$ considered as acting from
$\mathfrak{H}$ to $\mathfrak{G}.$ Since the operator ${\mathbf{U}}$ is self-
adjoint and bounded, the operator
$({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})$
is a self-adjoint bounded operator in $\mathfrak{H}$.

We return to $v={\mathbf{A}}^{\frac{1}{2}}u$. The expression in (2.5)
transforms to

${\mathbf{v}}[u]=(({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}){\mathbf{A}}^{\frac{1}{2}}u,{\mathbf{A}}^{\frac{1}{2}}u)_{\mathfrak{H}},\,u\in\mathfrak{H}^{1}.$

Now we consider the operator ${\mathbf{A}}_{\mathbf{V}}$ defined in
$\mathfrak{H}$ by means of the quadratic form (2.2), with ${\mathbf{v}}$ as in
(2.3). We suppose further on that the form ${\mathbf{a}}_{\mathbf{V}}[u]$ is
positive in $\mathfrak{H}^{1}$, namely,

${\mathbf{a}}_{\mathbf{v}}[u]\geq C{\mathbf{a}}[u],\,u\in\mathfrak{H}^{1},$
(2.6)

with some $C>0$. Under this condition, the operator
$1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})$
is positive, and therefore, we can write, for $u\in\mathfrak{H}^{1},$

$\displaystyle{\mathbf{a}}_{\mathbf{V}}[u]=({\mathbf{A}}^{\frac{1}{2}}u,{\mathbf{A}}^{\frac{1}{2}}u)_{\mathfrak{H}}+(({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}){\mathbf{A}}^{\frac{1}{2}}u,{\mathbf{A}}^{\frac{1}{2}}u)_{\mathfrak{H}}=$
(2.7)
$\displaystyle((1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})){\mathbf{A}}^{\frac{1}{2}}u,{\mathbf{A}}^{\frac{1}{2}}u)_{\mathfrak{H}}=$
$\displaystyle((1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{\frac{1}{2}}){\mathbf{A}}^{\frac{1}{2}}u,(1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{\frac{1}{2}}){\mathbf{A}}^{\frac{1}{2}}u)_{\mathfrak{H}}.$

The last expression in (2.7) still defines the quadratic form of the self-
adjoint operator ${\mathbf{A}}_{\mathbf{V}}$, and therefore we have the
representation for this operator as the composition of an operator and its
adjoint (such composition is always self-adjoint):

${\mathbf{A}}_{\mathbf{V}}=\left[(1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}))^{\frac{1}{2}}{\mathbf{A}}^{\frac{1}{2}}\right]^{*}\left[(1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}))^{\frac{1}{2}}{\mathbf{A}}^{\frac{1}{2}}\right].$
(2.8)

Note again that we do not (and need not to) describe explicitly the domain of
this operator. The operator ${\mathbf{A}}_{\mathbf{V}}$ is invertible, since
in (2.8) it is represented as a composition of two invertible operators.
Therefore, for the (bounded) inverse operator to ${\mathbf{A}}_{\mathbf{V}}$,
we obtain the representation

$\displaystyle({\mathbf{A}}_{\mathbf{V}})^{-1}=[(1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}))^{\frac{1}{2}}{\mathbf{A}}^{\frac{1}{2}}]^{-1}\times$
(2.9)
$\displaystyle[((1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}))^{\frac{1}{2}}{\mathbf{A}}^{\frac{1}{2}})^{*}]^{-1}=$
$\displaystyle{\mathbf{A}}^{-\frac{1}{2}}(1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}))^{-\frac{1}{2}}\times$
$\displaystyle(1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}))^{-\frac{1}{2}}{\mathbf{A}}^{-\frac{1}{2}}=$
$\displaystyle{\mathbf{A}}^{-\frac{1}{2}}(1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}))^{-1}{\mathbf{A}}^{-\frac{1}{2}}$
$\displaystyle\equiv{\mathbf{A}}^{-\frac{1}{2}}(1+{\mathbf{T}})^{-1}{\mathbf{A}}^{-\frac{1}{2}},\,\,{\mathbf{T}}\equiv{\mathbf{T}}_{\mathbf{V}}=({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}).$

We arrive here at the resolvent perturbation formula, which creates the base
of our further considerations.

######  Proposition 2.1.

Let ${\mathbf{A}}$ be a positive self-adjoint operator and let ${\mathbf{V}}$
be a perturbation as above. Then

$\displaystyle{\mathbf{R}}_{{\mathbf{V}}}\equiv{\mathbf{A}}^{-1}-({\mathbf{A}}_{\mathbf{V}})^{-1}={\mathbf{A}}^{-\frac{1}{2}}\left[1-(1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}))^{-1}\right]{\mathbf{A}}^{-\frac{1}{2}}\equiv$
(2.10)
$\displaystyle{\mathbf{A}}^{-\frac{1}{2}}\left[1-(1+{\mathbf{T}}_{{\mathbf{V}}})^{-1}\right]{\mathbf{A}}^{-\frac{1}{2}}.$

Operators of the form ${\mathbf{T}}$ and their analogies have been the base of
the study of discrete spectrum of Schrödinger-type operators since 60-s. It is
reasonable to call them ’Birman-Schwinger’ operators.

The representation (2.10) of the resolvent difference has been derived under
the condition that the operator

$1+{\mathbf{T}}_{{\mathbf{V}}}\equiv\left[1+({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})\right]$

is positive definite. If we replace the operator ${\mathbf{A}}$ by
${\mathbf{A}}-t,$ $t<0$, (2.6) still holds, and (2.9) gives the representation
for the inverse of ${\mathbf{A}}_{\mathbf{V}}-t,$ this means, for the
resolvent of the operator ${\mathbf{A}}_{\mathbf{V}}$ at the point $-t$. If
there is a family of quadratic forms ${\mathbf{v}}_{\epsilon}$ such that the
corresponding operators ${\mathbf{T}}_{{\mathbf{V}}_{\epsilon}}$ converge in
the norm sense to ${\mathbf{T}}_{{\mathbf{V}}}$ as $\epsilon\to 0$, then, as
soon as
$\|{\mathbf{T}}_{{\mathbf{V}}_{\epsilon}}-{\mathbf{T}}_{{\mathbf{V}}}\|$ is
small enough, the operator $1+{\mathbf{T}}_{{\mathbf{V}}_{\epsilon}}$ is
positive definite as well and the resolvent perturbation formula (2.10) is
valid.

###  2.2. Asymptotic spectral characteristics

We are going to study asymptotical spectral properties of compact operators in
a Hilbert space. Here we collect some basic notations and facts, mostly well
known. Let ${\mathbf{K}}$ be a compact self-adjoint operator in a Hilbert
space $\mathfrak{H}$. By $n_{\pm}(\lambda,{\mathbf{K}})$ we denote the
distribution function of positive (resp., negative) eigenvalues
$\pm\lambda^{\pm}_{j}({\mathbf{K}})$ of the operator ${\mathbf{K}}$:

$n_{\pm}(\lambda,{\mathbf{K}})=\\#\\{j:\lambda_{j}^{\pm}({\mathbf{K}})>\lambda\\}.$

For a given exponent $\theta>0$, the asymptotic characteristics of the
spectrum of ${\mathbf{K}}$ are defined as

${\mathbf{n}}^{\sup}_{\pm}({\mathbf{K}},\theta)=\limsup_{\lambda\to+0}n_{\pm}(\lambda,{\mathbf{K}})\lambda^{\theta},\,{\mathbf{n}}^{\inf}_{\pm}({\mathbf{K}},\theta)=\liminf_{\lambda\to+0}n_{\pm}(\lambda,{\mathbf{K}})\lambda^{\theta}$
(2.11)

and

${\mathbf{n}}_{\pm}({\mathbf{K}},\theta)=\lim_{\lambda\to+0}n_{\pm}(\lambda,{\mathbf{K}})\lambda^{\theta},$
(2.12)

provided these limits exist.

Similarly, for an _arbitrary_ compact operator ${\mathbf{K}}$, the
distribution function for its singular numbers $s_{j}({\mathbf{K}})$ and their
asymptotic characteristics are defined and denoted, as above, but with the
symbol $\pm$ deleted.

It follows immediately that for a bounded operator ${\mathbf{J}}$,

${\mathbf{n}}(\lambda,{\mathbf{J}}{\mathbf{K}})\leq{\mathbf{n}}(\lambda,\|{\mathbf{J}}\|{\mathbf{K}})={\mathbf{n}}(\|{\mathbf{J}}\|^{-1}\lambda,{\mathbf{K}}),$

therefore,

${\mathbf{n}}^{\sup}({\mathbf{J}}{\mathbf{K}},\theta)\leq\|{\mathbf{J}}\|^{\theta}{\mathbf{n}}^{\sup}({\mathbf{K}},\theta).$

The $\limsup$ form of the multiplicative Ky Fan inequality

$n(\lambda_{1}\lambda_{2},{\mathbf{K}}_{1}{\mathbf{K}}_{2})\leq
n(\lambda_{1},{\mathbf{K}}_{1})+n(\lambda_{2},{\mathbf{K}}_{2})$ (2.13)

implies in these terms

${\mathbf{n}}^{\sup}({\mathbf{K}}_{1}{\mathbf{K}}_{2},\theta)\leq
2{\mathbf{n}}^{\sup}({\mathbf{K}}_{1},\theta_{1})^{\frac{\theta}{\theta_{1}}}{\mathbf{n}}^{\sup}({\mathbf{K}}_{2},\theta_{2})^{\frac{\theta}{\theta_{2}}}.$
(2.14)

Similar inequalities are valid for the product of several operators,
${\mathbf{K}}=\prod_{j=1}^{m}{\mathbf{K}}_{j},$
$\theta^{-1}=\sum\theta_{j}^{-1}$:

${\mathbf{n}}^{\sup}({\mathbf{K}},\theta)\leq
m\prod\left({\mathbf{n}}^{\sup}({\mathbf{K}}_{j},\theta_{j})^{\frac{\theta}{\theta_{j}}}\right).$
(2.15)

Another simple consequence of the multiplicative Ky Fan inequality is the
following:

######  Lemma 2.2.

If ${\mathbf{n}}^{\sup}({\mathbf{K}}_{j},\theta_{j})<\infty,$ for $j=1,3$ and
${\mathbf{K}}_{2}$ is a compact operator, then

${\mathbf{n}}^{\sup}({\mathbf{K}}_{1}{\mathbf{K}}_{2}{\mathbf{K}}_{3},\theta)=0,\,\theta^{-1}=\theta_{1}^{-1}+\theta_{2}^{-1}.$

###  2.3. Ratios of quadratic forms

We will systematically use operators defined by the ratio of quadratic forms,
see, e.g., [23].

If ${\mathbf{k}}$ is a bounded Hermitian quadratic form in the Hilbert space
$\mathfrak{H}$, its eigenvalues (eigenvalues of the operator ${\mathbf{K}}$
defined by this quadratic form) are determined in variational way, in
particular,

$n_{\pm}(\lambda,{\mathbf{K}})=\min\\{\operatorname{codim\,}\sc\mbox{L}\hskip
1.0pt\subset\mathfrak{H}:\frac{\pm{\mathbf{k}}[u]}{{\mathbf{h}}[u]}>\lambda,u\in\sc\mbox{L}\hskip
1.0pt\setminus\\{0\\}\\},$

where ${\mathbf{h}}[u]:=\|u\|^{2}_{\mathfrak{H}}.$ It is convenient to denote
this distribution function by
$n_{\pm}(\lambda,\frac{{\mathbf{k}}}{{\mathbf{h}}})$. The notation of
asymptotic characteristics (2.11), (2.12) is modified accordingly.

When we handle several different Hilbert spaces (related, in concrete examples
later, with different domains in ${\mathbb{R}}^{\mathbf{N}}$), the following
statement is useful, see Lemma 1.2 in [23], which we reproduce here, in our
notations.

######  Lemma 2.3.

Let $\mathfrak{H}_{1},\mathfrak{H}_{2}$ be two Hilbert spaces, with squared
norm ${\mathbf{h}}_{1},{\mathbf{h}}_{2}$, and let
${\mathbf{K}}_{1},{\mathbf{K}}_{2}$ be self-adjoint compact operators in these
spaces. We denote by ${\mathbf{k}}_{1},{\mathbf{k}}_{2}$ the quadratic forms
of these operators,
${\mathbf{k}}_{\iota}[u]=({\mathbf{K}}_{\iota}u,u)_{\mathfrak{H}_{\iota}},\,\iota=1,2.$
Let ${\mathbf{J}}:{\mathcal{H}}_{1}\to{\mathcal{H}}_{2}$ be a bounded
operator, moreover, ${\mathbf{k}}_{1}[u]=0$ for all $u\in\hbox{{\rm
Ker}}\,({\mathbf{J}}).$ Suppose that for all $u\in\mathfrak{H}_{1}$,

$\pm\frac{{\mathbf{k}}_{1}[u]}{{\mathbf{h}}_{1}[u]}\leq\pm\frac{{\mathbf{k}}_{2}[{\mathbf{J}}u]}{{\mathbf{h}}_{2}[{\mathbf{J}}u]}$

for all $u\in\mathfrak{H}_{1},$ for which $\pm{\mathbf{k}}_{1}[u]>0.$ Then

$n_{\pm}(\lambda,{\mathbf{K}}_{1})\leq n_{\pm}(\lambda,{\mathbf{K}}_{2}),$

for all $\lambda>0,$ with the corresponding inequalities for asymptotical
spectral characteristics.

This statement will be used on two concrete occasions.

######  Corollary 2.4.

Let $\Omega_{2}\subset\Omega_{1}\subseteq{\mathbb{R}}^{\mathbf{N}}$.

  1. (1)

Let $\mathfrak{H}^{(l)}$ be some space of functions in
$H^{l}_{\operatorname{loc}}(\Omega_{1})$. Suppose that the operator
${\mathbf{J}}$ of _restriction_ of functions in $\mathfrak{H}^{(l)}$ to
$H^{l}(\Omega_{2})$ is bounded. We set
${\mathbf{h}}_{2}[u]=\|u\|_{H^{l}(\Omega_{2})}^{2}$,
${\mathbf{h}}_{1}[u]=\|u\|^{2}_{\mathfrak{H}^{(l)}(\Omega_{1})}$ and suppose
that ${\mathbf{k}}_{2}[{\mathbf{J}}u]={\mathbf{k}}_{1}[u].$ Then

$n_{\pm}(\lambda,{\mathbf{K}}_{1})\leq
n_{\pm}(\lambda,\|{\mathbf{J}}\|{\mathbf{K}}_{2})=n_{\pm}(\|{\mathbf{J}}\|^{-1}\lambda,{\mathbf{K}}_{2}).$
(2.16)

  2. (2)

Let $\Omega_{1}\subset\Omega_{2}\subseteq{\mathbb{R}}^{\mathbf{N}}$ and
${\mathbf{J}}$ be the bounded operator of _extension_ of functions in
$H^{l}(\Omega_{1})$ to $H^{l}(\Omega_{2})$; suppose that
${\mathbf{k}}_{2}[{\mathbf{J}}u]={\mathbf{k}}_{1}[u].$ Then (2.16) holds.

###  2.4. Asymptotic perturbations

The major tool in the study of _spectral asymptotics_ for problems involving
singularities of various kinds is the asymptotic perturbation Lemma by
M.Birman and M.Solomyak, see, e.g., [23], Lemma 1.5. We reproduce it here, in
our present notations.

######  Lemma 2.5.

Let ${\mathbf{K}}$ be a compact self-adjoint operator, and there exists a
sequence ${\mathbf{K}}_{\epsilon}$, $\epsilon\to 0,$ such that

(i):

for some $\theta>0,$ the limits
${\mathbf{n}}_{\pm}({\mathbf{K}}_{\epsilon},\theta)=\mathfrak{m}_{\pm,\epsilon}$
exist and are finite;

(ii):

$\lim_{\epsilon\to
0}{\mathbf{n}}^{\sup}({\mathbf{K}}-{\mathbf{K}}_{\epsilon},\theta)=0.$

Then the limits $\mathfrak{m}_{\pm}=\lim_{\epsilon\to
0}\mathfrak{m}_{\pm,\epsilon}$ exist and
${\mathbf{n}}_{\pm}({\mathbf{K}},\theta)=\mathfrak{m}_{\pm}.$
If the condition _(i)_ above is satisfied for only one of the signs $\pm,$ the
conclusion of the Lemma holds for this sign.

Note that often, say, in [41], Lemma 3.1, the condition (ii) above is stated
in a stronger form, requiring not the asymptotic but a uniform in $\lambda$
estimate for singular numbers of the difference
${\mathbf{K}}-{\mathbf{K}}_{\epsilon}$. In our study, it is the less
restrictive condition (ii) that can actually be guaranteed, while uniform
estimates may be not available.

###  2.5. Spectral estimates for form perturbations

In this subsection we reduce the task of finding eigenvalue estimates for the
perturbations of the resolvent to the one for Birman-Schwinger type operators.
We start with a simple transformation.

######  Lemma 2.6.

Let ${\mathbf{A}},$ ${\mathbf{A}}_{{\mathbf{V}}}$ be the operators constructed
in Section 2.1,

${\mathbf{T}}={\mathbf{T}}_{{\mathbf{V}}}=({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})$
(2.17)

Then the following identity holds

$\displaystyle{\mathbf{A}}^{-1}-{\mathbf{A}}_{{\mathbf{V}}}^{-1}\equiv{\mathbf{R}}_{{\mathbf{V}}}={\mathbf{A}}^{-\frac{1}{2}}{\mathbf{T}}{\mathbf{A}}^{-\frac{1}{2}}-{\mathbf{A}}^{-\frac{1}{2}}{\mathbf{T}}(1+{\mathbf{T}})^{-1}{\mathbf{T}}{\mathbf{A}}^{-\frac{1}{2}}$
(2.18)
$\displaystyle\equiv{\mathbf{R}}_{{\mathbf{V}}}^{(1)}-{\mathbf{R}}_{{\mathbf{V}}}^{(2)}.$

###### Proof.

The relation (2.18) follows immediately from the obvious identity
$1-(1+{\mathbf{T}})^{-1}={\mathbf{T}}-{\mathbf{T}}(1+{\mathbf{T}})^{-1}{\mathbf{T}}.$
∎

Next we transform the terms ${\mathbf{R}}_{{\mathbf{V}}}^{(1)}$ and
${\mathbf{R}}_{{\mathbf{V}}}^{(2)}$ in (2.18).

######  Lemma 2.7.

The terms in (2.18) are equal to

${\mathbf{R}}_{{\mathbf{V}}}^{(1)}=({\mathbf{F}}\gamma{\mathbf{A}}^{-1})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-1}),$
(2.19)

and

${\mathbf{R}}_{{\mathbf{V}}}^{(2)}=({\mathbf{F}}\gamma{\mathbf{A}}^{-1})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})(1+{\mathbf{T}})^{-1}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-1}).$
(2.20)

###### Proof.

All manipulations below are legal since we deal with bounded operators only.
For (2.19), we use that
${\mathbf{F}}\gamma{\mathbf{A}}^{-1}=({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}){\mathbf{A}}^{-\frac{1}{2}}$,
therefore,

$({\mathbf{F}}\gamma{\mathbf{A}}^{-1})^{*}={\mathbf{A}}^{-\frac{1}{2}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}.$

The same identity takes care of the first and the last factors in (2.20). ∎

Next, we have the eigenvalue estimate for ${\mathbf{R}}_{{\mathbf{V}}}$.

######  Lemma 2.8.

Suppose that for some $\theta>0$, the following asymptotic singular numbers
estimate is known:

${\mathbf{n}}^{\sup}({\mathbf{F}}\gamma{\mathbf{A}}^{-1},2\theta)=\mathfrak{m}<\infty,$

$\|{\mathbf{U}}\|\leq 1$, and the operator
${\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}$ is compact. Then

${\mathbf{n}}^{\sup}({\mathbf{R}}_{{\mathbf{V}}},\theta)\leq 2\mathfrak{m}.$

###### Proof.

We express (2.20) in the form

${\mathbf{R}}^{(2)}_{{\mathbf{V}}}={\mathbf{K}}_{1}^{*}{\mathbf{K}}_{2}{\mathbf{K}}_{1},\,{\mathbf{K}}_{1}={\mathbf{F}}\gamma{\mathbf{A}}^{-1},{\mathbf{K}}_{2}={\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})(1+{\mathbf{T}})^{-1}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}.$

The operator ${\mathbf{K}}_{2}$ is compact, and we can apply Lemma 2.2 and
then the Ky Fan inequality for the sum of compact operators. ∎

Next we establish a relation for operators containing ${\mathbf{A}}^{-1}$ to a
higher power. It is a direct consequence of the Ky-Fan inequality in the limit
form, see (2.13).

######  Lemma 2.9.

Let ${\mathbf{F}}_{1},{\mathbf{F}}_{2}$ be two operators in $\mathfrak{G}$
such that
${\mathbf{Q}}_{\iota}={\mathbf{F}}_{\iota}\gamma{\mathbf{A}}^{-l_{\iota}/2}$
are compact operators and
${\mathbf{n}}^{\sup}({\mathbf{Q}}_{\iota},2\theta_{\iota})=\mathfrak{m}_{\iota}<\infty$,
$\iota=1,2$; let also ${\mathbf{B}}$ be a bounded operator in $\mathfrak{G}$.
Then, for $\theta^{-1}=\theta_{1}^{-1}+\theta_{2}^{-1}$,

${\mathbf{n}}^{\sup}({\mathbf{Q}}_{1}^{*}{\mathbf{B}}{\mathbf{Q}}_{2},\theta)\leq
C\|{\mathbf{B}}\|^{\theta}\mathfrak{m}_{1}^{\frac{\theta}{\theta_{1}}}\mathfrak{m}_{2}^{\frac{\theta}{\theta_{2}}}.$

More generally, iterating, if we have the product of several operators,
${\mathbf{Q}}_{\iota}={\mathbf{F}}_{\iota}\gamma{\mathbf{A}}^{-l_{\iota}/2},\,{\mathbf{n}}^{\sup}({\mathbf{Q}}_{\iota},2\theta_{\iota})=\mathfrak{m}_{\iota}<\infty,$
$\iota=1,\dots,2m$ and bounded operators ${\mathbf{B}}_{\iota},$ then

${\mathbf{n}}^{\sup}(\prod_{\kappa=1}^{m}{\mathbf{Q}}_{2\kappa}^{*}{\mathbf{B}}_{\kappa}{\mathbf{Q}}_{2\kappa-1},\theta)\leq
C\prod\|{\mathbf{B}}_{\kappa}\|^{theta}\prod\mathfrak{m}_{\iota}^{\frac{\theta}{\theta_{\iota}}}.$

with $\theta^{-1}=\sum\theta_{\iota}^{-1}.$

###  2.6. The difference of perturbed resolvents

Let two quadratic forms, ${\mathbf{v}}_{1}[u],$ ${\mathbf{v}}_{2}[u]$ be
given, similarly to (2.3),

${\mathbf{v}}_{1}[u]=({\mathbf{F}}_{1}u,{\mathbf{U}}_{1}{\mathbf{F}}_{1}u)_{\mathfrak{G}},\,{\mathbf{v}}_{2}[u]=({\mathbf{F}}_{2}u,{\mathbf{U}}_{2}{\mathbf{F}}_{2}u)_{\mathfrak{G}},\,u\in\mathfrak{H}^{1},$

both satisfying the conditions in Sect.2.1, and, additionally, the positivity
condition, as in (2.6),

We are going to study the difference of the resolvents of the perturbed
operators,

${\mathbf{R}}^{(1,2)}={\mathbf{A}}_{{\mathbf{V}}_{1}}^{-1}-{\mathbf{A}}_{{\mathbf{V}}_{2}}^{-1};$
(2.21)

our main interests lies in eigenvalue estimates (and, if possible,
asymptotics) for ${\mathbf{R}}^{(1,2)}.$ Lemma to follow provides us with a
convenient representation of this difference.

######  Lemma 2.10.

Suppose that for the quadratic form
${\mathbf{v}}[u]={\mathbf{v}}_{1}[u]-{\mathbf{v}}_{2}[u]$, with
${\mathbf{v}}_{1},{\mathbf{v}}_{2}$ as in (2.3), $1+{\mathbf{T}}_{\iota}\equiv
1+({\mathbf{F}}_{\iota}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}_{\iota}({\mathbf{F}}_{\iota}\gamma{\mathbf{A}}^{-\frac{1}{2}}),$
$\iota=1,2$ are invertible. Then the following identity is valid

${\mathbf{R}}^{(1,2)}={\mathbf{A}}^{-\frac{1}{2}}({\mathbf{T}}_{1}-{\mathbf{T}}_{2}){\mathbf{A}}^{-\frac{1}{2}}-{\mathbf{Z}}_{1}+{\mathbf{Z}}_{2},$

where

${\mathbf{Z}}_{\iota}={\mathbf{A}}^{-\frac{1}{2}}({\mathbf{T}}_{\iota}(1+{\mathbf{T}}_{\iota})^{-1}{\mathbf{T}}_{\iota}){\mathbf{A}}^{-\frac{1}{2}}.$

###### Proof.

We express the operator (2.21) as

${\mathbf{R}}^{(1,2)}=\left(({\mathbf{A}}_{{\mathbf{V}}_{1}})^{-1}-{\mathbf{A}}^{-1}\right)-\left(({\mathbf{A}}_{{\mathbf{V}}_{2}})^{-1}-{\mathbf{A}}^{-1}\right)$
(2.22)

and apply to both term on the right in (2.22) the perturbation formula (2.18).
∎

From this Lemma, we derive the eigenvalue estimate for ${\mathbf{R}}^{(1,2)}.$

######  Lemma 2.11.

Suppose that for the quadratic forms ${\mathbf{v}}_{1},{\mathbf{v}}_{2}$, the
conditions of Lemma 2.8 are fulfilled,

${\mathbf{n}}^{\sup}({\mathbf{A}}^{-\frac{1}{2}}({\mathbf{T}}_{1}-{\mathbf{T}}_{2}){\mathbf{A}}^{-\frac{1}{2}},\theta)\leq\mathfrak{m}.$

and the operators
${\mathbf{T}}_{\iota}={\mathbf{F}}_{\iota}\gamma{\mathbf{A}}^{-\frac{1}{2}}$
are compact. Then

${\mathbf{n}}^{\sup}({\mathbf{R}}^{(1,2)},\theta)=\mathfrak{m}.$

###### Proof.

The result follows immediately from the relation
${\mathbf{n}}^{\sup}({\mathbf{Z}}_{j},\theta)=0,$ which is already established
in the proof of Lemma 2.8.∎

Note that this property imposes restrictions not only upon the difference
${\mathbf{T}}_{1}-{\mathbf{T}}_{2}$, but also upon the operators
${\mathbf{T}}_{1},{\mathbf{T}}_{2}$ separately.

##  3\. Elliptic operators with singular perturbations

###  3.1. Singular form-perturbations

Let $\Omega\subseteq{\mathbb{R}}^{\mathbf{N}}$ be a connected domain and
$\Sigma$ be a compact subset in $\overline{\Omega}$; let $\mu$ be a Radon
measure supported in $\Sigma$. Further, let ${\mathcal{L}}$ be a second order
elliptic operator in $\Omega$ with smooth coefficients. The corresponding
quadratic form is

${\mathbf{a}}[u]=\int_{\Omega}\left[\sum_{j,k}a_{jk}(X)\partial_{j}u\overline{\partial_{k}u}+t|u(X)|^{2}\right]dX,\,t>0,$
(3.1)

(where the zero order term is added in order to grant positivity) with domain
$\mathfrak{d}[{\mathbf{a}}]=H^{1}(\Omega)$; this is the _Neumann_ realization
of ${\mathcal{L}}$; if needed, this will be reflected in the notation of the
quadratic form, ${\mathbf{a}}^{\sc\mbox{N}\hskip 1.0pt}$, and the same for the
corresponding operator ${\mathbf{A}}={\mathbf{A}}^{\sc\mbox{N}\hskip 1.0pt}.$

Along with $H^{1}(\Omega),$ we will consider higher order Sobolev spaces
$H^{l}(\Omega),$ with integer $l>1$ as well as, for a domain $\Omega$ with
compact complement, Hilbert spaces $\mathfrak{H}_{{\mathcal{L}}}^{l}(\Omega)$
which locally, near $\partial\Omega,$ are contained in $H^{l}$

Under certain regularity conditions (specified later on) the restriction
$\gamma$ of functions in the Sobolev space $H^{l}(\Omega)$ to $\Sigma$ is a
well defined bounded operator from the Sobolev space $H^{l}(\Omega)$ to the
Hilbert space $\mathfrak{G}=L_{2,\mu}(\Sigma)$ with respect to the measure
$\mu,$ in particular, as explained in Sect.3.2, Haußdorff measure
${\mathcal{H}}^{d}$ of the proper dimension $d\leq{\mathbf{N}}$. Namely, this
restriction, the operator $\gamma$, defined initially on
$C(\overline{\Omega})\cap H^{l}(\Omega),$ can be extended by continuity to the
whole of $H^{l}(\Omega).$

For a given real-valued function $V$ on $\Sigma,$ measurable with respect to
the measure $\mu$ on $\Sigma$, the function $F$ is defined as
$F=|V|^{\frac{1}{2}}$, a real $\mu$\- measurable function on $\Sigma$. The
bounded function $U(X)$ on $\Sigma$ is defined as the sign of $V(X)$:
$U(X)=\hbox{{\rm sign}}\,V(X):=V(X)|V(X)|^{-1}$ if $V(X)\neq 0$ and $U(X)=0$
otherwise. In this setting, the perturbing quadratic form ${\mathbf{v}}[u]$ is
defined as

$\displaystyle{\mathbf{v}}[u]=\int_{\Sigma}V(X)|u(X)|^{2}\mu(dX)=(F\gamma
u,UF\gamma u)_{L_{2}(\Sigma,\mu)}:=$ $\displaystyle({\mathbf{F}}\gamma
u,{\mathbf{U}}{\mathbf{F}}\gamma u)_{\mathfrak{G}},\,u\in
H^{1}(\Omega),\mathfrak{G}=L_{2,\mu}.$

with ${\mathbf{F}},{\mathbf{U}}$ being the operators of multiplication by
$F,U$; this is a concrete realization of the abstract scheme in Sect. 2. In
order to use this scheme, we need to check the abstract properties required in
Sect. 2 for the particular setting, which will be done in this Section.

In this paper, we consider the measure $\mu$ having compact support. We will
show that if the ’density’ $V$ belongs to a certain $L_{p,\mu}$ class (with
$p$ depending on $d,{\mathbf{N}},l$) or, in some cases, the Orlicz class, then
the conditions (2.1) and (2.6) are fulfilled and the above construction fits
into the abstract setting of Section 2.

We consider two different cases in the study of the spectral properties of the
difference of resolvents. On the one hand $\Sigma$ may be a compact set in the
boundary $\Sigma\subset\Gamma=\partial\Omega$, of positive, or possibly, zero
surface measure and of Haußdorff dimension
$d\in({\mathbf{N}}-2,{\mathbf{N}}-1]$. The domain $\Omega$ may be bounded or
having bounded complement. Here, the perturbation ${\mathbf{V}}$ consists in
setting Robin type boundary conditions on $\Sigma,$ with measure $V\mu$
serving as weight, while keeping intact the Neumann boundary conditions on the
remaining part of $\Gamma$. In particular, if $V=0$ everywhere on $\Gamma,$
the quadratic form ${\mathbf{a}}_{{\mathbf{v}}}$ coincides with (3.1). Such,
unperturbed, operator is the operator of the Neumann problem,
${\mathbf{A}}={\mathbf{A}}^{\sc\mbox{N}\hskip 1.0pt}$. It turns out that the
eigenvalue estimates for $({\mathbf{A}}^{\sc\mbox{N}\hskip
1.0pt})^{-1}-({\mathbf{A}}_{\mathbf{V}})^{-1}$ have order depending on the
Haußdorff dimension of $\Sigma,$ in the special case when $d={\mathbf{N}}-1$
for $\Sigma$ with positive surface measure, the eigenvalue asymptotics holds.
These results are valid for the difference of resolvents of two Robin type
problems as well.

The second case concerns $\Omega={\mathbb{R}}^{{\mathbf{N}}}$ and
$\Sigma\subset{\mathbb{R}}^{{\mathbf{N}}}$ being a compact set of Haußdorff
dimension $d\in({\mathbf{N}}-2,{\mathbf{N}}]$ with a Haußdorff
${\mathcal{H}}^{d}$– measurable function $V$ on $\Sigma.$ Here, we consider
the difference of resolvents of Schrödinger like operators with, possibly
singular (for $d<{\mathbf{N}}$), potentials. In particular, the case fits in
this picture when the set $\Sigma$ is a Lipschitz hypersurface, and the
perturbation ${\mathbf{V}}$ consists, in fact, in setting ’transmission type’
condition on $\Sigma$. In this way, we include here $\delta$-interactions on a
hypersurface, $d={\mathbf{N}}-1$, systematically studied in the literature,
see, e.g., [10], but we fail to cover $\delta^{\prime}$-type interactions,
since this perturbation does not fit into our quadratic form approach. We
obtain spectral estimates for the difference of powers of the resolvents as
well and its eigenvalue asymptotics.

Further on, we specify the properties of the set $\Sigma$ and the weight
function $V$, which grant that the abstract conditions in Section 2 are
fulfilled.

###  3.2. Singular sets and measures

We recall the definition of Ahlfors regular sets and measures. Let $\Sigma$ be
a compact set in ${\mathbb{R}}^{\mathbf{N}}$ and $\mu$ be a Radon measure
supported in $\Sigma$. For $d\in(0,{\mathbf{N}}]$, the measure $\mu$ is called
_Ahlfors regular_ of dimension $d$ (sometimes the word ’$d$-set’ is used for
$\Sigma$) if for any $r\in(0,\operatorname{diam\,}(\Sigma)]$ and any point
$X\in\Sigma,$

${\mathcal{A}}_{-}r^{d}\leq\mu(\Sigma\cap B(X,r))\leq{\mathcal{A}}_{+}r^{d},$
(3.2)

with some positive constants
${\mathcal{A}}_{\pm}={\mathcal{A}}_{\pm}(\Sigma)$, $B(X,r)$ denoting the ball
with center at $X$ and radius $r$.

This notion is widely used in geometric measure theory and many other topics
in Analysis, see, e.g., [27], [71]. It is known that such measure is
equivalent to the $d$-dimensional Haußdorff measure ${\mathcal{H}}^{d}$ on
$\Sigma$. If the measure $\mu$ satisfies (3.2) we will write
$\mu\in\mathfrak{A}^{d},$ or $\Sigma\in\mathfrak{A}^{d}.$

For an integer dimension $d$, the most obvious example, the leading one in
this paper, is a compact Lipschitz surface or a finite union of such surfaces.
A more general construction, valid for fractional $d$ as well, is the
following (we refer to the exposition in [71], Ch.4,5).

Self-similar measures are multi-dimensional generalizations of the well-known
Cantor sets with Haußdorff measure of a proper dimension. An affine mapping
$S$ in ${\mathbb{R}}^{{\mathbf{N}}}$ is called _similitude_ with ratio $\rho$
if it is a composition of a translation, a rotation, and a homothety with
coefficient $\rho<1$. Let us have a finite collection of similitudes
${\mathcal{S}}=\\{S_{j}\\},j=1,\dots,N$, with ratios $\rho_{j}.$ The compact
set ${\mathcal{K}}$ is called invariant with respect to ${\mathcal{S}}$ if
${\mathcal{K}}=\cup_{j}S_{j}{\mathcal{K}}$. For any such collection of
similitudes, there exists a unique invariant compact set ${\mathcal{K}}$. This
set, for every fixed $m$, satisfies

${\mathcal{K}}=\bigcup_{{\mathbf{j}}=(j_{1},\dots,j_{m})}S_{j_{1}}\dots
S_{j_{m}}{\mathcal{K}}.$

Let $d$ be the Haußdorff dimension of ${\mathcal{K}}.$ The set ${\mathcal{K}}$
is called _selfsimilar_ , if, moreover,

${\mathcal{H}}^{d}(S_{j}{\mathcal{K}}\cap
S_{j^{\prime}}({\mathcal{K}}))=0,\,\,j\neq j^{\prime}.$ (3.3)

A strengthening of (3.3) is the _open set condition_ : there exists an open
set ${\mathcal{O}}$ containing ${\mathcal{K}}$ such that

$S_{j}({\mathcal{O}})\cap S_{j^{\prime}}({\mathcal{O}})=\varnothing,j\neq
j^{\prime},\,\,\mbox{and}\,\bigcup_{j}S_{j}({\mathcal{O}})\subset{\mathcal{O}}.$

The property of our interest is that if the open set condition is satisfied,
then the Haußdorff dimension $d$ of ${\mathcal{K}}$ equals the only positive
solution of

$\sum_{j}\rho_{j}^{d}=1,$

and the Haußdorff measure ${\mathcal{H}}^{d}$ restricted to ${\mathcal{K}}$
belongs to $\mathfrak{A}^{d}$. Obviously, a bilipschitz image of a $d$-set is
a $d$-set again. More examples one can get by considering a disjoint union of
$d$-sets, finite, and under some conditions, infinite.

It was found in [67] that sometimes, when studying spectral estimates, only
one of the inequalities in (3.2) is needed. In this connection, we call the
measure (the set) lower-, resp., upper-regular of order $d$ if the left,
resp., the right inequality in (3.2) is satisfied. The corresponding classes
of measures will be denoted $\mathfrak{A}^{d}_{-},$ resp.,
$\mathfrak{A}^{d}_{+}$, so
$\mathfrak{A}^{d}=\mathfrak{A}^{d}_{+}\cap\mathfrak{A}^{d}_{-}$. (If the
support of the measure $\mu$ is not compact, the inequalities in (3.2) are
supposed to hold for all $r>0$, we, however, restrict our considerations to
compact sets.)

###  3.3. Regularity of the boundary

For different types of spectral problems under consideration, we will require
different regularity level of the domain
$\Omega\subset{\mathbb{R}}^{\mathbf{N}}.$ For the Schrödinger type problems,
where only the local regularity of solutions of the elliptic equations is
used, $\Omega$ is an arbitrary domain without additional restrictions, and we
can consider the whole ${\mathbb{R}}^{\mathbf{N}}$. For boundary problems with
Robin boundary conditions, when considering the difference of resolvents, we
suppose that the boundary is present, is compact and is of class $C^{1,1},$
this means that the boundary can be represented, locally in properly rotated
local co-ordinates $X=(x_{1},x^{\prime})$, as the graph of a function
$x_{1}=\psi(x^{\prime})$ where first order partial derivatives of the function
$\psi$ belong to the Lipschitz class. In the latter case, the following local
regularity property plays the key role.

######  Proposition 3.1.

Let $\Omega$ be a domain of class $C^{1,1}$ in ${\mathbb{R}}^{\mathbf{N}}$ and
let ${\mathcal{L}}$ be a Hermitian elliptic second order operator in $\Omega$
with smooth coefficients. Let $\Omega^{\prime}\subset\Omega$ be a bounded
domain with $C^{1,1}$ boundary such that the boundary of $\Omega$ lies in
$\overline{\Omega^{\prime}}$. Then for any $f\in L_{2}(\Omega),$ the (unique)
solution $u$ of the Neumann boundary problem

${\mathcal{L}}u+tu=f,\,\,\partial_{\nu({\mathcal{L}})}u_{|_{\partial\Omega}}=0,t>0,$
(3.4)

belongs to the Sobolev space $H^{2}(\Omega^{\prime})$ and

$\|u\|_{H^{2}(\Omega^{\prime})}\leq C\|f\|_{L^{2}(\Omega^{\prime})}.$

For a smooth boundary, this is the general elliptic regularity fact, presented
in a number of standard sources. For the boundary of finite smoothness, as
above, this result can be found, with different degree of detalization, in
certain advanced treaties on elliptic boundary problems, sometimes in the
version for the Dirichlet boundary conditions, with referral to a similar
reasoning for the Neumann conditions. We can cite here, e.g., [38], Theorem
2.5.1.1.

For higher order operators, we suppose, for the sake of brevity, that the
boundary is infinitely smooth. For a bounded domain $\Omega,$ the general
elliptic regularity results, see, e.g., [51], imply that the operator
$({\mathbf{A}}^{\sc\mbox{N}\hskip 1.0pt}+t)^{-l/2}$ acts continuously from
$L_{2}(\Omega)$ to $H^{l}(\Omega)$. For an _exterior_ domain $\Omega$ with
compact boundary, some more discussion is needed.

We suppose that the operator ${\mathcal{L}}$ is uniformly elliptic in $\Omega$
and has coefficients with bounded derivatives of all orders. Let
${\mathbf{A}}^{\sc\mbox{N}\hskip 1.0pt}$ be the Neumann operator for
${\mathcal{L}}+t$ in $\Omega.$ The domain of $({\mathbf{A}}^{\sc\mbox{N}\hskip
1.0pt})^{\frac{1}{2}}$ is the Sobolev space $H^{1}(\Omega).$ The domain of a
higher power, $({\mathbf{A}}^{\sc\mbox{N}\hskip 1.0pt})^{\frac{l}{2}}$, of our
operator (we denote this domain by $\mathfrak{H}_{{\mathcal{L}}}^{l}$, with
graph norm), may be hard to describe explicitly as soon as it concerns the
behavior at infinity. Locally, as follows from the elliptic regularity,
functions in $\mathfrak{H}_{{\mathcal{L}}}^{l}$ belong to $H^{l}.$ A detailed
analysis of the boundary regularity of functions in
$\mathfrak{H}_{{\mathcal{L}}}^{l}$ is performed, e.g., in [40], [53] (see,
especially, Sect.4, Proposition 4.9 in the latter paper.) It follows, in
particular, that the functions in $\mathfrak{H}_{{\mathcal{L}}}^{l}$ belong to
$H^{l}$ near the boundary. These results can be summarized as the following.

######  Proposition 3.2.

Let $\Omega\subset{\mathbb{R}}^{\mathbf{N}}$ be a domain with smooth compact
boundary $\Gamma=\partial\Omega$. Let $\Omega^{\prime}$ be a bounded subdomain
in $\Omega$ such that $\Gamma\subset\overline{\Omega^{\prime}}$. Then the
operator ${\mathbf{J}}$ of restriction of functions in
$\mathfrak{H}_{{\mathcal{L}}}^{l}$ to $\Omega^{\prime}$ acts boundedly to
$H^{l}(\Omega^{\prime}).$

It follows, in particular, that the operator $({\mathbf{A}}^{\sc\mbox{N}\hskip
1.0pt})^{-l/2}$ acts as a bounded operator from $L_{2}(\Omega^{\prime})$ to
$H^{l}(\Omega^{\prime}).$

Another fact, important in our study, is the possibility of continuation of
solutions of the equation (3.4) from $\Omega$ across the boundafy, with
preservation of smoothness.

######  Lemma 3.3.

If $\Omega$ is a $C^{0,1}$ domain with compact boundary then there exists a
bounded extension operator ${\mathbf{J}}:H^{l}(\Omega)\to
H^{l}({\mathbb{R}}^{\mathbf{N}})$ for all $l$.

This is a particular case of the classical Stein extension theorem, see
Theorem 5, Chapter 6, in [69].

Thus, for a smooth boundary, under the conditions of Proposition 3.1, there
exist a continuation operator ${\mathbf{J}}$ in $H^{l}$ for all solution of
the Neumann problem.

###  3.4. Singular quadratic forms in the space $H^{l}$

The conditions we impose on the functions defined on singular sets will depend
on the exponent $\theta>0.$ For the sake of the uniformity of formulations, we
introduce the following notation. For a given exponent $\theta$ and a finite
measure $\mu$, we denote by $L_{(\theta),\mu}$ the space $L_{\theta,\mu}$ for
$\theta>1$, $L_{1,\mu}$ for $\theta<1$, and, for $\theta=1$, the Orlicz space
$L_{\Psi,\mu}$ corresponding to the function $\Psi(s)=(s+1)\log(s+1)-s:$
$\int\Psi(|V(X))|\mu(dX)<\infty.$ By $\|F\|_{(\theta),\mu}$ we denote the norm
of a function $F$ in this space. There are several equivalent ways to
introduce a norm in the Orlicz space; we will use the Luxemburg norm (see,
e.g., Chapter 8 in [69]).

The statement to follow presents some basic, mostly known, facts about the
properties of the quadratic form

${\mathbf{a}}_{V\mu}[u]=\int V(X)|u(X)|^{2}\mu(dX)$ (3.5)

considered in the Sobolev spaces $H^{l}({\mathbb{R}}^{\mathbf{N}})$. We will
use it for integer values of $l$, while $l=1$ and $l=2$ are the most important
ones.

First of all, if ${\mathbf{N}}<2l$, the Sobolev space
$H^{l}({\mathbb{R}}^{\mathbf{N}})$ is embedded in
$C({\mathbb{R}}^{\mathbf{N}}),$ therefore the quadratic form
${\mathbf{a}}_{V\mu}[u]$ is well defined and bounded in
$H^{l}({\mathbb{R}}^{\mathbf{N}})$ for any finite measure $\mu$ and for any
$V\in,L{1,\mu}.$ The case ${\mathbf{N}}\geq 2l$ is considerably more delicate.

######  Lemma 3.4.

Let ${\mathbf{N}}\geq 2l$ and let the measure $\mu$ belong to
$\mathfrak{A}_{+}^{d}$, for some $d$,
${\mathbf{N}}-2l<d\leq{\mathbf{N}}$ if ${\mathbf{N}}>2l$,
$\mu\in\mathfrak{A}^{d}$, $d>0$ if ${\mathbf{N}}=2l$. Suppose that $V\in
L_{(\theta),\mu}$, $\theta=\frac{d}{d-{\mathbf{N}}+2l}$. Then

  1. (1)

the quadratic form (3.5), defined initially for $u\in
H^{l}({\mathbb{R}}^{\mathbf{N}})\cap C({\mathbb{R}}^{\mathbf{N}})$, is bounded
in the $H^{l}({\mathbb{R}}^{\mathbf{N}})$-metric and admits a unique bounded
extension to the whole Sobolev space $H^{l}({\mathbb{R}}^{{\mathbf{N}}});$

$\left|\int|u(X)|^{2}V(X)\mu(dX)\right|\leq
C_{0}\|V\|_{L_{(\theta),\mu}}\|u\|^{2}_{H^{l}({\mathbb{R}}^{\mathbf{N}})},\,u\in
H^{l}({\mathbb{R}}^{\mathbf{N}});$ (3.6)

  2. (2)

the quadratic form

${\mathbf{a}}_{V\mu}=\int_{{\mathbb{R}}^{\mathbf{N}}}|\nabla^{l}u(X)|^{2}dX+\int|u(X)|^{2}V(X)\mu(dX),u\in
H^{l}({\mathbb{R}}^{\mathbf{N}}),$ (3.7)

is lower semibounded in $H^{l}({\mathbb{R}}^{\mathbf{N}})$:

$\displaystyle\int_{{\mathbb{R}}^{\mathbf{N}}}|\nabla^{l}u(X)|^{2}dX+\int|u(X)|^{2}V(X)\mu(dX)+t\int_{{\mathbb{R}}^{\mathbf{N}}}|u(X)|^{2}dX\geq$
(3.8) $\displaystyle
c\int_{{\mathbb{R}}^{\mathbf{N}}}|\nabla^{l}u(X)|^{2}dX,\,u\in
H^{l}({\mathbb{R}}^{\mathbf{N}})$

for some $t\in{\mathbb{R}}^{1}$.

The same statements are valid, with ${\mathbb{R}}^{\mathbf{N}}$ replaced in
the formulation by a domain $\Omega$ with compact $C^{0,1}$ boundary.

###### Proof.

It suffices to prove Lemma for the whole space; the case of a domain follows
immediately by using the Stein extension theorem. Statement (1) can be found,
e.g., in [65] for the case ${\mathbf{N}}=2l$ and in [67] for
${\mathbf{N}}>2l$; both are direct consequences of trace theorems in the book
by V. Maz’ya, [55].

After this, the starting point in the proof of Statement (2) is the classical
multiplicative inequality in the Sobolev space. We reproduce its version as a
particular case of Theorem 1.4.7 in [55]:
_Let the measure $\mu$ belong to $\mathfrak{A}^{d}_{+}$ with
$d>{\mathbf{N}}-2l.$ Then for any function $u$ in $\sc\mbox{D}\hskip
1.0pt({\mathbb{R}}^{\mathbf{N}})$,_

$\|u\|_{L_{q,\mu}}\leq
C_{0}\|\nabla^{l}u\|_{L_{2}({\mathbb{R}}^{\mathbf{N}})}^{\tau}\|u\|_{L_{2}({\mathbb{R}}^{\mathbf{N}})}^{1-\tau},\,q\geq
2,\,\frac{d}{q}>\frac{{\mathbf{N}}}{2l}-1,\tau=\frac{{\mathbf{N}}}{2l}-\frac{d}{q}.$
(3.9)

_with constant $C$ depending on the measure $\mu$ and parameters
${\mathbf{N}},q,d.$ _

It follows from (3.9) that for any $\epsilon>0$, there exists $C_{\epsilon}$
such that

$\|u\|^{2}_{L_{q,\mu}}\leq\epsilon\|\nabla^{l}u\|^{2}_{L_{2}({\mathbb{R}}^{\mathbf{N}})}+C_{\epsilon}\|u\|^{2}_{L_{2}({\mathbb{R}}^{{\mathbf{N}}})}.$
(3.10)

Now, for proving (3.8), for a given $V\in L_{\theta,\mu},$
$\theta=\frac{d}{d-{\mathbf{N}}+2l}$ and given $\delta>0$, we find such
$h=h(\delta)>0$ so that

$V=V_{1}+V_{2},\,|V_{1}(X)|\leq h,\|V_{2}\|_{\theta,\mu}<\delta.$

Correspondingly, the quadratic form ${\mathbf{a}}_{V\mu}$ splits into the sum,
${\mathbf{a}}_{V\mu}[u]={\mathbf{a}}_{V_{1}\mu}[u]+{\mathbf{a}}_{V_{2}\mu}[u].$
For ${\mathbf{a}}_{V_{1}\mu}[u],$ we apply (3.10) with $q=2,$
$\tau=\frac{{\mathbf{N}}-d}{2l}$, which gives

$\displaystyle|{\mathbf{a}}_{V_{1}\mu}[u]|=\left|\int
V_{1}(X)|u(X)|^{2}\mu(dX)\right|\leq$ $\displaystyle h\|u\|_{L_{2,\mu}}\leq
h\epsilon\int_{{\mathbb{R}}^{\mathbf{N}}}|\nabla^{l}u|^{2}dX+hC_{\epsilon}\int_{{\mathbb{R}}^{\mathbf{N}}}|u(X)|^{2}dX.$

To estimate the second term, we apply (3.10) for
$q=\frac{2d}{{\mathbf{N}}-2l},$ i.e., $\tau=1$. We obtain, by the Hölder
inequality

$|{\mathbf{a}}_{V_{1}\mu}[u]|\leq\|V_{2}\|_{L_{\theta,\mu}}\|u\|^{2}_{L_{q,\mu}}\\\
\leq C\delta\int_{{\mathbb{R}}^{\mathbf{N}}}|\nabla^{l}u(X)|^{2}dX.$

We take $\epsilon,\delta$ so small that $a\epsilon<\frac{1}{4},$
$C\delta<\frac{1}{4},$ therefore,

$\left|\int
V(X)|u(X)|^{2}\mu(dX)\right|\leq\frac{1}{2}\int_{{\mathbb{R}}^{\mathbf{N}}}|\nabla^{l}u|^{2}dX+C^{\prime}\|u\|^{2}_{L_{2}({\mathbb{R}}^{\mathbf{N}})},$

and, finally,

$\int_{{\mathbb{R}}^{\mathbf{N}}}|\nabla^{l}u|^{2}dX+\int
V(X)|u(X)|^{2}\mu(dX)\geq\frac{1}{2}\int_{{\mathbb{R}}^{\mathbf{N}}}|\nabla^{l}u|^{2}dX-C^{\prime}\|u\|^{2}_{L_{2}({\mathbb{R}}^{{\mathbf{N}}})},$

which justifies the semiboundedness inequality.

In dimension ${\mathbf{N}}=2l,$ a similar reasoning proves the required
semiboundedness of the quadratic form (3.7), with the weight $V$ belonging to
the Orlicz class $L_{\Psi,\mu}.$ The trace inequality replacing (3.9), (3.10)
is Theorem 11.8 in [55] and its $H^{l}$ version, see detains in [65], [62]. ∎

In the next section, we will use these results for $l=1$ and $l=2$. The
results for higher order will be used in the treatment of the difference of
powers of resolvents.

###  3.5. A singular measure on the boundary and the operator for the Robin
problem

Here we discuss the same kind of inequalities for the measure supported at the
boundary of $\Omega.$ We recall that the classical Robin boundary problem in a
domain $\Omega\subset{\mathbb{R}}^{{\mathbf{N}}}$ for the elliptic operator
${\mathcal{L}}$ is

${\mathcal{L}}u+tu=f\,\mbox{in}\,\,\Omega,\,\partial_{\nu({\mathcal{L}})}u(X)+V(X)u(X)=0\,\,\mbox{on}\,\,\partial\Omega,$
(3.11)

with a ’nice’ function $V.$ This problem can be expressed in the variational
form, see, e.g., Sect.5 in [8], see also [53]. The operator which maps $u$ to
$f$ is defined by the quadratic form

$\displaystyle{\mathbf{a}}_{R,V}=\int_{\Omega}\langle\mathfrak{a}(X)\nabla
u(X),\nabla u(X)\rangle dX+t\int_{\Omega}|u(X)|dX+$ (3.12)
$\displaystyle\int_{\partial\Omega}V(X)|(\gamma u)(X)|^{2}\sigma(dX),u\in
H^{1}(\Omega),$

where $\mathfrak{a}(X)$ is the matrix of leading coefficients of
${\mathcal{L}}$ and $\sigma$ is the natural surface measure on
$\partial\Omega.$ The quadratic form is considered on the Sobolev space
$H^{1}(\Omega)$ equipped with the metric defined by the first two terms in
(3.12). Note that for $u\in H^{1}(\Omega)$, the trace $\gamma u$ on the
boundary is well defined as an element in $L_{2}(\partial\Omega)$ and even as
an element in the Sobolev space $H^{\frac{1}{2}}(\partial\Omega)$, as soon as
the boundary is Lipschitz. It is in this variational form that we will
introduce the generalization of the classical Robin problem (3.11). In more
recent papers, a more general setting was considered, with the function $V$ in
(3.11) replaced by a pseudodifferential operator $\Theta$ on $\partial\Omega$,
acting in some convenient way between Sobolev spaces on $\partial\Omega$, see,
e.g., [18], [15], [16], [53]. In our case, this operator will be replaced by a
measure on $\partial\Omega,$ possibly, singular with respect to $\sigma$, with
summability conditions imposed.

Namely, having a measure $\mu$ supported in $\Gamma=\partial\Omega$ and a
$\mu$-measurable function $V$, we can consider the quadratic form
${\mathbf{a}}_{V\mu}[u]=\int V(X)|\gamma u(X)|^{2}\mu(dX)$ instead of the one
on the second line in (3.12).

For such quadratic form, the results similar to the ones in the previous
subsection are valid, under the condition that the boundary
$\Gamma=\partial\Omega$ possesses the $H^{l}$-extension property; in our case,
recall, we suppose that it is a Lipschitz one.

######  Lemma 3.5.

Let the measure $\mu$ be supported in $\Gamma$. Suppose that the conditions of
Lemma 3.4 are satisfied, just with the inequality
${\mathbf{N}}-2l<d\leq{\mathbf{N}}$ replaced by
${\mathbf{N}}-2l<d\leq{\mathbf{N}}-1$. Then the statements in Lemma 3.4 are
correct.

###### Proof.

We consider the bounded _extension_ operator ${\mathbf{J}}:H^{l}(\Omega)\to
H^{l}({\mathbb{R}}^{\mathbf{N}}),$ $\|{\mathbf{J}}\|=C_{1}$. and obtain, using
the fact that ${\mathbf{a}}_{V\mu}[{\mathbf{J}}u]={\mathbf{a}}_{V\mu}[u]$

${\mathbf{v}}_{V\mu}[u]\leq
C_{0}\|{\mathbf{J}}u\|^{2}_{H^{l}(\Omega^{\prime})}\leq
C_{0}C_{1}\|u\|^{2}_{H^{l}(\Omega)},$ (3.13)

where $C_{0}$ is the constant in (3.6). The same reasoning takes care of the
lower semiboundedness of the quadratic form ${\mathbf{a}}_{V\mu}.$ ∎

.

##  4\. Spectral estimates for singular measures

In this section, we adapt the results of the papers [65] and [67] to our
concrete setting. We consider two main cases, $l=1$, corresponding to
quadratic forms in the Sobolev space $H^{1}(\Omega)$, and $l=2$, corresponding
to the Sobolev space $H^{2}(\Omega)$. Then we discuss also the case of a
general integer $l$, used later to treat the difference of powers of
resolvents.

###  4.1. Upper spectral estimates

In the papers [65], [67] spectral estimates of weighted pseudodifferential
operators with singular measure acting as weight were obtained. Let $\mu$ be a
measure with support (the smallest closed set of full $\mu$-measure)
$\Sigma\subset\overline{\Omega}\subseteq{\mathbb{R}}^{\mathbf{N}}$ and let
$V(x)$ be a $\mu$-measurable function on $\Sigma$ (supposed, if necessary,
being extended by zero to the whole of $\Omega$ or to the whole of
${\mathbb{R}}^{\mathbf{N}}$); we denote by $\boldsymbol{\pi}$ the measure
$V\mu$.

First, we will obtain spectral estimates for an operator
${{\mathbf{S}}}={{\mathbf{S}}}(\boldsymbol{\pi})={{\mathbf{S}}}(\boldsymbol{\pi},l,\Omega)$
defined by the quadratic form
${\mathbf{v}}[u]=\int|u(X)|^{2}\boldsymbol{\pi}(dX)$ in the Sobolev space
$H^{l}(\Omega).$ The decisive property of these estimates is that they depend
not on the $L_{\infty}$ norm of the weight function $V$, but on its norm in
some integral metric. Since the groundbreaking papers by M. Birman and M.
Solomyak, such estimates keep being, probably, the only way to prove spectral
_asymptotics_ for operators with singularities.

The boundedness of ${\mathbf{S}}$, equivalently, the boundedness of the
quadratic form ${\mathbf{v}}[u]$ in $H^{l}(\Omega)$, was established in Sect.
3. Under the conditions below, the operator ${\mathbf{S}}$ turns out to be
compact. We will use the notations set in Sect. 2 for the characteristics of
the distribution of the spectrum of ${\mathbf{S}}$.

The following results were established in [65], [67]. The conditions imposed
on the measure look differently for the following three cases:

$\displaystyle\left\\{\begin{array}[]{ll}\mbox{subcritical},&{2l<{\mathbf{N}},\,\mu\in\mathfrak{A}^{d}_{+},\,d>{\mathbf{N}}-2l;}\\\
\mbox{critical},&{2l={\mathbf{N}},\,\mu\in\mathfrak{A}^{d},\,0<d\leq{\mathbf{N}};}\\\
\mbox{supercritical},&{2l>{\mathbf{N}},\,\mu\in\mathfrak{A}^{d}_{-},\,0<d\leq{\mathbf{N}}.}\end{array}\right.$
(4.4)

######  Theorem 4.1.

_[_ Theorem 3.3 in [65], Theorem 3.3 and Theorem 3.9 in [67]_]_ and let
$\theta=\frac{d}{d+2l-{\mathbf{N}}}$. Let the weight function $V$ belong to
$L_{(\theta),\mu}$. Then for the corresponding case in (4.4) above, for the
operator ${\mathbf{S}}$, the following estimates hold:

$\displaystyle{\mathbf{n}}^{\sup}({\mathbf{S}},\theta)\leq
C\|V\|_{(\theta),\mu}^{\theta},$ (4.5) and
$\displaystyle{\mathbf{n}}_{\pm}^{\sup}({\mathbf{S}},\theta)\leq
C\|V_{\pm}\|_{(\theta),\mu}^{\theta},$

with constant $C$ not depending on $V$ in the corresponding space. In the
sources cited above, the dependence of the constants in the spectral estimates
on the measure $\mu$ and, especially, on the constants in (3.2), are described
as well, however, since the measure in our considerations is fixed, we do not
need such results here. In the critical case, in [65], a different, ’averaged’
norm in the Orlicz space is used. Using the Luxemburg norm instead changes
only the constant $C$ in (4.5).

Eigenvalue estimates in Theorem 4.1 extend easily to operators
${\mathbf{S}}(\boldsymbol{\pi},l,{\mathcal{L}},\Omega)$ generated by the
quadratic form ${\mathbf{v}}[u]$ in the space
$\mathfrak{H}_{{\mathcal{L}}}^{l},$ the domain of the operator
${\mathbf{A}}^{l/2},$ for the case when the measure $\mu$ is supported
strictly _inside_ $\Omega.$

######  Corollary 4.2.

Let, in conditions of Theorem 4.1, ${\mathcal{L}}$ be a second order elliptic
operator in $\Omega$ with smooth coefficients. Then for the operator
${\mathbf{S}}(\boldsymbol{\pi},l,{\mathcal{L}},\Omega)$ spectral estimates
(4.5) hold.

###### Proof.

We use Lemma 2.3 and its corollary. First, we take a bounded domain
$\Omega_{1}\subset\Omega$ with smooth boundary, $\hbox{{\rm
supp}}\,\boldsymbol{\pi}\subset\Omega_{1}$. By the local elliptic regularity,
the operator ${\mathbf{J}}_{1}$, the restriction of functions in
$\mathfrak{H}_{{\mathcal{L}}}^{l}$ to $H^{l}(\Omega_{1})$ is bounded. Next,
the Stein operator ${\mathbf{J}}_{2}$ of extension from $H^{l}(\Omega)$ to
$H^{l}({\mathbb{R}}^{\mathbf{N}})$ is bounded as well. By Corollary 2.4, the
spectral estimates survive, probably, with some controlled degradation in the
constants. ∎

We recall here that the eigenvalue estimates for the operator
${\mathbf{S}}(\boldsymbol{\pi},l,{\mathcal{L}},\Omega)$ are equivalent to
eigenvalue estimates for the operator

${\mathbf{W}}_{l}=({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{l}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{l}{2}}),$

see Sect.2. These estimates will be discussed in the next section.

###  4.2. Estimates for measures supported at the boundary of $\Omega$

In the study of the Robin boundary problem, we will need eigenvalue estimates
for the case when the measure $\mu$ is supported in the boundary of the domain
$\Omega\subset{\mathbb{R}}^{{\mathbf{N}}}$, $\mu\in\mathfrak{A}^{d}$ with
${\mathbf{N}}-2<d\leq{\mathbf{N}}-1$.

For a bounded domain $\Omega\subset{\mathbb{R}}^{\mathbf{N}}$ with boundary of
class $C^{1,1}$, for the elliptic operator ${\mathcal{L}}$ with coefficients
smooth in $\overline{\Omega}$, we know that the domain of
$({\mathbf{A}}^{\sc\mbox{N}\hskip 1.0pt})^{\frac{1}{2}}$ coincides with
$H^{1}(\Omega)$ and the domain of ${\mathbf{A}}^{\sc\mbox{N}\hskip 1.0pt}$
coincides with $H^{2}(\Omega).$ Therefore, the eigenvalue estimates for the
operator defined by the quadratic form ${\mathbf{v}}$ in these Hilbert spaces
are the same as estimates in the corresponding Sobolev spaces, i.e., are the
same as in Theorem 4.1. For higher values of $l,$ we require that the boundary
$\Gamma$ is smooth, and the reasoning goes in the same way, using the elliptic
regularity.

For an unbounded domain $\Omega$ with compact boundary, we use the boundary
elliptic regularity, described in Sect. 3.3. The functions in the domain of
$({\mathbf{A}}^{\sc\mbox{N}\hskip 1.0pt})^{\frac{l}{2}}$ belong to $H^{l}$ in
a neighborhood of the boundary, and we can repeat the reasoning in Corollary
4.2, where Corollary 2.4 is used again, reducing the estimation of eigenvalues
of the operator ${\mathbf{S}}(\boldsymbol{\pi},l,{\mathcal{L}},\Omega)$ to
estimates in the Sobolev space, found in Theorem 4.1.

###  4.3. Lower estimates

We present here some lower spectral estimates for operators of the form
${\mathbf{S}},$ with sign-definite, say, positive weight function $V$. Here,
unlike the upper estimates above, the measure $\mu$ is supposed to be two-
sided Ahlfors-regular, $\mu\in\mathfrak{A}^{d},$ for all relations between $l$
and ${\mathbf{N}}$, therefore, we may suppose that $\mu$ coincides with the
Haußdorff measure ${\mathcal{H}}^{d}$. We reproduce here the results
established in the book [71]. The particular case we are interested in is
$l=2.$ For this case, Theorem 28.6 in [71], with $m=2,$ $\varkappa=4,$ gives
the following result. We formulate it in our present notations.

######  Theorem 4.3 (Triebel’s lower estimate.).

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{\mathbf{N}}$, let $\Sigma$
be a compact set in $\Omega,$ Ahlfors $d$-regular
${\mathbf{N}}-4<d\leq{\mathbf{N}}$, $m=1,\varkappa=2$, $\mu={\mathcal{H}}^{d}$
on $\Sigma$. Consider the operator ${\mathbf{S}}={\mathbf{S}}(\mu,2,\Omega)$,
defined by the quadratic form $\int_{\Sigma}|u(X)|^{2}{\mathcal{H}}^{d}(dX)$
in the Sobolev space $H^{2}(\Omega)$. Then

$n^{\inf}({\mathbf{S}},\theta)>0,\theta=\frac{d}{4-{\mathbf{N}}+d}.$

######  Remark.

The equivalence of our formulation with H.Triebel’s one is explained in [71],
see (28.27) there. Theorem 4.3 does not provide estimates for operators with
non-sign-definite $V$ and we refer to it only in order to give a que that our
_upper_ eigenvalue estimates are order sharp.
As it concerns _upper_ spectral estimates established in [71], the interested
Reader is addressed to the Appendix in [67], where the non-sharpness of these
estimates is discussed, therefore, we used more sharp estimates in Theorem 4.1
instead.

##  5\. The resolvent of a singular Schrödinger operator

Here we apply the general results on the spectrum of singular quadratic form
perturbations.

###  5.1. The Schrödinger operator with singular potential

Let $\Omega$ be an unbounded domain in ${\mathbb{R}}^{{\mathbf{N}}}$ with
$C^{1,1}$ boundary and compact complement. Suppose that $\Sigma$ is a compact
subset in $\Omega$ and let $\mu$ be a Radon measure supported in $\Sigma$ and
$V(X)$ be a real $\mu$-measurable function. Let further on, ${\mathcal{L}}$ be
a uniformly elliptic formally self-adjoint second order operator in $\Omega,$

${\mathcal{L}}=-\sum_{j,j^{\prime}}\partial_{j}a_{j,j^{\prime}}(X)\partial_{j^{\prime}},$
(5.1)

with bounded smooth coefficients $a_{j,j^{\prime}}$ With the operator (5.1),
we associate the quadratic form

${\mathbf{a}}[u]=\int_{\Omega}\sum_{j,j^{\prime}}a_{j,j^{\prime}}(X)\partial_{j}u(X)\overline{\partial_{j^{\prime}}u(X)}dX+t\int_{\Omega}|u(X)|^{2}dX.$
(5.2)

with $t$ large enough. This form, considered for $u\in H^{1}(\Omega),$ defines
the _Neumann_ operator ${\mathbf{A}}={\mathbf{A}}^{\sc\mbox{N}\hskip 1.0pt}$
in $L_{2}(\Omega)$. For $t>0$, this operator is positive in $H^{1}(\Omega)$,
so that ${\mathbf{a}}[u]$ is equivalent to the squared norm in
$H^{1}(\Omega)$; we suppose that this is already done and keep our notations.

Let now the measure $\mu$ and the weight function $V$ satisfy the conditions
of Lemma 3.4 for $l=1$, $d\in({\mathbf{N}}-2,{\mathbf{N}}].$ Denote by
$\gamma$ the restriction operator, bounded as acting from $H^{1}(\Omega)$ to
$L_{2}(\Gamma)$. By this Lemma, the perturbing quadratic form
${\mathbf{v}}[u],$

${\mathbf{v}}[u]=\int_{\Sigma}V(X)|u(X)|^{2}\mu(dX)=({\mathbf{U}}{\mathbf{F}}\gamma
u,{\mathbf{F}}\gamma u)_{L_{2}(\Sigma,\mu)},\,u\in H^{1}(\Omega),$

is bounded in $H^{1}(\Omega),$ and defines in $H^{1}(\Omega)$ a bounded
operator ${{\mathbf{S}}}_{{\mathbf{v}}}$, for which the spectral estimates in
Theorem 4.1 are valid. The operator ${\mathbf{A}}^{\frac{1}{2}}$ is an
isomorphism between $H^{1}(\Omega)$ and $L_{2}(\Omega)$. Thus, setting
$v={\mathbf{A}}^{\frac{1}{2}}u\in L_{2}(\Omega),$ we obtain the quadratic form

${\mathbf{v}}[u]={\mathbf{v}}[{\mathbf{A}}^{-\frac{1}{2}}v]=({\mathbf{U}}{\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}v,{\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}v)_{L_{2}(\Sigma,\mu)},v\in
L_{2}(\Omega).$

This quadratic form defines the operator
${\mathbf{T}}={\mathbf{T}}_{{\mathbf{v}}}$ in $L_{2}(\Omega)$, unitarily
equivalent to ${{\mathbf{S}}}$,

${\mathbf{T}}={\mathbf{T}}_{{\mathbf{v}}}=\left[{\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}]^{*}{\mathbf{U}}[{\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}\right].$

In this way, we have arrived to the setting of our Sect.2, with a compact
operator ${\mathbf{T}}$ (Theorem 4.1 with $l=1$ gives order sharp eigenvalue
estimates, but at the moment, only compactness is needed.)

Following Sect.2, we will also need spectral estimates for the operator

${\mathbf{W}}\equiv{\mathbf{W}}_{{\mathbf{v}},2}={\mathbf{A}}^{-1/2}{\mathbf{T}}{\mathbf{A}}^{-\frac{1}{2}}=\left[{\mathbf{F}}\gamma{\mathbf{A}}^{-1}\right]^{*}{\mathbf{U}}\left[{\mathbf{F}}\gamma{\mathbf{A}}^{-1}\right]$
(5.3)

in $L_{2}(\Omega).$ This operator is defined by the quadratic form

${\mathbf{w}}[v]={\mathbf{v}}[{\mathbf{A}}^{-\frac{1}{2}}v]=\int
U(X)|F(X)|^{2}|({\mathbf{A}}^{-1}v)(X)|^{2}\mu(dX),v\in L_{2}(\Omega).$ (5.4)

By setting ${\mathbf{A}}^{-1}v=w,$ we transform (5.4) to

${\mathbf{w}}[v]={\mathbf{s}}[w]=\int U(X)|F(X)|^{2}|\gamma
w(X)|^{2}\mu(dX),\,\,w\in{\mathcal{D}}({\mathbf{A}}).$ (5.5)

The quadratic form (5.5) is considered in the Hilbert space
${\mathcal{D}}({\mathbf{A}})=H^{2}_{{\mathcal{L}}}$, the domain of the self-
adjoint operator ${\mathbf{A}}$. For an unbounded $\Omega,$ this domain,
generally, is rather hard, if possible, to describe. This circumstance was
taken care of in Sect.4, by means of describing the local regularity of
functions in $H^{2}_{{\mathcal{L}}}$

The corresponding result is just a re-formulation of Theorem 4.1. Before
stating it, we recall the restrictions imposed by this theorem on the measure
$\mu$ and the weight function $V$ for the most important values $l=1$ and
$l=2$. These conditions should be satisfied both for $l=1$, in order to be
able to define the operator ${\mathbf{T}}$ and grant its compactness, and for
$l=2,$ in order to estimate the eigenvalues of ${\mathbf{W}}:$

$\displaystyle\mu\in\mathfrak{A}^{d},\,0<d\leq 2\,\mbox{for}\,{\mathbf{N}}=2;$
(5.6)
$\displaystyle\mu\in\mathfrak{A}^{d},\,{\mathbf{N}}-2<d\leq{\mathbf{N}},\,\mbox{for}\,{\mathbf{N}}=3,\,\mbox{or}\,{\mathbf{N}}=4;$
$\displaystyle\mu\in\mathfrak{A}^{d}_{+},\,{\mathbf{N}}-2<d\leq{\mathbf{N}},\,\mbox{for
}\,{\mathbf{N}}>4,$ and $\displaystyle V\in
L_{(\vartheta),\mu},\vartheta=\frac{d}{d+2-{\mathbf{N}}}.$

######  Theorem 5.1.

Let the measure $\mu$ and the weight function $V$ satisfy the above
conditions. Then for the operator
${\mathbf{W}}={\mathbf{A}}^{-\frac{1}{2}}{\mathbf{T}}{\mathbf{A}}^{-\frac{1}{2}}$
the following estimate holds for $\theta=\frac{d}{d-{\mathbf{N}}+4}$

${\mathbf{n}}^{\sup}({\mathbf{W}},\theta)\leq
C\|V\|_{L_{(\theta),\mu}}^{\theta},,$
${\mathbf{n}}_{\pm}^{\sup}({\mathbf{W}},\theta)\leq
C\|V_{\pm}\|_{L_{(\theta),\mu}}^{\theta},$

Similarly, the eigenvalue estimates for other types of problems, obtained in
Sect.4 can be re-written in terms of ’Birman-Schwinger operators.’

######  Theorem 5.2.

(1) Let $\Omega\subset{\mathbb{R}}^{{\mathbf{N}}}$ be an open set and $\mu$ be
a Radon measure compactly supported in $\Omega$, $l\geq 1$ and let
${\mathbf{A}}={\mathbf{A}}^{{\mathcal{N}}}$ be the Neumann realization of the
second order elliptic operator ${\mathcal{L}}$ in $\Omega.$ Suppose that the
measure $\mu$ satisfies the condition (4.4) corresponding to the relation
between $l$ and ${\mathbf{N}}$ and $V\in L_{(\theta),\mu},$
$\theta=\frac{d}{d+4l-{\mathbf{N}}}$. Then for the operator

${\mathbf{W}}_{l}={\mathbf{W}}_{l,V,\mu}={\mathbf{A}}^{-(l-1/2)}{\mathbf{T}}{\mathbf{A}}^{-(l-1/2)}=(F\gamma{\mathbf{A}}^{-l})^{*}U,(F\gamma{\mathbf{A}}^{-l}),\,F=|V|^{\frac{1}{2}},\,U=\hbox{{\rm
sign}}\,(V),$ (5.7)

the following estimate is valid:

${\mathbf{n}}^{\sup}({\mathbf{W}}_{l},\theta)\leq
C\|V\|_{(\theta),\mu}^{\theta}$ (5.8)

(2) Let $\Omega$ be a domain with compact $C^{1,1}$ boundary, for $l=1,2$, or
a domain with compact smooth boundary, for $l>2$. Let $\mu$ be a measure on
$\Gamma=\partial\Omega,$ satisfying the condition (4.4) corresponding to the
relation between $l$ and ${\mathbf{N}}$ and $V\in L_{(\theta),\mu},$
$\theta=\frac{d}{d+2l-{\mathbf{N}}}$. Then for the operator
${\mathbf{W}}_{l}$, the estimate (5.8) is valid.

Theorem 5.2 implies spectral estimates for some non-self-adjoint operators
arising in the study of the difference of powers of resolvents.

First, we consider the operator ${\mathbf{Q}}_{l}={\mathbf{Q}}_{l,G,\mu}$, for
an integer $l>0$ or $l=\frac{1}{2}$.

${\mathbf{Q}}_{l}u=G\gamma{\mathbf{A}}^{-l}u;{\mathbf{Q}}_{l}:L_{2}({\mathbb{R}}^{\mathbf{N}})\to
L_{2,\mu},$

where ${\mathbf{A}}$ is a second order elliptic operator in
${\mathbb{R}}^{\mathbf{N}}$ and $\mu$ is a measure with compact support,
$\mu\in\mathfrak{A}^{d}.$

######  Lemma 5.3.

Let $G\in L_{(\eta),\mu},$ where $\eta=\frac{2d}{d-{\mathbf{N}}+4l}$. Then

${\mathbf{n}}^{\sup}({\mathbf{Q}}_{l},\eta)\leq C\|G\|_{(\eta),\mu}^{\eta}.$
(5.9)

###### Proof.

Let $l$ be an integer, first. Since for the operator ${\mathbf{W}}_{l}$ in
(5.7), ${\mathbf{W}}_{l}={\mathbf{Q}}_{l}^{*}{\mathbf{Q}}_{l}$ with
$V=|G|^{2},$ the singular numbers of ${\mathbf{Q}}_{l}$ ar square roots of the
singular numbers of ${\mathbf{W}}_{l}$, and the required estimate follows
immediately from (5.8). If $l=\frac{1}{2}$, the same reasoning goes through,
using ${\mathbf{Q}}_{l}^{*}{\mathbf{Q}}_{l}={\mathbf{T}}$ and applying the
estimate in Theorem 4.1. ∎

Next we need a spectral estimate for the operator

${\mathbf{Z}}={\mathbf{Z}}_{l_{1},l_{2},G_{1},G_{2},\mu}=({\mathbf{Q}}_{l_{1},G_{1},\mu})^{*}({\mathbf{Q}}_{l_{2},G_{2},\mu}),\,l_{1},l_{2}\geq
1,$ (5.10)

this means for the product of two operators in Lemma 5.3, with different $l,$
again with an integer $l_{\iota}$ or $l_{\iota}=\frac{1}{2}.$ We apply Lemma
5.3 to each factor in (5.10) and then the Ky Fan inequality (2.14), which
gives the following estimate.

######  Lemma 5.4.

Let $\mu$ be a measure in $\mathfrak{A}^{d},$
${\mathbf{N}}-2<d\leq{\mathbf{N}},$ and let the weight functions $G_{1},G_{2}$
satisfy the conditions of Lemma 5.3, correspondingly, with $l=l_{1}$ and
$l=l_{2}$. Set $\eta_{\iota}=\frac{2d}{d-{\mathbf{N}}+4l_{\iota}},$
$\theta^{-1}=\eta_{1}^{-1}+\eta_{2}^{-1}=\frac{d-{\mathbf{N}}+2(l_{1}+l_{2})}{d}.$
Then

${\mathbf{n}}^{\sup}({\mathbf{Z}},\theta)\leq
C\|G_{1}\|_{(\eta_{1}),\mu}^{\theta}\|G_{2}\|_{(\eta_{2}),\mu}^{\theta}.$
(5.11)

The crucial feature of Lemma 5.4 is that the order $\theta$ in the eigenvalue
estimate of the product of operators is determined by the sum $l_{1}+l_{2}$,
and not by $l_{1},l_{2}$ separately.

Lemma 5.4 extends in the natural way to the product of several operators of
the form ${\mathbf{Z}}.$

By means of transformations applied above, in Sect.4, the same estimate is
justified for the measure $\mu$ supported on the boundary of a domain in
${\mathbb{R}}^{{\mathbf{N}}}.$

###  5.2. Spectrum of the difference of resolvents

Now we combine the results of Sect.2 and Sect. 5.1 to obtain spectral
estimates for the difference of resolvents of singular perturbed elliptic
operators.

In conditions of Sect. 5.1, we consider the difference of resolvents. Let
$\Omega\subseteq{\mathbb{R}}^{\mathbf{N}}$ be a connected open set. Let the
operator ${\mathbf{A}}$ in $\Omega$ be defined by the quadratic form
${\mathbf{a}}$ in (5.2) with some form-domain
$\mathfrak{d}[{\mathbf{a}}]\subset H^{1}(\Omega)$. Further, let $\Sigma$ be a
compact subset in $\Omega,$ and let $\mu$ be a measure supported in $\Sigma$,
and let the ’$\mu-$ measurable weight function’ $V$ satisfy (5.6). Then the
quadratic form ${\mathbf{v}}[u]=\int V(X)|u(X)|^{2}\mu(dX)$ is bounded in
$\mathfrak{d}[{\mathbf{a}}]$ and, following the procedure described in Sect.2,
the perturbed self-adjoint operator ${\mathbf{A}}_{{\mathbf{V}}}$ is defined.

According to the results of Sect.2, for the difference
${\mathbf{R}}_{\mathbf{V}}={\mathbf{A}}^{-1}-{\mathbf{A}}_{\mathbf{V}}^{-1}$
of resolvents of these operators, formula (2.18) is valid, with operator
${\mathbf{T}}\equiv{\mathbf{T}}_{{\mathbf{V}}}$ defined in (2.17). Having our
results about eigenvalue estimates for the operators ${\mathbf{T}}$ in Theorem
4.1 and Theorem 5.1, together with Lemma 2.8, we arrive at the following
estimate.

######  Theorem 5.5.

Under the assumption that (5.6) is satisfied, for the resolvent difference
${\mathbf{R}}_{\mathbf{V}}={\mathbf{A}}^{-1}-{\mathbf{A}}_{\mathbf{V}}^{-1},$
the following spectral estimate is valid

${\mathbf{n}}^{\sup}({\mathbf{R}}_{\mathbf{V}},\theta)\leq
C\|V\|_{L_{(\theta),\mu}}^{\theta},\,\theta=\frac{d}{d-{\mathbf{N}}+4},$
(5.12)

with constant $C$ depending on the measure $\mu,$ the operator ${\mathbf{A}}$
and the domain $\Omega,$ but not depending on $V$.

Next we consider the resolvent difference for two singular perturbations,
following Lemma 2.11

######  Theorem 5.6.

Suppose that the measure $\mu$ and the weight functions $V_{1},V_{2}$ satisfy
(5.6). We construct two perturbed operators
${\mathbf{A}}_{{\mathbf{V}}_{1}},{\mathbf{A}}_{{\mathbf{V}}_{2}}$. Then for
the difference of resolvents of these perturbed operators,
${\mathbf{R}}^{(1,2)}={\mathbf{A}}_{{\mathbf{V}}_{1}}^{-1}-{\mathbf{A}}_{{\mathbf{V}}_{2}}^{-1}$,
the following spectral estimate holds

${\mathbf{n}}^{\sup}({\mathbf{R}}^{(1,2)},\theta)\leq
C\|V_{1}-V_{2}\|_{L_{(\theta),\mu}}^{\theta},\theta=\frac{d}{d-{\mathbf{N}}+4},$
(5.13)

again, with constant $C$ depending on the measure $\mu,$ the operator
${\mathbf{A}}$ and the domain $\Omega,$ but not depending on $V.$

Since
${\mathbf{v}}_{1}[u]-{\mathbf{v}}_{2}[u]=\int_{\Sigma}(V_{1}(X)-V_{2}(X))|u(X)|^{2}\mu(dX),$
the proof follows immediately from Proposition 2.11 and Theorem 5.1 applied to
$V=V_{1}-V_{2}$.

It follows from Theorem 4.3 that the decay order of eigenvalues in (5.13) is
sharp, at least for a sign-definite weight function $V_{1}-V_{2}.$

######  Remark.

A natural question here is how Theorem 5.6 should be modified if the
perturbations ${\mathbf{V}}_{1},{\mathbf{V}}_{2}$ are defined by two
_different_ measures, $\mu_{1}$, $\mu_{2}$, with corresponding support
$\Sigma_{1},\Sigma_{2}$. First, if both measures belong to $\mathfrak{A}^{d}$,
with the same $d,$ we can consider their sum $\mu=\mu_{1}+\mu_{2}$ with
support $\Sigma=\Sigma_{1}\cup\Sigma_{2}$ which belongs to $\mathfrak{A}^{d}$
as well, extend the weight functions $V_{j},\,j=1,2$ outside $\Sigma_{j}$ by
zero this reduces the problem to the case of one measure.

Suppose now that
$\mu_{1}\in\mathfrak{A}^{d_{1}},\mu_{2}\in\mathfrak{A}^{d_{2}},$ with
different dimension, $d_{1}<d_{2}$. Then the perturbation with larger $\theta$
makes a stronger contribution to the eigenvalue estimate. The result depends
on the dimension of the space ${\mathbb{R}}^{{\mathbf{N}}}$. If
${\mathbf{N}}>4,$ we have
$\theta_{1}=\frac{d_{1}}{d_{1}-{\mathbf{N}}+4}>\theta_{2}=\frac{d_{2}}{d_{2}-{\mathbf{N}}+4},$
therefore, in (2.22), the perturbation $V_{2}\mu_{2}$ gives a weaker
contribution to the spectral estimate. On the contrary, for ${\mathbf{N}}<4$,
we have $\theta_{1}<\theta_{2}$, and therefore, it is the perturbation
$V_{2}\mu_{2}$ that gives a stronger contribution. The most delicate case is
the dimension ${\mathbf{N}}=4,$ where one can see that
$\theta_{1}=\theta_{2}=1.$ Here, the contributions of two perturbations are of
the same order in the eigenvalue estimate. In this case, unlike the case of
$d_{1}=d_{2},$ above, the contributions of positive and negative parts of
$V_{\iota}$ to eigenvalue estimates do not cancel. These estimates have the
form

${\mathbf{n}}^{\sup}_{\pm}({\mathbf{D}}^{(1,2)},1)\leq
C((\|(V_{1})_{\pm}\|_{(1),\mu_{1}}+\|(V_{2})_{\mp}\|_{(1),\mu_{2}})$

(see details in [62]).

###  5.3. Estimates: some special cases

We consider two examples.

######  Example 5.7.

Let $d={\mathbf{N}};$ this means that the measure $\mu$ is equivalent to the
Lebesgue measure ${\mathcal{H}}^{\mathbf{N}}.$ Let the densities $V_{1},V_{2}$
be compactly supported in $\Omega.$ Then the exponent $\theta$ equals
$\frac{{\mathbf{N}}}{4}$ and Theorem 5.6, gives us the estimate

${\mathbf{n}}^{\sup}({\mathbf{A}}_{{\mathbf{V}}_{1}}^{-1}-{\mathbf{A}}_{{\mathbf{V}}_{2}}^{-1},\theta,)\leq
C\|V_{1}-V_{2}\|_{(\theta)}^{\theta},\theta=\frac{{\mathbf{N}}}{4},$

under the condition $V_{1},V_{2}\in L_{(\theta)}.$ In particular, if
$V_{2}=0,$ we have

${\mathbf{n}}^{\sup}({\mathbf{A}}^{-1}-{\mathbf{A}}_{{\mathbf{V}}_{1}}^{-1},\theta)\leq
C\|V_{1}\|_{(\theta)}^{\theta},\theta=\frac{{\mathbf{N}}}{4}.$

Essentially, these results are not new, see, in particular, [73] and
references therein. Note that in the literature on the scattering theory, it
is the behavior of $V$ at infinity, $\Omega={\mathbb{R}}^{\mathbf{N}}$, that
is of main interest, the topic we do not touch upon in _this_ paper.

######  Example 5.8.

Let $d={\mathbf{N}}-1$. The condition $\mu\in\mathfrak{A}^{{\mathbf{N}}-1}$ is
satisfied, for example, for the case of the Haußdorff measure
$\mu={\mathcal{H}}^{d}$ on a Lipschitz hypersurface $\Sigma\subset\Omega.$
Then the spectral problem for the operator ${\mathbf{A}}_{{\mathbf{V}}}$ is
the transmission problem, consisting of finding the solution of the equation
${\mathcal{L}}u=\lambda u$ with jump condition on $\Sigma$:
$[\partial_{\nu({\mathcal{L}})}u]_{|_{\Sigma}}(X)-V(X)u(X)=0,\,X\in\Sigma,$
where brackets denote the jump in the conormal derivative of $u$ when crossing
$\Sigma$, while the direction of the differentiation coincides with the
direction of crossing. From a formally different point of view, this problem
was considered in [12], [10], [14], where such perturbation was understood as
a weighted $\delta$-interaction supported on a hypersurface $\Sigma$. The
conditions in these papers were rather restrictive, in particular, both the
hypersurface $\Sigma$ and the weight function were supposed to be smooth. The
correct order for the spectral asymptotics was found there.

Using the results of this paper, due to Theorem 5.6, the exponent $\theta$
equals $\frac{N-1}{3}$, and our general eigenvalue estimates take the form

${\mathbf{n}}^{\sup}({\mathbf{A}}_{{\mathbf{V}}_{1}}^{-1}-{\mathbf{A}}_{{\mathbf{V}}_{2}}^{-1},\theta)\leq
C\|V_{1}-V_{2}\|_{(\theta)}^{\theta},\theta=\frac{{\mathbf{N}}-1}{3}.$

We return to this setting in the next Section, where we justify the spectral
asymptotics for
${\mathbf{A}}_{{\mathbf{V}}_{1}}^{-1}-{\mathbf{A}}_{{\mathbf{V}}_{2}}^{-1}$

###  5.4. Singular Robin problem

We recall that the classical Robin problem for the second order elliptic
operator ${\mathcal{L}}$ in a domain $\Omega\subset{\mathbb{R}}^{\mathbf{N}}$
consists in finding the solution of the equation ${\mathcal{L}}u=f$ with
boundary conditions on $\Gamma=\partial\Omega$:

$\partial_{\nu({\mathcal{L}})}(X)u(X)-V(X)u(X)=0,\,X\in\Gamma,$ (5.14)

where $\partial_{\nu({\mathcal{L}})}(X)$ is the conormal boundary derivative
associated with the operator ${\mathcal{L}}$. As mentioned in the
Introduction, under certain regularity conditions, the singular numbers
asymptotics for the difference of resolvents of two operators of this form,
was found in [41]; we return to this issue later on.

In the present subsection we consider a more general spectral problem, with
the density $V$ in (5.14) replaced by a measure $\boldsymbol{\pi}=V\mu$ on the
boundary. For the statement of the problem and initial estimates, this
boundary is supposed to be of the class $C^{0,1}$. Namely, we consider the
spectral problem

${\mathbf{A}}_{\Gamma,{\mathbf{V}}}u=\lambda u,$

with ${\mathbf{A}}_{\Gamma,{\mathbf{V}}}$ being the operator ${\mathcal{L}}$
in $\Omega$ with, formally, Robin type boundary conditions on $\partial\Omega$

$[\partial_{\nu({\mathcal{L}})}u]_{|\Gamma}(X)-V(X)u(X)\mu=0,$ (5.15)

where $\mu$ is a measure on $\Gamma$, possibly, singular, and $V$ is a real
$\mu$-measurable function on $\Gamma.$ This operator is defined variationally,
by means of a quadratic form. Namely, generalizing [15], we associate with the
Robin problem (5.15) the quadratic form

${\mathbf{a}}_{{\mathbf{v}}}[u]={\mathbf{a}}[u]+{\mathbf{v}}[u],\,u\in
H^{1}(\Omega),$ (5.16)

where the perturbing quadratic form is

${\mathbf{v}}[u]=\int_{\Gamma}V(X)|\gamma u(X)|^{2}\mu(dX);$ (5.17)

where $\gamma$, as before, is the restriction operator of functions in
$H^{1}(\Omega)$ to the boundary. Since the quadratic form (5.16) is defined on
the space $H^{1}(\Omega)$, the Robin problem, thus stated, is a quadratic form
perturbation of the Neumann operator
${\mathbf{A}}={\mathbf{A}}^{{\mathcal{N}}}$.

######  Proposition 5.9.

Suppose that the boundary $\Gamma$ is Lipschitz and the measure $\mu$ on
$\Gamma$ belongs to $\mathfrak{A}^{d},$ ${\mathbf{N}}-2<d\leq{\mathbf{N}}-1$.
Let $V\in L_{(\vartheta),\mu},$ $\vartheta=\frac{d}{d-{\mathbf{N}}+2}$. Then
the quadratic form (5.17) is bounded in $H^{1}(\Omega)$, the quadratic form
(5.16) is semibounded. The operator ${\mathbf{T}}$ defined in (2.17), where,
as previously, ${\mathbf{F}}$ is the operator of multiplication by the
function $F=|V|^{\frac{1}{2}}$ and ${\mathbf{U}}$ is the operator of
multiplication by $U=\hbox{{\rm sign}}\,V$, is compact and for its spectrum
the estimates hold:

${\mathbf{n}}_{\pm}^{\sup}({\mathbf{T}},\vartheta)\leq
C\|V_{\pm}\|_{L_{(\vartheta),\mu}}^{\vartheta},\,{\mathbf{n}}^{\sup}({\mathbf{T}},\vartheta)\leq
C\|V\|_{L_{(\vartheta),\mu}}^{\vartheta}$ (5.18)

###### Proof.

Two first statements follow directly from Lemma 3.5, with use of the Stein
extension operator ${\mathbf{J}}:H^{1}(\Omega)\to
H^{1}({\mathbb{R}}^{{\mathbf{N}}})$. The same extension operator takes care of
carrying over eigenvalue estimates in the whole space to the ones in the
domain $\Omega,$ using Lemma 2.3. ∎

Next, we obtain spectral estimates for the operator

${\mathbf{W}}={\mathbf{A}}^{-\frac{1}{2}}{\mathbf{T}}{\mathbf{A}}^{-\frac{1}{2}}=({\mathbf{F}}\gamma{\mathbf{A}}^{-1})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-1})$
(5.19)

in $L_{2}(\Omega).$ Here we will need some more regularity of the boundary of
$\Omega.$

######  Lemma 5.10.

Let the boundary $\Gamma$ belong to $C^{1,1}$ and the measure $\mu$ and the
weight $V$ on $\Omega$ satisfy (5.6). Then the singular numbers of
${\mathbf{W}}$ satisfy estimate

${\mathbf{n}}^{\sup}({\mathbf{W}},\theta)\leq
C\|V\|_{(\theta),\mu}^{\theta},\,\theta=\frac{d}{d-2N+4},$

and similar estimates are valid for positive and negative eigenvalues of
${\mathbf{W}}$ separately.

###### Proof.

The operator ${\mathbf{W}}$ is unitarily equivalent to the operator defined by
the quadratic form ${\mathbf{v}}[u]$ in ${\mathcal{D}}({\mathbf{A}})$. Due to
the $C^{1,1}$ regularity of the boundary, by Lemma 3.1, the domain
${\mathcal{D}}({\mathbf{A}})$ of the Neumann operator ${\mathbf{A}}$ is
contained in $H^{2}$ in a neighborhood of the boundary. We apply, first, the
restriction to this neighborhood and then the $H^{2}$-extension to the whole
${\mathbb{R}}^{\mathbf{N}},$ both times using Lemma 2.3. Therefore the task of
estimating eigenvalues of the operator ${\mathbf{W}}$ is reduced to estimating
the singular numbers of the operator defined by the quadratic form (5.17) in
the space $H^{2}({\mathbb{R}}^{\mathbf{N}}),$ and the required estimate is
provided by Theorem 5.6 with $l=2$. ∎

Having established the required estimates for the operators ${\mathbf{T}}$ and
${\mathbf{W}},$ we can now apply the abstract results of Sect. 2.

######  Theorem 5.11.

Let the boundary $\Gamma=\partial\Omega$, the measure $\mu$ and the weight
functions $V_{1}$, $V_{2}$ satisfy the conditions of Lemma 5.9 and Lemma 5.10.
Then for the generalized Robin operators, the following estimates hold with
$\theta=\frac{d}{d-{\mathbf{N}}+4}$:

(1):

For the operator
${\mathbf{R}}_{{\mathbf{V}}_{1}}={\mathbf{A}}^{-1}-{\mathbf{A}}_{{\mathbf{V}}_{1}}^{-1}$,

${\mathbf{n}}_{\pm}^{\sup}({\mathbf{R}}_{{\mathbf{V}}_{1}},\theta)\leq
C\|V_{1,\pm}\|_{{(\theta)},\mu}^{\theta};$

(2):

For the operator
${\mathbf{R}}^{(1,2)}={\mathbf{A}}_{{\mathbf{V}}_{1}}^{-1}-{\mathbf{A}}_{{\mathbf{V}}_{2}}^{-1}$

${\mathbf{n}}_{\pm}^{\sup}({\mathbf{R}}^{(1,2)},\theta)\leq
C\|(V_{1}-V_{2})_{\pm}\|_{{(\theta)},\mu}^{\theta}.$

######  Remark.

The same discussion as in Remark Remark clarifies the situation with two
different measures for two generalized Robin problems.

For the case when the measure $\mu$ coincides with the surface measure on the
boundary, $d={\mathbf{N}}-1$, we obtain the eigenvalue estimate for the
classical Robin problem in terms of integral characteristics of the density,
improving the results of [22], [12], [41], [53].

######  Corollary 5.12.

Let $\mu$ be the surface measure on $\Gamma=\partial{\Omega},$
$d={\mathbf{N}}-1,$ and the conditions of Lemma 5.9 and 5.10 be satisfied.
Then $\theta=\frac{d}{d-{\mathbf{N}}+4}=\frac{{\mathbf{N}}-1}{3}$ and

${\mathbf{n}}_{\pm}^{\sup}({\mathbf{A}}_{{\mathbf{V}}_{1}}-{\mathbf{A}}_{{\mathbf{V}}_{2}},\frac{{\mathbf{N}}-1}{3})\leq
C\|(V_{1}-V_{2})_{\pm}\|_{L_{(\theta)},\mu}^{\theta},\,\theta=\frac{d}{d-{\mathbf{N}}+4}=\frac{{\mathbf{N}}-1}{3}.$
(5.20)

##  6\. Eigenvalue asymptotics for resolvent difference.

For the case when $\Sigma$ is a Lipschitz hypersurface in
${\mathbb{R}}^{\mathbf{N}}$, this means of dimension $d={\mathbf{N}}-1,$ and
$\mu$ is the surface measure on $\Sigma$, thus, belonging to
$\mathfrak{A}^{{\mathbf{N}}-1}$, we can obtain the asymptotics of eigenvalues
of the operator
${\mathbf{R}}^{(1,2)}={\mathbf{A}}^{-1}_{{\mathbf{V}}_{1}}-{\mathbf{A}}^{-1}_{{\mathbf{V}}_{2}}$.
Moreover, when $\mu$ is the Haußdorff measure ${\mathcal{H}}^{{\mathbf{N}}-1}$
(and we can restrict ourselves to this case), such asymptotics can be proved
for certain more general sets $\Sigma$.

In our initial motivation, namely, for the difference of Robin resolvents with
different densities, we can prove Weyl-type asymptotic formulas for a measure
$\mu\in\mathfrak{A}^{{\mathbf{N}}-1}$, however under the condition that the
boundary is smooth. This latter condition is explained by the fact that we use
a perturbation approach based upon Lemma 2.5. This means that we approximate
the ’singular’ problem under consideration by more regular ones, where the
eigenvalue asymptotics is already known. When we consider the perturbation
contained strictly inside our domain, we can use as approximating, the results
on the asymptotics for Lipschitz surfaces, obtained in [66]. On the other
hand, for the Robin problem, the only approximating results available are the
ones in [41], where the boundary is smooth. It is unclear at the moment, how
far one can relax these smoothness conditions, using either the
pseudodifferential approach in [42] or the approximation approach developed in
[63].

###  6.1. $\delta$-potentials

We consider first a more simple case, namely, when the measure $\mu$ is in
$\mathfrak{A}^{d}$, $d={\mathbf{N}}-1$, on some compact set $\Sigma$ inside
$\Omega$; equivalently, $\mu$ is the Haußdorff measure
${\mathcal{H}}^{{\mathbf{N}}-1}$ on $\Sigma$. In other words, we study the
difference of resolvents of two Schrödinger type operators

${\mathbf{R}}^{(1,2)}={\mathbf{A}}_{{\mathbf{V}}_{1}}^{-1}-{\mathbf{A}}_{{\mathbf{V}}_{2}}^{-1},$

in $L_{2}(\Omega)$, with ${\mathcal{H}}^{d}$-measurable functions
$V_{1},V_{2}$ on $\Sigma$. When the set $\Sigma$ is a hypersurface in
$\Omega,$ say Lipschitz one, this setting corresponds to considering
$\delta$-potentials on $\Sigma,$ with weights $V_{1},V_{2}.$

The corresponding expression is formally the same as in [12], [14]. We are
pretty sure that for sufficiently regular $\Sigma$ and piecewise smooth
functions $V_{1},V_{2},$ it might be proved following the approach in [41],
[39], [1] with use of the calculus of nonsmooth singular Green operators. We,
however, aim for obtaining the eigenvalue asymptotics under considerably less
restrictive conditions. We follow essentially the pattern elaborated in [67],
[62], [65]; therefore, we skip technical details and restrict ourselves to
explaining main steps only.

For a point $X$ on a Lipschitz hypersurface $\Sigma,$ the density $\omega(X)$
is determined by the principal symbol
$\mathfrak{a}(X;\xi)=\sum_{j,j^{\prime}}a_{j,j^{\prime}}(X)\xi_{j}\xi_{j^{\prime}}$
of the operator ${\mathcal{L}}$ at this point. This density is defined
$\mu$-almost everywhere on $\Sigma,$ namely, at those points $X\in\Sigma$
where there exists the tangent hyperplane $\mathrm{T}_{X}\Sigma$, with the
tangent unit sphere $\mathrm{S}_{X}\Sigma$, and the corresponding normal
vector $\boldsymbol{\nu}(X):$

$\displaystyle\omega(X)=\frac{1}{d(2\pi)^{d}}\int_{S_{X}\Sigma}{\mathbf{r}}(X,\xi^{\prime})^{\theta}d\sigma(\xi^{\prime}),\,\theta=\frac{{\mathbf{N}}-1}{3};$
(6.1)
$\displaystyle{\mathbf{r}}(X,\xi^{\prime})=(2\pi)^{-1}\int_{-\infty}^{\infty}\mathfrak{a}(X;\xi^{\prime},\eta\boldsymbol{\nu}(X))^{-2}d\eta,$

see [67], where $\sigma$ is the natural measure on the unit cotangent sphere
in ${\mathbb{R}}^{d-1}$, see Theorem 6.2 in [66] (for ${\mathbf{N}}\neq 4$)
and Theorem 2.4 in [65] (for ${\mathbf{N}}=4$).

######  Theorem 6.1.

Let $\Sigma\subset\Omega$ be a Lipschitz hypersurface in $\Omega$, and
$V_{1},V_{2}$ be functions in $L_{(\vartheta),\mu},$
$\vartheta={{\mathbf{N}}-1}.$ Then for the difference of resolvents
${\mathbf{R}}^{(1,2)}={\mathbf{A}}_{{\mathbf{V}}_{1}}^{-1}-{\mathbf{A}}_{{\mathbf{V}}_{2}}^{-1}$
the asymptotic formulas hold:

${\mathbf{n}}_{\pm}({\mathbf{R}}^{(1,2)},\theta)=(2\pi)^{-d}\int_{\Sigma}\omega(X)(V_{2}(X)-V_{1}(X))_{\pm}^{\theta}{\mathcal{H}}^{d}(dX),\,\theta=\frac{{\mathbf{N}}-1}{3},$

where $\omega(X),$ $X\in\Sigma$ is the density determined by the coefficients
of the operator ${\mathcal{L}}$ at $X,$ see (6.1).

###### Proof.

As explained above, the asymptotics of eigenvalues of ${\mathbf{R}}^{(1,2)}$
is the same as the one for the compact operator

${\mathbf{W}}={\mathbf{W}}^{(1,2)}=(|V_{1}-V_{2}|^{\frac{1}{2}}\gamma{\mathbf{A}}^{-1})^{*}{\mathbf{U}}(|V_{1}-V_{2}|^{\frac{1}{2}}\gamma{\mathbf{A}}^{-1}),$

${\mathbf{U}}=\hbox{{\rm sign}}\,(V_{1}-V_{2})$; such operators were
considered systematically in [65], [67], [62]. In fact, the statement in the
theorem above is a particular case of Theorem 2.4 in [65] and Theorem 6.6 in
[62] for the critical dimension ${\mathbf{N}}=4$, and Theorem 6.4 in [67] for
the non-critical dimension, ${\mathbf{N}}\neq 4$, granted that the estimate in
Theorem 5.11 is already established.

For the benefit of the Readers, we explain here the structure of the proof,
addressing to the above sources for technical details.

We will use Lemma 2.5 systematically. Having obtained spectral estimates, as
in Theorem 5.11, involving _integral norm_ of the weight functions, we
approximate _in this integral norm over $\Sigma$_ the given weight functions
by more and more regular or more convenient ones, arriving at last to the
situation where the eigenvalue asymptotics is already known. We denote
$V=V_{1}-V_{2}$ and perform the following steps.

  1. (1)

The function $V(X)$ is approximated (in the above integral norm) by the
restriction to $\Sigma$ of a _smooth_ function
$V^{(1)}(X),\,X\in{\mathbb{R}}^{{\mathbf{N}}}$ belonging to $C_{0}^{\infty}$
in a neighborhood of $\Sigma.$

  2. (2)

The function $V^{(1)}$ is approximated (in the above integral norm, again) by
a smooth function $V^{(2)}$ such that the set where it is positive and the set
where it is negative are separated, ${\rm dist}\,(\hbox{{\rm
supp}}\,(V^{(2)}_{+}),\hbox{{\rm supp}}\,(V^{(2)}_{-}))>0.$ After this, the
property ’sign separation’ established in the papers cited above, is used: for
_such_ weight function $V^{(2)}$, the leading order asymptotic characteristics
of positive eigenvalues of ${\mathbf{W}}$ do not depend on the negative part,
$V^{(2)}_{-}$, and vice versa. Therefore, we can restrict our considerations
to a smooth non-negative function $V^{(2)}\geq 0$.

  3. (3)

Consider now the spectrum of the (nonnegative) operator

${\mathbf{W}}^{(2)}=[(V^{(2)})^{\frac{1}{2}}\gamma{\mathbf{A}}^{-1}]^{*}[(V^{(2)})^{\frac{1}{2}}\gamma{\mathbf{A}}^{-1}].$

Here we may commute the (smooth!) function $(V^{(2)})^{\frac{1}{2}}$ and the
restriction operator $\gamma,$

${\mathbf{W}}^{(2)}=\left[\gamma(V^{(2)})^{\frac{1}{2}}{\mathbf{A}}^{-1}\right]^{*}\left[\gamma(V^{(2)})^{\frac{1}{2}}{\mathbf{A}}^{-1}\right].$
(6.2)

The nonzero (in fact, positive) eigenvalues of the operator (6.2) in
$L_{2}(\Omega)$ coincide with nonzero eigenvalues of the operator

${\mathbf{W}}^{(3)}=[\gamma(V^{(2)})^{\frac{1}{2}}{\mathbf{A}}^{-1}][\gamma(V^{(2)})^{\frac{1}{2}}{\mathbf{A}}^{-1}]^{*}$
(6.3)

acting in $L_{2}(\Sigma,\mu)$.

  4. (4)

The operator $(V^{(2)})^{\frac{1}{2}}{\mathbf{A}}^{-1}$ is, in fact, up to a
smoother remainder, an integral operator in $\Omega$ with kernel
$(V^{(2)})^{\frac{1}{2}}(X)G_{-2}(X,Y),$ where $G_{-2}$ is the fundamental
solution of the operator ${\mathbf{A}}$. The leading term of this fundamental
solution is the Fourier transform of the inverse symbol
$\mathfrak{a}(X,\xi)^{-1}$ in ${\mathbb{R}}^{\mathbf{N}}$ and, therefore, is
the integral kernel with singularity at the diagonal. On this stage, we pass
from the pseudodifferential setting, where certain regularity of symbols and
surfaces is required, to the setting involving integral operators, where
considerably less regularity is needed. After the restriction to $\Sigma$,
this means, after the application of $\gamma$, it follows that
$\gamma\circ(V^{(2)})^{\frac{1}{2}}{\mathbf{A}}^{-1}$ is the integral operator
with the same kernel, but acting from $L_{2}(\Omega)$ to $L_{2}(\Sigma,\mu)$.
The adjoint, $[\gamma\circ(V^{(2)})^{\frac{1}{2}}{\mathbf{A}}^{-1}]^{*}$, is,
therefore, the integral operator with the kernel
$(V^{(2)})^{\frac{1}{2}}(Y)G_{-2}(X,Y)$ acting from $L_{2}(\Sigma,\mu)$ to
$L_{2}(\Omega)$. This kernel has power type or (for ${\mathbf{N}}=2$)
logarithmic singularity at the diagonal $X=Y$. Finally, for the composition in
(6.3), we see that ${\mathbf{W}}^{(3)}$ is the integral operator on $\Gamma,$
with leading singularity in the kernel:

${\mathcal{W}}(X,Y)=(V^{(2)}(X))^{\frac{1}{2}}\int_{\Omega}G_{-2}(X,Z)G_{-2}(Y,Z)dZ(V^{(2)}(Y))^{\frac{1}{2}}.$
(6.4)

  5. (5)

The composition of two copies of the fundamental solution of ${\mathcal{L}}$
in (6.4) produces, again, up to some smoother remainder terms, the fundamental
solution of the _fourth_ order elliptic operator ${\mathcal{L}}^{2}$. The
leading term in this fundamental solution has the power, or $\log$-power for
${\mathbf{N}}=2$, or logarithmic for ${\mathbf{N}}=4$, singularity at the
diagonal $X=Y.$

  6. (6)

We can apply now the results of the paper [66], where for integral operators
on a Lipschitz surface, with power or $\log$-power singularity at the
diagonal, the eigenvalue asymptotic formulas were obtained. This takes care of
the statement of Theorem 6.1 for a single Lipschitz surface.

∎

In the papers [62], [67], asymptotic formulas for operators of the type
${\mathbf{W}}$ were established for the case when the Lipschitz surface
$\Sigma$ is replaced by a more general, uniformly rectifiable, set.

######  Definition 6.2.

Let $d={\mathbf{N}}-1.$ The set $\Sigma\subset{\mathbb{R}}^{\mathbf{N}}$ is
called _uniformly rectifiable_ (of dimension $d$) if it is, up to a set of
zero Haußdorff measure ${\mathcal{H}}^{d}$, the union of a countable
collection of Lipschitz surfaces.

A detailed description of uniformly rectifiable sets and an extensive
discussion of their properties can be found in the books [54], [27], [32] and
other sources on the geometric measure theory.

In the papers [62], [67], the eigenvalue asymptotic formulas were obtained for
the case when the operator ${\mathcal{L}}$ is the Laplacian, where the density
$\omega(X)$ in (6.1) can be calculated explicitly. For a general second order
elliptic operator ${\mathcal{L}},$ the expression for this density is quite
unwieldy.

In the proof, in order to pass to the set $\Sigma$ consisting of a _finite_
collection of Lipschitz surfaces, one more approximation is used, reducing the
problem to disjoint surfaces; here the spectrum of the integral operator,
again, up to a weaker term, is the union of the spectra operators on separate
surfaces, which justifies the asymptotic formula for this case. Finally, for a
uniformly rectifiable set, the final approximation is performed, reducing the
problem to the previous case, using, for the last time, the estimate in
Theorem 5.11. The detailed procedure is described in [62], [67].

###  6.2. Regular Schrödinger operator

For completeness, we explain how our approach works when the measure $\mu$ is
the Lebesgue measure on ${\mathbb{R}}^{{\mathbf{N}}}$, and thus the potentials
$V_{\iota}\mu$, $\iota=1,2$ can be considered as usual functions. In this
case, $d={\mathbf{N}}$ and the exponent $\theta$ equals $\frac{d}{4},$ while
$\vartheta$ equals $\frac{d}{2}$.

In these conditions, by Theorems 5.5, 5.6, the estimate

${\mathbf{n}}^{\sup}({\mathbf{W}},\theta)\leq
C\|V_{1}-V_{2}\|_{(\theta)}^{\theta},\theta=\frac{{\mathbf{N}}}{4},$

is valid, under the condition that $V_{1},V_{2}\in L_{(\vartheta)}.$ Now,
similarly to our reasoning above, we obtain the asymptotics

${\mathbf{n}}_{\pm}({\mathbf{R}}^{(1,2)},\theta)=C_{\mathbf{N}}\int(V_{1}(X)-V_{2}(X))_{\pm}^{\frac{d}{4}}\det(\mathfrak{a}(X))^{-\frac{1}{4}}dX.$

For the Laplacian, such result seems to be folklore since early 1970-s, hinted
for by Birman-Solomak, Simon, Rozenblum.

###  6.3. The Robin problem

Here, we suppose that the boundary is $C^{\infty}$-smooth. This restriction is
caused by the fact that the existing results on the eigenvalue asymptotics for
the difference of Robin resolvents are established for smooth boundaries only.
As soon as the asymptotic formulas are proved for less smooth boundary and for
a smooth weight function, the asymptotic formula for nonsmooth weights will
follow automatically.

######  Theorem 6.3.

Let $\Omega$ be a domain with smooth boundary and let ${\mathcal{L}}$ be a
second order elliptic operator. We suppose that $\mu$ is the Haußdorff measure
${\mathcal{H}}^{d}$ on $\partial\Omega$ and, finally, $V_{1},V_{2}$ satisfy
(5.6). Then for the eigenvalues of the resolvent difference of Robin operators
in $\Omega$ with densities $V_{1},V_{2}$, the asymptotic formulas hold, with
$d={\mathbf{N}}-1$, $\theta=\frac{d}{3}$:

${\mathbf{n}}_{\pm}({\mathbf{R}}^{(1,2)},\theta)=\frac{1}{d(2\pi^{d})}\int_{\partial\Omega}\omega_{R}(X)(V_{1}(X)-V_{2}(X))_{\pm}^{\frac{d}{3}}dX,$
(6.5)

where $\omega_{R}(X)$ is given by the integrand in Theorem 3.5 in [41].

The proof, up to the step 3, follows the one in Theorem 6.1. The difference is
that in the expression for the operator ${\mathbf{W}}$ in (5.19), the
unperturbed operator ${\mathbf{A}}$ is not an elliptic operator in
${\mathbb{R}}^{\mathbf{N}}$ but the Neumann operator in $\Omega.$ Under the
condition that $\partial\Omega$ is of class $C^{1,1},$ the spectral estimate
for the operator ${\mathbf{W}}$ in Theorem 5.11 has the same form as for the
interior potential; it involves some integral norm of $V=V_{1}-V_{2}.$ Using
this estimate, we can repeat the reasoning in Theorem 6.1, this means,
separate positive and negative parts of the density $V$ and then perform the
approximation by smooth positive densities, using Lemma 2.5.

###  6.4. The case of a fractional dimension $d$

A natural question arises, whether there are asymptotic eigenvalue formulas
for the operator ${\mathbf{R}}^{(1,2)}$ with measure $\mu,$ Ahlfors regular
with a fractional dimension $d$. Our reference to the result by H.Triebel [71]
shows that, generally, the asymptotic _order_
$\theta=\frac{d}{d-{\mathbf{N}}+4}$ is sharp.

In the literature, there are a number of studies of the spectrum of operators
of the form ${\mathbf{T}}$ above; sometimes they are called Krein-Feller
operators. Main interest there is directed to the cases when the measure $\mu$
and the density $V$ have some regular fractal structure; in such cases, the
spectrum can be studied using arithmetic and algebraic relations. The leading
examples of measures in $\mathfrak{A}^{d}$ with fractional $d$ (the
construction is valid for integer $d$ as well, of course) are the ones
obtained as self-similar measures, see Sect. 3.2.

The results existing in the literature concern the spectrum of operators of
the form ${\mathbf{T}}.$ The author failed while searching for the results for
the operator ${\mathbf{W}}.$

For ${\mathbf{T}},$ even in the one-dimensional case, the asymptotics for
$n(\lambda)$ can be non-power-like. Namely, see, e.g., [68], where, depending
on the arithmetic structure of the self-similar singular measure, the class of
measures was described, where the eigenvalue asymptotics is

$n({\mathbf{T}},\lambda)\sim\lambda^{-d}\Phi(\log\lambda)+o(\lambda^{-d}),$
(6.6)

with $d$ being the Haußdorff dimension and $\Phi$ being a bounded periodic
function. By passing to the direct product, one obtains similar examples with
non-power asymptotics in higher dimensions. This effect was also studied in
[59], [47] and many other papers in an arbitrary dimension. Thus, at least for
now, it seems to be non-realistic to search for eigenvalue asymptotics in a
general case, without structural regularity. Nevertheless, our results in
Sect. 3 demonstrate that the upper eigenvalues estimates are order sharp.

##  7\. The difference of powers of resolvents

The reasoning leading to the eigenvalue estimates above, produces spectral
estimates for the difference of powers of resolvents. In order to simplify the
presentation, we discuss Schrödinger-like operators in
${\mathbb{R}}^{\mathbf{N}}$ with singular potentials only. The case of Robin
operators is treated similarly.

We recall the expressions for ${\mathbf{A}}_{{\mathbf{V}}}^{-1}$ in (2.9),

$({\mathbf{A}}_{{\mathbf{V}}})^{-1}={\mathbf{A}}^{-\frac{1}{2}}(1+{\mathbf{T}})^{-1}{\mathbf{A}}^{-\frac{1}{2}},\,{\mathbf{T}}=({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}})^{*}{\mathbf{U}}({\mathbf{F}}\gamma{\mathbf{A}}^{-\frac{1}{2}}),$
(7.1)

thus,

$(1+{\mathbf{T}})^{-1}=1-{\mathbf{T}}+{\mathbf{T}}(1+{\mathbf{T}})^{-1}{\mathbf{T}}\equiv
1-{\mathbf{T}}+{\mathbf{T}}^{\prime}.$ (7.2)

We consider the operator

$({\mathbf{A}}_{{\mathbf{V}}})^{-m}=\left[{\mathbf{A}}^{-\frac{1}{2}}(1-{\mathbf{T}}+{\mathbf{T}}^{\prime}){\mathbf{A}}^{-\frac{1}{2}}\right]^{m},$
(7.3)

for an integer $m>1.$ The operator in (7.3) splits into the sum of terms of
the following form:

H1:

${\mathbf{A}}^{-m}$;

H2:

$m$ terms containing one factor
${\mathbf{W}}={\mathbf{A}}^{-\frac{1}{2}}{\mathbf{T}}{\mathbf{A}}^{-\frac{1}{2}}$
and $m-1$ factors ${\mathbf{A}}^{-1}$;

H3:

terms containing two factors ${\mathbf{W}}$ and $m-2$ factors
${\mathbf{A}}^{-1}$

H4:

the remaining terms, those that have at least one factor
${\mathbf{A}}^{-\frac{1}{2}}{\mathbf{T}}^{\prime}{\mathbf{A}}^{-\frac{1}{2}}$
and those that have at least three factors ${\mathbf{W}}$.

In calculation of the resolvent difference
${\mathbf{A}}_{{\mathbf{V}}}^{-m}-{\mathbf{A}}^{-m}$, the term of the form H1
cancels. Consider some term of the form H2. Such term is obtained from H1 by
means of replacing one of $m$ factors ${\mathbf{A}}^{-1}$ by ${\mathbf{W}}$,
so it has the form

${\mathbf{R}}_{m_{1},m_{2}}{\mathbf{A}}^{-m_{1}}{\mathbf{A}}^{-1/2}{\mathbf{T}}{\mathbf{A}}^{-1/2}{\mathbf{A}}^{-m_{2}}={\mathbf{Q}}_{1}^{*}{\mathbf{Q}}_{2},\,m_{1}+m_{2}=m-1,$
(7.4)

where ${\mathbf{Q}}_{1}=UF\gamma{\mathbf{A}}^{-m_{1}-1},$
${\mathbf{Q}}_{2}=F\gamma{\mathbf{A}}^{-m_{2}-1}$, and as previously,
$F=|V|^{\frac{1}{2}}$, $U=\hbox{{\rm sign}}\,(V).$

The singular numbers of such operator are estimated by means of Lemma 5.4,
with $l_{1}=m_{1}+1,$ $l_{2}=m_{2}+1.$ This gives

$\displaystyle n^{\sup}({\mathbf{R}}_{m_{1},m_{2}},\theta)\leq C$ (7.5)
$\displaystyle\|F_{1}\|^{\theta}_{(\theta_{1}),\mu}\|F_{2}\|^{\theta}_{(\theta_{2}),\mu},$
$\displaystyle\theta^{-1}=\frac{d-{\mathbf{N}}+2(m_{1}+m_{2}+2)}{d}=\frac{d-{\mathbf{N}}+2(m+1)}{d}.$

Note that all terms of this form produce the same order of eigenvalue
estimates, with $\theta=\frac{d}{d-{\mathbf{N}}+2(m+1)}$. This will be the
leading term in the spectral estimate for
${\mathbf{A}}_{{\mathbf{V}}}^{-1}-{\mathbf{A}}^{-1}.$

Next, we consider terms of type H3; each such term is now obtained by
replacing _two_ factors ${\mathbf{A}}^{-1}$ in ${\mathbf{A}}^{-m}$ by
${\mathbf{W}}$. So, such operator can be represented as

${\mathbf{R}}_{m_{1},m_{2},m_{3}}={\mathbf{A}}^{-m_{1}}{\mathbf{A}}^{-1/2}{\mathbf{T}}{\mathbf{A}}^{-1/2}{\mathbf{A}}^{-m_{2}}{\mathbf{A}}^{-1/2}{\mathbf{T}}{\mathbf{A}}^{-1/2}{\mathbf{A}}^{-m_{3}},\,m_{1}+m_{2}+m_{3}=m-2.$
(7.6)

Such operator can be represented as the product of two operators of the form
H2,

${\mathbf{R}}_{m_{1},m_{2},m_{3}}=\left[{\mathbf{A}}^{-m_{1}}{\mathbf{A}}^{-1/2}{\mathbf{T}}{\mathbf{A}}^{-1/2}{\mathbf{A}}^{-m_{2}^{\prime}}\right]\left[{\mathbf{A}}^{-m_{2}^{\prime\prime}}{\mathbf{A}}^{-1/2}{\mathbf{T}}{\mathbf{A}}^{-1/2}{\mathbf{A}}^{-m_{3}}\right].$
(7.7)

with $m_{2}^{\prime}+m_{2}^{\prime\prime}=m_{2}$. We apply the previous
calculation, using Lemma 5.4, for each of factors in (7.7) and then the Ky Fan
inequality for the product of operators, to obtain

$n(\lambda,{\mathbf{R}}_{m_{1},m_{2},m_{3}})=O(\lambda^{-\theta^{(3)}}),\,\theta^{(3)}=\frac{d}{d-{\mathbf{N}}+2(m+2)}<\theta=\frac{d}{d-{\mathbf{N}}+2(m+1)}.$
(7.8)

For terms of type H4, similar calculation shows that the more factors
${\mathbf{A}}^{-1/2}{\mathbf{T}}{\mathbf{A}}^{-1/2}$ are inserted in the term
of the expansion of ${\mathbf{A}}_{{\mathbf{V}}}^{-m},$ the faster is the
singular numbers decay for this term.

Finally, if some factor ${\mathbf{A}}^{-1}$ is replaced by
${\mathbf{T}}^{\prime}={\mathbf{T}}(1+{\mathbf{T}})^{-1}{\mathbf{T}}$, the
resulting operator ${\mathbf{R}}^{\prime}$ can be represented as

$\displaystyle{\mathbf{R}}^{\prime}={\mathbf{A}}^{-m_{1}-\frac{1}{2}}{\mathbf{T}}^{\prime}{\mathbf{A}}^{-m_{2}-\frac{1}{2}}={\mathbf{Q}}_{1}^{*}{\mathbf{Q}}_{2}(1+{\mathbf{T}})^{-1}{\mathbf{Q}}_{2}*^{\mathbf{Q}}_{3},$
$\displaystyle{\mathbf{Q}}_{1}=F\gamma{\mathbf{A}}^{-m_{1}-1},{\mathbf{Q}}_{2}=F\gamma{\mathbf{A}}^{-\frac{1}{2}},{\mathbf{Q}}_{3}=F\gamma{\mathbf{A}}^{-m_{2}-1}.$

To estimate the singular numbers of the latter operator we again apply Lemma
5.4 and then the Ky Fan inequality, which gives, again,

$n(\lambda,{\mathbf{R}}^{\prime})=O(\lambda^{-\theta^{3}}.$

Terms of the type H4 containing more additional compact factors ${\mathbf{T}}$
or ${\mathbf{T}}^{\prime}$, have, by Lemma 2.2 a faster singular numbers decay
than the main term H2.

The passage to estimates for the resolvent difference for two perturbations,
${\mathbf{V}}_{1}$ and ${\mathbf{V}}_{2}$ follows the pattern of Theorem 5.6,
with the same use of Lemma 2.11. As a result, we have the following estimate
for eigenvalues of the difference of powers of resolvents.

######  Theorem 7.1.

Let the measure $\mu$ and the weight functions $V_{1},V_{2}$ satisfy condition
(5.6). Then for the singular numbers of the operator
${\mathbf{A}}_{V_{1}}^{-m}-{\mathbf{A}}_{{\mathbf{V}}_{2}}^{-m}$ the estimate
holds

${\mathbf{n}}^{\sup}({\mathbf{A}}_{V_{1}}^{-m}-{\mathbf{A}}_{{\mathbf{V}}_{2}}^{-m},\theta)\leq
C\|V_{1}-V_{2}\|_{(\theta),\mu}^{\theta},\,\theta=\frac{d}{d-{\mathbf{N}}+2(m+1)}.$
(7.9)

It follows, in particular, that for $m$ large enough, the difference of
resolvents belongs to the trace class, which leads to standard consequences of
the scattering theory.

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