# Body resonances for classical waves

Andrea Mantile  and  Andrea Posilicano  Laboratoire de Mathématiques de Reims,
UMR9008 CNRS et Université de Reims Champagne-Ardenne, Moulin de la Housse BP
1039, 51687 Reims, France  DiSAT, Sezione di Matematica, Università
dell’Insubria, via Valleggio 11, I-22100 Como, Italy  andrea.mantile@univ-
reims.fr<EMAIL_ADDRESS>
###### Abstract.

We provide a detailed study of the spectral properties of the linear operator
$H(\varepsilon)=-(\varepsilon^{2}\chi_{\Omega_{\varepsilon}}+\chi_{\Omega^{c}_{\varepsilon}})\Delta$
modeling, through the wave equation $(\partial_{tt}+H(\varepsilon))u=0$, the
dynamics of acoustic waves in the presence of a small inhomogeneity of size
$\varepsilon$ having high contrast $\varepsilon^{-2}$. In particular, we give
precise results on the localization of the resonances of $H(\varepsilon)$ and
their first-order $\varepsilon$-expansions; the latter are explicitly
expressed in terms of the eigenvalues and eigenvectors of the Newton potential
operator of the set $\Omega$ whose rescaling of size $\varepsilon$ defines
$\Omega_{\varepsilon}$.

##  1\. Introduction

We consider the $\varepsilon$-dependent linear operator

(1.1) $\begin{array}[c]{ccccc}H(\varepsilon):H^{2}(\mathbb{R}^{3})\subset
L^{2}(\mathbb{R}^{3})\rightarrow
L^{2}(\mathbb{R}^{3})\,,&&H(\varepsilon):=-b_{\varepsilon}\Delta\,,&&b_{\varepsilon}:=\varepsilon^{2}\chi_{\Omega_{\varepsilon}}+\chi_{\Omega_{\varepsilon}^{c}}\,,\end{array}$

where $0<\varepsilon<1$ is a small parameter fixing the size of the contracted
set

$\Omega_{\varepsilon}:=\left\\{x\in\mathbb{R}^{3}:x=\varepsilon y\,,\
y\in\Omega\right\\}\,,$

built from the reference set $\Omega\subset\mathbb{R}^{3}$ which is assumed to
be open, bounded, connected and containing the origin;
$\chi_{\Omega_{\varepsilon}}$ is the characteristic function of
$\Omega_{\varepsilon}$ and the contrast function $b_{\varepsilon}$ in (1.1)
models a sharp discontinuity of a medium across the interface
$\partial\Omega_{\varepsilon}$. $H(\varepsilon)$ generates the dynamics of
acoustic pressure waves by the equation

(1.2) $b_{\varepsilon}^{-1}(x)\,\frac{\partial^{2}u}{\partial
t^{2}}(t,x)=\Delta u(t,x,t)\,.$

Here $\varepsilon=\sqrt{1_{\Omega_{\varepsilon}}b_{\varepsilon}}$ defines the
relative speed of propagation inside $\Omega_{\varepsilon}$; hence, a small
value of $\varepsilon$ corresponds to a regime of small relative speed of
propagation. While our work is focused on the $3D$ case, it is worth
mentioning that this equation in 2D is also associated to electromagnetic
waves propagation in an isotropic medium having unit magnetic permeability and
a dielectric permittivity given by $b_{\varepsilon}$. In this connection, we
expect that our analysis may provide qualitative insights for a wider class of
physical systems. It is also worth mentioning that the stationary (time-
harmonic) problem corresponding to (1.2) has been considered in recent works
as a simplified scattering model of a dielectric microresonator (see, e.g.,
[3], [4] and related references). In this connection, $H(\varepsilon)$ can be
considered as the generator of the dynamics for such specific resonant-
scattering models.

A small-$\varepsilon$ approximation for the solutions of the Cauchy problem
corresponding to (1.2) has been recently discussed in [10], where a point-
scatterer effective dynamics is validated in the case the initial data are
sufficiently regular and supported outside the inhomogeneity
$\Omega_{\varepsilon}$. It is important to underline that such effective
dynamics has a rather not-standard definition, in the sense that it does not
corresponds to the one generated by a standard point-perturbation of the
Laplacian (as given in [2, Chapter I.1]), but expresses in terms of
infinitely-many independent $1D$ dynamical systems, each fed by the Cauchy
data of the problem and by a single eigenvalue of the Newton potential
operator of $\Omega$.

As well known from the theory of second-order Cauchy problems, the wave
dynamics generated by $H(\varepsilon)$ is defined in terms of the cosine and
sine operators associated to its resolvent (see [10, Section 3]). In
particular, the computation of such operators involve integrals of the
resolvent along the parabola $\\{z=(c+i\gamma)^{2}\,,\ \gamma\in\mathbb{R}\\}$
with arbitrary $c>0$. The analysis developed in [10] requires resolvent
estimates sufficiently far away from the spectrum, depending on $c>0$, and the
main technical issues are concerned with the control of both the small scale
limit and the parabola’s infinite branches contributions (involving high-
energies resolvent estimates) on a large time. Such analysis provides a
formula, for the dominant part of the wave, involving an infinite sum over
localized states (Green’s functions) at energies corresponding to the
reciprocals of the eigenvalues of the Newtonian operator on $\Omega$. The
asymptotic formulae presented in [10] suggest that the dynamical system (1.2)
could be driven, in the small scale regime, by a (possibly finite) number of
resonant states. However, a detailed analysis of the resonances involved in
the dynamics and their behavior as $\varepsilon\ll 1$ have not been considered
yet. The aim of this work is the study the generalized spectral problem for
$H(\varepsilon)$, in particular the localization of its resonances, as
$\varepsilon\searrow 0$.

###  1.1. Notation

$\bullet$ $|\Omega|$ denotes the volume of $\Omega$ and $d_{\Omega}$ is its
diameter.

$\bullet$ $\|\cdot\|$ denotes the norm in a space of square integrable
functions like $L^{2}(\Omega)$ and $L^{2}(\mathbb{R}^{3})$ with scalar product
$\langle\cdot,\cdot\rangle$; $\|\cdot\|$ also denotes the operator norm for
bounded linear operators acting between such spaces. Norms in different spaces
are specified with the appropriate subscripts.

$\bullet$ ${\mathscr{B}}(X,Y)$ denotes the Banach space of bounded operators
from the Banach space $X$ to the Banach space $Y$;
${\mathscr{B}}(X,X)\equiv{\mathscr{B}}(X)$.

$\bullet$ ${\mathfrak{S}}_{\infty}(X,Y)\subseteq{\mathscr{B}}(X,Y)$ denotes
the set of compact operators.

$\bullet$ $H^{2}(\mathbb{R}^{3})$, denotes the Hilbert-Sobolev space of order
$2$, i.e. the space of square integrale functions
$f:\mathbb{R}^{3}\to{\mathbb{C}}$ such that $\Delta f$ is square integrable.

$\bullet$ $\mathcal{G}_{\kappa}$ denotes the kernel function of
$(-\Delta-\kappa^{2})^{-1}$, i.e.,
$\mathcal{G}_{\kappa}(x)=\frac{e^{i\kappa|x|}}{4\pi|x|}$.

$\bullet$ $\varrho(A)$ and $\sigma(A)$ denote the resolvent set and the
spectrum of the self-adjoint operator $A$; $\sigma_{p}(A)$,
$\sigma_{disc}(A)$, $\sigma_{ess}(A)$, $\sigma_{cont}(A)$, $\sigma_{ac}(A)$,
$\sigma_{{sc}}(A)$, denote the point, discrete, essential, continuous,
absolutely continuous and singular continuous spectra.

$\bullet$ ${\mathbb{C}}_{\pm}:=\\{z\in{\mathbb{C}}:\pm\text{Im}(z)>0\\}$.

$\bullet$ $D_{r}(z)\subset{\mathbb{C}}$ denotes the open disc of radius $r$
and center $z$; $D_{r}\equiv D_{r}(0)$.

###  1.2. Results

We investigated the spectrum and resonances of $H\left(\varepsilon\right)$ in
the small-$\varepsilon$ regime. As regards the spectrum, in Lemma 2.3 and
Corollary 2.6 we show that, for $\varepsilon\in(0,1)$, one has

$\begin{array}[c]{ccc}\sigma_{ac}\left(H(\varepsilon)\right)=\left[0,+\infty\right)\,,&\text{and}&\sigma_{p}\left(H\left(\varepsilon\right)\right)=\sigma_{sc}\left(H(\varepsilon)\right)=\varnothing\,,\end{array}$

which gives, as regards the resolvent set,
$\varrho(H(\varepsilon)):=\mathbb{C}\backslash[0,+\infty)$. In particular,
$H(\varepsilon)$ has no embedded eigenvalues and in Theorem 2.5 we give a
limiting absorption principle for $H(\varepsilon)$: namely, we show that the
limits

$\lim_{\delta\searrow 0}\,\big{(}H(\varepsilon)-(\lambda\pm
i\delta)\big{)}^{-1}\,,$

exist in appropriate weighted spaces for all $\lambda\in\mathbb{R}$. As
regards the resonances, after introducing the operator

$M_{\kappa}(\varepsilon):=1-(1-\varepsilon^{2})\kappa^{2}N_{\varepsilon\kappa}\,,\qquad
N_{\kappa}u(x):=\frac{1}{4\pi}\int_{\Omega}\frac{e^{i\kappa|x-y|}}{|x-y|}\,u(y)\,dy\,,\quad\kappa\in{\mathbb{C}}\,,$

using the resolvent formula (2.5) together with (3.1), (3.5) and Lemma 3.1, we
identify the resonant set of $H(\varepsilon)$ with the exceptional points
$\kappa^{2}$ where $M_{\kappa}(\varepsilon)$ is not invertible, i.e.

(1.3)
$\mathsf{e}(H(\varepsilon))=\\{\kappa_{\circ}^{2}(\varepsilon):\ker(M_{\kappa_{\circ}(\varepsilon)}(\varepsilon))\not=\\{0\\},\
\kappa_{\circ}(\varepsilon)\in{\mathbb{C}}_{-}\\}\,,$

(see the definition (3.6) and the subsequent remark). Then, the analysis is
carried on the corresponding limit operator

$M_{\kappa}\left(0\right):=1-\kappa^{2}N_{0}\,,$

whose characteristic points are determined in terms of the spectrum of the
Newton potential operator of the reference set $\Omega$, i.e.,

$N_{0}u(x)=\frac{1}{4\pi}\int_{\Omega}\frac{u(y)\,dy}{|x-y|}\,.$

It is well known that $N_{0}$ is an Hilbert-Schmidt not-negative operator
having a $L^{2}(\Omega)$-complete orthonormal eigensystem
$\\{(\lambda_{n},e_{n})\\}_{1}^{+\infty}$, where the eigenvalues $\lambda_{n}$
accumulate at zero. Building on these properties and using a
small-$\varepsilon$ expansion, we show that $M_{\kappa}(\varepsilon)$ is
invertible depending on the distance of $\kappa^{2}$ from
$\sigma(N_{0}^{-1}))$: in any bounded region of ${\mathbb{C}}$,
$\mathsf{e}(H(\varepsilon))$ is contained inside the disjoint union of small
disks of radius of order $\varepsilon$ centered at points
$\lambda^{-1}\in\sigma(N_{0}^{-1})$ (see Theorem 3.4 and Corollary 3.5). This
yields the localization of the resonances as $\varepsilon\searrow 0$. Our main
result, whose proof is given in Section 3, shows that in a small neighborhood
of each $\lambda^{-1}\in\sigma(N_{0}^{-1})$ there are as many resonances
(counted with multiplicity) as the multiplicity of $\lambda$ and provides
their first-order $\varepsilon$-expansions:

######  Theorem 1.1.

Let $\lambda_{\circ}\in\sigma_{disc}(N_{0})$ with
$\dim(\ker(\lambda_{\circ}-N_{0}))=m$; let $e_{j}^{\circ}$, $1\leq j\leq m$,
be the corresponding orthonormal eigenvectors. Then, whenever $\varepsilon$ is
sufficiently small, close to $\lambda_{\circ}^{-1}$ there are $m$ (not
necessarily distinct) resonances
$\kappa_{\circ,j}^{2}(\varepsilon)\in\mathsf{e}(H(\varepsilon))$, the
functions $\varepsilon\mapsto\kappa_{\circ,j}^{2}(\varepsilon)$ are analytic
in a neighborhood of zero and

(1.4)
$\kappa_{\circ,j}^{2}(\varepsilon)=\frac{1}{\lambda_{\circ}}-i\varepsilon\
\frac{|\langle
1,e_{j}^{\circ}\rangle|^{2}}{4\pi\lambda_{\circ}^{5/2}}+O(\varepsilon^{2})\,.$

It is known that the quantitative analysis of the resolvent on the non-
physical sheet requires the perturbation theory of eigenvalues of non self-
adjoint operators. In this setting, even if the perturbation is analytic, the
eigenvalue expansions are given as Puiseux series. However, in our specific
case, an implicit function argument allows us to reduce the problem to the
perturbative analysis of $N_{\kappa}$ for $\kappa$ close to
$0\in{\mathbb{C}}$. Since $N_{i\kappa}$ is self-adjoint, an adaptation of the
Rellich classical arguments leads to the power series expansion in (1.4).

The asymptotic analysis of the resonances of $H(\varepsilon)$, in particular
(1.4), provides a new insight on the small-scale behavior of the system in
(1.2). As recalled above, an interesting outcome of [10] is that the main
contribution to the waves dynamics expresses in terms of the eigenvalues in
$\sigma(N_{0}^{-1})$ which, according to (1.4), correspond to the small-scale
limit values of the resonances. This motivates to reconsider the time-domain
problem to enlighten the role of resonances. Since the time propagator of
(1.2) requires inverse Laplace transforms of the resolvent of
$H(\varepsilon)$, we expect that the resolvent analysis provided here will be
a key ingredient in computing resonant-modes expansions in the asymptotic
regime.

The spectral analysis of model operators related to micro-resonators in wave
dynamics has not been investigated so far. Such models are characterized by
the enhancement of the scattered field occurring at specific resonant
frequencies and, due to their application perspectives, they have attracted an
increasing attention in recent years. These resonant phenomena have been shown
to emerge in different physical systems where stationary-waves interact with
highly-contrasted small inhomogeneities (see, e.g., [5]). Depending on the
micro-resonator design, the resonant modes (subwavelenght resonances in
physical literature), arise from the excitation of specific spectral points of
the volume potential or of the surface potential involved in the integral
representation of the scattering equation. In the time-harmonic problem
corresponding to our setting, a Lippmann-Schwinger representation of the
scattering allows to express the solution in terms of the Newton potential
operator: in this case, the resonant scattering is determined by volume modes,
also referred to as body-resonances.

It is a commonly accepted idea that subwavelenght resonances should correspond
to resonances in the proper generalized-spectral sense. At this concern, we
remark that the expansions in (1.4) are consistent with the asymptotic
formulae presented in [4] in the time-harmonic case. In this connection, our
results extend the previous analysis, providing the expected spectral
characterization of body resonances.

##  2\. The model operator and its spectrum

The linear operator (1.1) re-writes as an additive perturbation of the
Laplacian:

(2.1)
$\begin{array}[c]{ccc}H(\varepsilon)=-\Delta+T(\varepsilon)\,,&&T(\varepsilon):=(1-\varepsilon^{2})\chi_{\Omega_{\varepsilon}}\Delta\,.\end{array}$

We introduce the free resolvent

$R_{\kappa}\in\mathscr{B}(L^{2}(\mathbb{R}^{3}),H^{2}(\mathbb{R}^{3}))\,,\qquad
R_{\kappa}:=(-\Delta-\kappa^{2})^{-1}\,,\qquad\kappa\in\mathbb{C}_{+}\,.$

In the following, we will use the equivalence, holding in the Banach space
sense,

(2.2)
$L^{2}(\mathbb{R}^{3}\\!,\varepsilon):=L^{2}(\mathbb{R}^{3};b_{\varepsilon}^{-1}dx)\simeq
L^{2}(\mathbb{R}^{3})\,.$

By [10, Theorem 3.2], a Rellich-Kato type resolvent formula holds:

######  Theorem 2.1.

$H(\varepsilon)$ is a closed operator in $L^{2}(\mathbb{R}^{3})$ and a non-
negative self-adjoint operator in $L^{2}(\mathbb{R}^{3}\\!,\varepsilon)$ with
resolvent

(2.3)
$R_{\kappa}(\varepsilon):=(H(\varepsilon)-\kappa^{2})^{-1}=R_{\kappa}-R_{\kappa}\Lambda_{\kappa}(\varepsilon)R_{\kappa}\,,\qquad\kappa\in{\mathbb{C}}_{+}\,,$

where

$\Lambda_{\kappa}(\varepsilon):=(1+T(\varepsilon)R_{\kappa})^{-1}T(\varepsilon)\,,\qquad(1+T(\varepsilon)R_{\kappa})^{-1}\in{{\mathscr{B}}}(L^{2}(\mathbb{R}^{3}))\,,\qquad\kappa\in{\mathbb{C}}_{+}\,.$

######  Remark 2.2.

Let $\kappa\in{\mathbb{C}}_{+}$. By

(2.4)
$R_{\kappa}(1+T(\varepsilon)R_{\kappa})^{-1}=R_{\kappa}-R_{\kappa}(1+T(\varepsilon)R_{\kappa})^{-1}T(\varepsilon)R_{\kappa}\,,$

one has

(2.5) $R_{\kappa}(\varepsilon)=R_{\kappa}(1+T(\varepsilon)R_{\kappa})^{-1}$

and so, by (2.3), there holds

$R_{\kappa}(\varepsilon)=R_{\kappa}-R_{\kappa}(\varepsilon)T(\varepsilon)R_{\kappa}\,.$

Then, by

(2.6)
$T(\varepsilon)R_{\kappa}=(1-\varepsilon^{2})1_{\Omega_{\varepsilon}}\Delta
R_{\kappa}=-(1-\varepsilon^{2})1_{\Omega_{\varepsilon}}(1+\kappa^{2}R_{\kappa})\,,$

one gets the relation

(2.7)
$R_{\kappa}(\varepsilon)-R_{\kappa}=(1-\varepsilon^{2})R_{\kappa}(\varepsilon)1_{\Omega_{\varepsilon}}(1+\kappa^{2}R_{\kappa})\,.$

######  Lemma 2.3.

$\sigma_{p}(H(\varepsilon))=\varnothing$.

###### Proof.

Since $H(\varepsilon)\geq 0$, it suffices to show that
$\sigma_{p}(H(\varepsilon))\cap[0,+\infty)=\varnothing$. Since
$b_{\varepsilon}(x)\not=0$ for any $x\in\mathbb{R}^{3}$, $H(\varepsilon)u=0$
is equivalent to $-\Delta u=0$; this gives
$0\notin\sigma_{p}(H(\varepsilon))$. Let us now take $\lambda>0$ and let us
seek for a solution $u\in
H^{2}(\mathbb{R}^{3})\subset\mathcal{C}(\mathbb{R}^{3})$ of the equation

(2.8) $(H(\varepsilon)-\lambda)u=0\,.$

We aim to show that $u=0$. By the definition of $H(\varepsilon)$, (2.8)
implies

$(\Delta+\lambda)u(x)=0\,,\qquad
x\in\mathbb{R}^{3}\backslash\overline{\Omega}_{\varepsilon}\,.$

By the unique continuation property (see, e.g., [12, Theorem XIII.63]), one
obtains: $u=0$ in $\mathbb{R}^{3}\backslash\overline{\Omega}_{\varepsilon}$.
Henceforth (2.8) recasts as

$\begin{cases}(\varepsilon^{2}\Delta+\lambda)u(x)=0\,,&x\in\Omega_{\varepsilon}\,,\\\
u(x)=0\,,&x\in\partial\Omega_{\varepsilon}\,.\end{cases}$

This is a boundary value problem in $\Omega_{\varepsilon}$ for the Helmholtz
equation with zero Dirichlet trace; by the uniqueness of its solution, we get
$u=0$ in $\overline{\Omega}_{\varepsilon}$. ∎

######  Lemma 2.4.

$\sigma_{ess}(H(\varepsilon))=[0,+\infty)$.

###### Proof.

We use the resolvent difference formula (2.7). The adjoint
$(R_{\kappa}(\varepsilon)1_{\Omega_{\varepsilon}})^{\ast}$ in the Hilbert
space $L^{2}(\mathbb{R}^{3}\\!,\varepsilon)$ is
$(R_{\kappa}(\varepsilon)1_{\Omega_{\varepsilon}})^{\ast}=1_{\Omega_{\varepsilon}}R_{\bar{z}}(\varepsilon)$.
By the equivalence (2.2) and by the mapping properties of
$R_{\kappa}(\varepsilon)$, one gets
$(R_{\kappa}(\varepsilon)1_{\Omega_{\varepsilon}})^{\ast}\in{{\mathscr{B}}}(L^{2}(\mathbb{R}^{3}),H^{2}(\Omega_{\varepsilon}))$.
By the compact embedding $H^{2}(\Omega_{\varepsilon})\hookrightarrow
L^{2}(\Omega_{\varepsilon})$, the operator
$(R_{\kappa}(\varepsilon)1_{\Omega_{\varepsilon}})^{\ast}\in{{\mathscr{B}}}(L^{2}(\mathbb{R}^{3}))$
is compact, which implies the compactness of
$R_{\kappa}(\varepsilon)1_{\Omega_{\varepsilon}}\in{{\mathscr{B}}}(L^{2}(\mathbb{R}^{3}))$
and, hence, the compactness of the r.h.s. of (2.7). Therefore, the Weyl
theorem applies (see, e.g., [12, Th. XIII.14]) and
$\sigma_{ess}(H(\varepsilon))=\sigma_{ess}(-\Delta)=[0,+\infty)$. ∎

###  2.1. The limiting absorption principle

Here, we prove that there is no singular continuous spectrum, so that the
spectrum of $H(\varepsilon)$ is purely absolutely continuous. To this end we
at first show that a limiting absorption principle holds for $H(\varepsilon)$.
As it is well known, a limiting absorption principle holds for the free
Laplacian (see, e.g., [1, Section 4]):

For any $\lambda\in\mathbb{R}\backslash\\{0\\}$ and for any
$\alpha>{\frac{1}{2}}$, the limits

(2.9) $\lim_{\delta\searrow 0}(-\Delta-(\lambda\pm i\delta))^{-1}$

exist in
$\mathscr{B}(L^{2}_{\alpha}(\mathbb{R}^{3}),H^{2}_{-\alpha}(\mathbb{R}^{3}))$;
the same hold true in the case $\lambda=0$ whenever $\alpha>3/2$. Here,
$H^{k}_{\alpha}(\mathbb{R}^{3})$, $k\in\mathbb{N}$, $\alpha\in\mathbb{R}$,
($L^{2}_{\alpha}(\mathbb{R}^{3})\equiv H^{0}_{\alpha}(\mathbb{R}^{3})$)
denotes the weighted Sobolev spaces with weight $w(x)=(1+|x|^{2})^{\alpha}$
(see [1, Section 2] for more details).

As regards $H(\varepsilon)$, the result is of the same kind:

######  Theorem 2.5.

The limits

(2.10) $\lim_{\delta\searrow 0}(H(\varepsilon)-(\lambda\pm i\delta))^{-1}\,,$

exist in
${{\mathscr{B}}}(L^{2}_{\alpha}(\mathbb{R}^{3}),L^{2}_{-\alpha}(\mathbb{R}^{3}))$
for any $\alpha>1/2$ and any $\lambda\in\mathbb{R}\backslash\\{0\\}$; the same
hold true in the case $\lambda=0$ whenever $\alpha>3/2$.

###### Proof.

Introducing, for any $\lambda\in\mathbb{R}$ and
$z\in{\mathbb{C}}\backslash[0,+\infty)$, the abbreviated notations

${\mathcal{R}}^{\pm}_{\lambda}\equiv\lim_{\delta\searrow
0}{\mathcal{R}}_{\lambda\pm
i\delta}\,,\qquad{\mathcal{R}}_{z}\equiv(-\Delta-z)^{-1}\,,\qquad{\mathcal{R}}_{z}(\varepsilon)\equiv(H(\varepsilon)-z)^{-1}\,,$

by [12, Lemma 1, page 170], one has

(2.11)
$\mathcal{R}_{z}\in\mathscr{B}(L_{\alpha}^{2}(\mathbb{R}^{3}))\,,\qquad\alpha\in\mathbb{R}\,.$

Then, by
$1_{\Omega_{\varepsilon}}\in{\mathscr{B}}(L_{\alpha}^{2}(\mathbb{R}^{3}))$, by
(2.11), by (2.3) and (2.6),

(2.12)
$\mathcal{R}_{z}(\varepsilon)\in\mathscr{B}(L_{\alpha}^{2}(\mathbb{R}^{3}))\,,\qquad\alpha\in\mathbb{R}\,.$

By arguing as in the proof of Lemma 2.4 and by the continuous embedding
$L^{2}(\Omega_{\varepsilon})\hookrightarrow L_{\beta}^{2}(\mathbb{R}^{3})$,
one gets
$\mathcal{R}_{z}(\varepsilon)1_{\Omega_{\varepsilon}}\in{\mathfrak{S}}_{\infty}(L_{\alpha}^{2}(\mathbb{R}^{n}),L_{\beta}^{2}(\mathbb{R}^{3}))$
for any real $\alpha$ and $\beta$. Hence, in particular, by (2.7),

(2.13)
$\mathcal{R}_{z}(\varepsilon)-\mathcal{R}_{z}\in{\mathfrak{S}}_{\infty}(L^{2}(\mathbb{R}^{n}),L_{\beta}^{2}(\mathbb{R}^{3}))\,,\quad\beta>2\alpha\,.$

Furthermore, according to [7, Corollary 5.7(b)], for all compact subset
$K\subset(0,+\infty)$ there exists a constant $c_{K}>0$ such that, for
$\lambda\in K$ one has

(2.14) $\forall u\in
L_{2\alpha}^{2}(\mathbb{R}^{n})\cap\ker(\mathcal{R}_{\lambda}^{+}-\mathcal{R}_{\lambda}^{-}),\quad\|\mathcal{R}_{\lambda}^{\pm}u\|_{L^{2}(\mathbb{R}^{3})}\leq
c_{K}\|u\|_{L_{2\alpha}^{2}(\mathbb{R}^{3})}\,.$

The relations (2.11)-(2.14) permit us to apply the abstract results provided
in [14], where, with respect to the notations there, $H_{1}=-\Delta$,
$\mathcal{H}_{1}=L^{2}(\mathbb{R}^{3})$, $H_{2}=H(\varepsilon)$,
$\mathcal{H}_{2}=L^{2}(\mathbb{R}^{3}\\!,\varepsilon)$, $J_{1}$ the identity
in $\mathcal{H}_{1}$, $J_{2}:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$ the
multiplication by $b_{\varepsilon}^{1/2}$, $X=L_{\alpha}^{2}(\mathbb{R}^{3})$
: hypothesis (T1) and (E1) in [14, page 175] corresponds to our (2.9), (2.14)
and (2.13) respectively; then, by [14, Proposition 4.2], the latter imply
hypotheses (LAP) and (E) in [14, page 166], i.e. (2.9) again and

$J_{2}^{\ast}\mathcal{R}_{z}(\varepsilon)J_{2}-\mathcal{R}_{z}\in{\mathfrak{S}}_{\infty}(L_{-\alpha}^{2}(\mathbb{R}^{3}),L_{\alpha}^{2}(\mathbb{R}^{3}))\,,$

and hypothesis (T) in [14, page 168], a technical variant of (2.14). By [14,
Theorem 3.5], these last hypotheses, together with $H(\varepsilon)\geq 0$ and
(2.11)-(2.12) (i.e. hypothesis (OP) in [14, page 165]), give the existence of
the limits (2.10) outside $\sigma_{p}(H(\varepsilon))$; the latter is empty by
Lemma 2.3. ∎

From the selfadjointness of $H(\varepsilon)$ on
$L^{2}(\mathbb{R}^{3}\\!,\varepsilon)$, we have the orthogonal decomposition

$L^{2}(\mathbb{R}^{3}\\!,\varepsilon)=(L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{c}\oplus(L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{pp}\,,$

where $(L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{c}$ and
$(L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{pp}$ respectively denote the
continuous and pure point subspaces of $H(\varepsilon)$. Let
$P_{\lambda}(\varepsilon)$ be the spectral resolution of the identity
associated with $H(\varepsilon)$. Recall that if
$f\in(L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{c}=((L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{pp})^{\bot}$,
$\lambda\mapsto\left\langle f,P_{\lambda}(\varepsilon)f\right\rangle$ is a
continuous function on $\sigma_{c}(H(\varepsilon))$. The continuous subspace
$(L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{c}$ further decomposes as an
orthogonal sum of the absolutely continuous and singular subspaces of
$H(\varepsilon)$

$(L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{c}=(L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{ac}\oplus(L^{2}(\mathbb{R}^{3}))^{sc}\,.$

The absolute continuous part of the spectrum ${\sigma}_{{ac}}(H(\varepsilon))$
is defined by the condition that, if
$f\in(L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{ac}$, then
$\lambda\mapsto\left\langle f,P_{\lambda}(\varepsilon)f\right\rangle$ is
absolutely continuous on $\sigma_{{ac}}(H(\varepsilon))$. Hence, the absence
of singular continuous spectrum corresponds to the identity

$(L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{ac}=((L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{pp})^{\bot}\,.$

By Theorem 2.5, we get

######  Corollary 2.6.

$\begin{array}[c]{ccc}\sigma_{ac}(H(\varepsilon))=[0,+\infty)\,,&&\sigma_{sc}(H(\varepsilon))=\varnothing\,.\end{array}$

###### Proof.

Arguing as in [1, Proof of Theorem 6.1], in order to show that
$\sigma_{c}(H(\varepsilon))=\sigma_{ac}(H(\varepsilon))$, it suffices to show
that $\lambda\mapsto\left\langle f,P_{\lambda}(\varepsilon)f\right\rangle$ is
absolutely continuous in $\mathbb{R}\backslash\\{0\\}$ for any
$f\in((L^{2}(\mathbb{R}^{3}\\!,\varepsilon))^{pp})^{\bot}$. Let
$[a,b]\subset\mathbb{R}\backslash\\{0\\}$ and $f\in
L^{2}_{\alpha}(\mathbb{R}^{3})$; by Theorem 2.5 and by Stone’s formula one has

$\left\langle
f,(P_{b}(\varepsilon)-P_{a}(\varepsilon))f\right\rangle=\frac{1}{2\pi
i}\int_{a}^{b}\left\langle
f,(\mathcal{R}_{\lambda}^{+}(\varepsilon)-\mathcal{R}_{\lambda}^{-}(\varepsilon))f\right\rangle
d\lambda\,,$

where we used the abbreviated notation

$\mathcal{R}_{\lambda}^{\pm}(\varepsilon)\equiv\lim_{\delta\searrow
0}(H(\varepsilon)-(\lambda\pm i\delta))^{-1}\,.$

Hence $\lambda\mapsto\left\langle f,P_{\lambda}(\varepsilon)f\right\rangle$ is
continuously differentiable at each point in $\mathbb{R}\backslash\\{0\\}$ for
any $f\in L^{2}_{\alpha}(\mathbb{R}^{3})$. By the same argument as in [1,
Proof of Theorem 6.1], the property extends to the whole
$((L^{2}(\mathbb{R}^{3}))^{pp})^{\bot}$. ∎

######  Corollary 2.7.

$\sigma(H(\varepsilon))=\sigma_{ac}(H(\varepsilon))=[0,+\infty)$

and ${\mathbb{C}}_{+}\ni\kappa\mapsto R_{\kappa}(\varepsilon)$ is an analytic
${\mathscr{B}}(L^{2}(\mathbb{R}^{3}))$-valued map.

###### Proof.

By Lemma 2.3 and Corollary 2.6,

${\sigma}(H(\varepsilon))=\sigma_{p}(H(\varepsilon))\cup\sigma_{cont}(H(\varepsilon))=\sigma_{cont}(H(\varepsilon))=\sigma_{ac}(H(\varepsilon))\,.$

By Lemma 2.3 and by
$\sigma_{{disc}}(H(\varepsilon))\subseteq\sigma_{p}(H(\varepsilon))$,

${\sigma}(H(\varepsilon))=\sigma_{{disc}}(H(\varepsilon))\cup\sigma_{ess}(H(\varepsilon))=[0,+\infty)\,.$

Finally, since $H(\varepsilon)$ is a closed operator, the map
${\mathbb{C}}_{+}\ni\kappa\mapsto(H(\varepsilon)-\kappa^{2})^{-1}$ is
analytic. ∎

##  3\. Localization and $\varepsilon$-expansions of the resonances

In the following, we use the identification

$L^{2}(\mathbb{R}^{3})\equiv L^{2}(\Omega^{c}_{\varepsilon})\oplus
L^{2}(\Omega_{\varepsilon})\,,$

provided by the unitary map $u\mapsto 1_{\Omega_{\varepsilon}^{c}}u\oplus
1_{\Omega_{\varepsilon}}$. Here and below, given the measurable domain
$D\subset\mathbb{R}^{3}$ we denote by $1_{D}:L^{2}(\mathbb{R}^{3})\to
L^{2}(D)$ the bounded linear operator given by the restriction to $D$; then,
the adjoint $1^{*}_{D}:L^{2}(D)\to L^{2}(\mathbb{R}^{3})$ corresponds to the
extension by zero.

In such a framework, see [10, Section 4],

$1+T(\varepsilon)R_{\kappa}:L^{2}(\Omega^{c}_{\varepsilon})\oplus
L^{2}(\Omega_{\varepsilon})\to L^{2}(\Omega^{c}_{\varepsilon})\oplus
L^{2}(\Omega_{\varepsilon})\,,$
$1+T(\varepsilon)R_{\kappa}=\begin{bmatrix}1&0\\\
-(1-\varepsilon^{2})\kappa^{2}1_{\Omega_{\varepsilon}}R_{\kappa}1^{*}_{\Omega^{c}_{\varepsilon}}&\varepsilon^{2}-(1-\varepsilon^{2})\kappa^{2}1_{\Omega_{\varepsilon}}R_{\kappa}1^{*}_{\Omega_{\varepsilon}}\end{bmatrix}\,.$

Hence, for any $\kappa\in{\mathbb{C}}_{+}$, introducing the operator

$S_{\kappa}(\varepsilon):=\varepsilon^{2}-(1-\varepsilon^{2})\kappa^{2}1_{\Omega_{\varepsilon}}R_{\kappa}1^{*}_{\Omega_{\varepsilon}}\,,$

one has, by (2.1),

$S_{\kappa}(\varepsilon)^{-1}\in{\mathscr{B}}(L^{2}(\Omega_{\varepsilon}))\,,\qquad\kappa\in{\mathbb{C}}_{+}\,,$

and

(3.1) $(1+T(\varepsilon)R_{\kappa})^{-1}=\begin{bmatrix}1&0\\\
-(1-\varepsilon^{2})S_{\kappa}(\varepsilon)^{-1}\kappa^{2}1_{\Omega_{\varepsilon}}R_{\kappa}1^{*}_{\Omega_{\varepsilon}^{c}}&S_{\kappa}(\varepsilon)^{-1}\end{bmatrix}\,.$

Introducing the unitary dilation operator

$U(\varepsilon):L^{2}(\mathbb{R}^{3})\to L^{2}(\mathbb{R}^{3})\,,\qquad
U(\varepsilon)u(x):=\varepsilon^{3/2}u(\varepsilon x)\,,$

one has

$U(\varepsilon)\chi_{\Omega_{\varepsilon}}=\chi_{\Omega}U(\varepsilon)\,,\qquad
U(\varepsilon)\chi_{\Omega^{c}_{\varepsilon}}=\chi_{\Omega^{c}}U(\varepsilon)$

and

$U(\varepsilon)R_{\kappa}=\varepsilon^{2}R_{\varepsilon\kappa}U(\varepsilon)\,.$

Hence, defining

(3.2) $N_{\kappa}:=1_{\Omega}R_{\kappa}1^{*}_{\Omega}:L^{2}(\Omega)\to
L^{2}(\Omega)\,,$

i.e.,

$N_{\kappa}u(x)=\frac{1}{4\pi}\int_{\Omega}\frac{e^{ik|x-y|}}{|x-y|}\,u(y)\,dy\,,$

one has

(3.3)
$S_{\kappa}(\varepsilon)=\varepsilon^{2}1_{\Omega_{\varepsilon}}U^{*}_{\varepsilon}1^{*}_{\Omega}M_{\kappa}(\varepsilon)1_{\Omega}U_{\varepsilon}1^{*}_{\Omega_{\varepsilon}}\,,$

where

(3.4)
$M_{\kappa}(\varepsilon):=1-(1-\varepsilon^{2})\kappa^{2}N_{\varepsilon\kappa}\,.$

Furthermore,

(3.5)
$S_{\kappa}(\varepsilon)^{-1}=\varepsilon^{-2}1_{\Omega_{\varepsilon}}U^{*}_{\varepsilon}1^{*}_{\Omega}M_{\kappa}(\varepsilon)^{-1}1_{\Omega}U_{\varepsilon}1^{*}_{\Omega_{\varepsilon}}\,.$

######  Lemma 3.1.

The map ${\mathbb{C}}\ni\kappa\mapsto M_{\kappa}(\varepsilon)$ is a
$\mathscr{B}(L^{2}(\Omega))$-valued analytic map and
${\mathbb{C}}\ni\kappa\mapsto M_{\kappa}(\varepsilon)^{-1}$ is a
$\mathscr{B}(L^{2}(\Omega))$-valued meromorphic map with poles of finite rank.
Such poles are the points
$\kappa(\varepsilon)\in{\mathbb{C}}\backslash{\mathbb{C}}_{+}$ such that
$\ker(M_{\kappa(\varepsilon)}(\varepsilon))\not=\\{0\\}$.

###### Proof.

By Sobolev’s inequality (see, e.g., [11, Example 3, Section IX.4]),
$N_{\kappa}$ is Hilbert-Schmidt and hence compact. Since the analytic map
$\kappa\mapsto N_{\kappa}$ has values in the space of compact operators in
$\mathscr{B}(L^{2}(\Omega))$, the thesis is consequence of the meromorphic
Fredholm theory (see, e.g., [12, Theorem XIII.13]. ∎

The set of resonances of $H(\varepsilon)$ is here defined by

(3.6)
$\mathsf{e}(H(\varepsilon)):=\\{\kappa_{\circ}^{2}(\varepsilon):\kappa\mapsto
S_{\kappa}(\varepsilon)^{-1}\ \text{has a pole at
$\kappa_{\circ}(\varepsilon)\in\mathbb{C}_{-}$}\\}\,.$

######  Remark 3.2.

Since $T(\varepsilon)$ is an additive perturbation with compact support on
$\Omega_{\varepsilon}$, the analogy with the case of Schrödinger operators
suggests to define the resonances of $H(\varepsilon)$ as the poles of the
meromorphic extension of the map $z\mapsto
1_{\Omega_{\varepsilon}}(-H(\varepsilon)-z)^{-1}1^{*}_{\Omega_{\varepsilon}}$
in the non-physical sheet where $\text{Im}(\sqrt{z})<0$. As well known, the
unperturbed resolvent $z\mapsto
1_{\Omega_{\varepsilon}}(-\Delta-z)^{-1}1^{*}_{\Omega_{\varepsilon}}$ has an
analytic extension to the whole complex plane; then, reasoning in terms of
$\kappa=\sqrt{z}$ and using the formula (2.3), we identify
$\mathsf{e}(H(\varepsilon))$ with the set of $\kappa^{2}$ such that
$\kappa\in\mathbb{C}_{-}$ is a pole of the meromorphic extension of
$\kappa\mapsto
1_{\Omega_{\varepsilon}}(1+T(\varepsilon)R_{\kappa})^{-1}1^{*}_{\Omega_{\varepsilon}}$.
Using (3.1), this corresponds to the definition (3.6); furthermore, our
definition of resonance set (3.6) is in agreement with the one given in [2,
Appendix B] (see (3.7) below).

In view of (3.5) and of the result of Lemma 3.1, the set
$\mathsf{e}(H(\varepsilon))$ is discrete and the right hand side of (3.6)
recasts as

$\mathsf{e}(H(\varepsilon))=\\{\kappa_{\circ}^{2}(\varepsilon):\kappa\mapsto
M_{\kappa}(\varepsilon)^{-1}\ \text{has a pole at
$\kappa_{\circ}(\varepsilon)\in{\mathbb{C}}_{-}$}\\}\,,$

which is equivalent to

(3.7)
$\mathsf{e}(H(\varepsilon))=\\{\kappa_{\circ}^{2}(\varepsilon):\ker(M_{\kappa_{\circ}(\varepsilon)}(\varepsilon))\not=\\{0\\},\
\kappa_{\circ}(\varepsilon)\in{\mathbb{C}}_{-}\\}\,.$

The _Newton potential operator_ $N_{0}$ of an open bounded domain
$\Omega\subset\mathbb{R}^{3}$ is the not negative, symmetric operator $N_{0}$
in $L^{2}(\Omega)$ defined by

$N_{0}:L^{2}(\Omega)\to L^{2}(\Omega)\,,\qquad
N_{0}u(x):=\frac{1}{4\pi}\int_{\Omega}\frac{u(y)\,dy}{|x-y|}\,.$

By Sobolev’s inequality (see, e.g., [11, Example 3, Section IX.4]), $N_{0}$ is
Hilbert-Schmidt and hence compact. Since $-\Delta N_{0}u=u$, one has
$\ker(N_{0})=\\{0\\}$ and $N_{0}^{-1}$ is a unbounded self-adjoint operator.
By the spectral theory of compact symmetric operators (see, e.g., [8, Section
6]),

$\sigma_{disc}(N_{0})=\sigma(N_{0})\backslash\\{0\\}$

and the orthonormal sequence $\\{e_{n}\\}_{1}^{+\infty}$ of eigenvectors
corresponding to the discrete spectrum is an orthonormal base of
$\ker(N_{0})^{\perp}=L^{2}(\Omega)$. We denote by
$\\{\lambda_{n}\\}_{1}^{+\infty}$, the set of eigenvalues (counting
multiplicities) indexed in decreasing order. One has

$\|N_{0}\|=\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{n}\searrow 0\,.$

Furthermore,

$\sigma(N_{0}^{-1})=\sigma_{disc}(N_{0}^{-1})=\\{\lambda^{-1}:\lambda\in\sigma_{disc}(N_{0})\\}.$

Let us notice that, depending on $\Omega$, the eigenvalues of $N_{0}^{-1}$ may
not be simple (whenever $\Omega$ is a ball, see [9]).

Now, we introduce the operator

$M_{\kappa}(0):=1-\kappa^{2}N_{0}\,.$

By $M_{\kappa}(0)=(N_{0}^{-1}-\kappa^{2})N_{0}$, one has
$M_{\kappa}(0)^{-1}=N_{0}^{-1}(N_{0}^{-1}-\kappa^{2})^{-1}\in{\mathscr{B}}(L^{2}(\Omega))$
for any $\kappa\in{\mathbb{C}}$ such that
$\kappa^{2}\in\varrho(N_{0}^{-1})\supset{\mathbb{C}}\backslash[\lambda_{1}^{-1},+\infty)$.
Since $N_{0}$ is compact, ${\mathbb{C}}\ni\kappa\mapsto M_{k}(0)^{-1}$ is
meromorphic with poles of finite rank at the points $\kappa_{\circ}$ such that
$\kappa_{\circ}^{2}\in\sigma(N_{0}^{-1})$.

######  Lemma 3.3.

For any $\kappa\in{\mathbb{C}}$ such that
$\kappa^{2}\not\in\sigma(N_{0}^{-1})$, one has

$\|M_{\kappa}(0)^{-1}\|\leq\sqrt{2}\,\,\frac{\max\\{\lambda_{1}^{-1},{\text{\rm
Re}(\kappa^{2})}\\}}{\text{\rm dist}(\kappa^{2},\sigma(N_{0}^{-1}))}.$

###### Proof.

Given $\delta\in(0,1)$, let

$\displaystyle\Lambda_{\delta}(\kappa^{2}):=\\{\lambda\geq\lambda_{1}^{-1}:|\lambda-\kappa^{2}|\geq\delta\lambda\\}$
$\displaystyle=$
$\displaystyle\\{\lambda\geq\lambda_{1}^{-1}:(1-\delta^{2})\lambda^{2}-2\lambda\text{\rm
Re}(\kappa^{2})+\text{\rm Re}(\kappa^{2})^{2}+\text{\rm
Im}(\kappa^{2})^{2}\geq 0\\}\,.$

Hence,

$\Lambda_{\delta}(\kappa^{2})=[\lambda_{1}^{-1},+\infty)$

whenever

(3.8) either $\text{\rm Re}(\kappa^{2})<0$ or $0\leq\delta\text{\rm
Re}(\kappa^{2})\leq(1-\delta^{2})^{1/2}|\text{\rm Im}(\kappa^{2})|$

and

$\displaystyle\Lambda_{\delta}(\kappa^{2})=[\lambda_{1}^{-1},+\infty)$
$\displaystyle\cap\Bigg{(}\left\\{\lambda\leq\frac{\text{\rm
Re}(\kappa^{2})-\sqrt{\delta^{2}\text{\rm
Re}(\kappa^{2})^{2}-(1-\delta^{2})\text{\rm
Im}(\kappa^{2})^{2}}}{1-\delta^{2}}\right\\}$
$\displaystyle\cup\left\\{\lambda\geq\frac{\text{\rm
Re}(\kappa^{2})+\sqrt{\delta^{2}\text{\rm
Re}(\kappa^{2})^{2}-(1-\delta^{2})\text{\rm
Im}(\kappa^{2})^{2}}}{1-\delta^{2}}\right\\}\Bigg{)}$

whenever

(3.9) $\delta\text{\rm Re}(\kappa^{2})>(1-\delta^{2})^{1/2}|\text{\rm
Im}(\kappa^{2})|$

Then, by

$\|M_{\kappa}(0)^{-1}\|=\|N_{0}^{-1}(N_{0}^{-1}-\kappa^{2})^{-1}\|=\sup_{\lambda\in\sigma(N_{0}^{-1})}\,\frac{\lambda}{|\lambda-\kappa^{2}|}\,,$

one gets

$\displaystyle\|M_{\kappa}(0)^{-1}\|=\sup_{\lambda\in\sigma(N_{0}^{-1})\cap\Lambda_{\delta}(\kappa^{2})}\,\frac{\lambda}{|\lambda-\kappa^{2}|}\leq\frac{1}{\delta}\,,$

whenever (3.8) holds and

$\displaystyle\|M_{\kappa}(0)^{-1}\|=$
$\displaystyle\max\left\\{\sup_{\lambda\in\sigma(N_{0}^{-1})\cap\Lambda_{\delta}(\kappa^{2})}\,\frac{\lambda}{|\lambda-\kappa^{2}|}\
,\sup_{\lambda\in\sigma(N_{0}^{-1})\backslash(\Lambda_{\delta}(\kappa^{2})\cap\sigma(N_{0}^{-1}))}\,\frac{\lambda}{|\lambda-\kappa^{2}|}\right\\}$
$\displaystyle\leq$ $\displaystyle\max\left\\{\frac{1}{\delta}\
,\frac{\sup\\{\sigma(N_{0}^{-1})\backslash(\Lambda_{\delta}(\kappa^{2})\cap\sigma(N_{0}^{-1}))\\}}{\text{\rm
dist}(\kappa^{2},\sigma(N_{0}^{-1}))}\right\\}$

whenever (3.9) holds. In the latter case, since

$\\{\lambda\in\sigma(N_{0}^{-1}):\delta\lambda\leq\text{\rm
dist}(\kappa^{2},\sigma(N_{0}^{-1}))\\}\subseteq\Lambda_{\delta}(\kappa^{2})\cap\sigma(N_{0}^{-1})\,,$

one has

$\\{\sigma(N_{0}^{-1})\backslash(\Lambda_{\delta}(\kappa^{2})\cap\sigma(N_{0}^{-1}))\subseteq\\{\lambda\in\sigma(N_{0}^{-1}):\delta\lambda>\text{\rm
dist}(\kappa^{2},\sigma(N_{0}^{-1}))\\}$

and so,

$\displaystyle\|M_{\kappa}(0)^{-1}\|\leq\frac{\sup\\{\sigma(N_{0}^{-1})\backslash(\Lambda_{\delta}(\kappa^{2})\cap\sigma(N_{0}^{-1}))\\}}{\text{\rm
dist}(\kappa^{2},\sigma(N_{0}^{-1}))}$ $\displaystyle\leq$
$\displaystyle\,\frac{1}{\text{\rm
dist}(\kappa^{2},\sigma(N_{0}^{-1}))}\,\,\max\left\\{\frac{1}{\lambda_{1}}\
,\frac{\text{\rm Re}(\kappa^{2})+\sqrt{\delta^{2}\text{\rm
Re}(\kappa^{2})^{2}-(1-\delta^{2})\text{\rm
Im}(\kappa^{2})^{2}}}{1-\delta^{2}}\right\\}$ $\displaystyle\leq$
$\displaystyle\,\frac{1}{\text{\rm
dist}(\kappa^{2},\sigma(N_{0}^{-1}))}\,\,\max\left\\{\frac{1}{\lambda_{1}}\
,\frac{\text{\rm Re}(\kappa^{2})}{1-\delta}\right\\}\,.$

By taking $\delta=1/{\sqrt{2}}$, one gets

$\|M_{\kappa}(0)^{-1}\|\leq\begin{cases}\sqrt{2}&\text{\rm
Re}(\kappa^{2})\leq|\text{\rm Im}(\kappa^{2})|\\\ \text{\rm
dist}(\kappa^{2},\sigma(N_{0}^{-1}))^{-1}\max\left\\{\lambda_{1}^{-1},{\text{\rm
Re}(\kappa^{2})}\right\\}&\text{\rm Re}(\kappa^{2})>|\text{\rm
Im}(\kappa^{2})|\end{cases}\quad.$

The proof is concluded by noticing that, whenever $\text{\rm
Re}(\kappa^{2})\leq|\text{\rm Im}(\kappa^{2})|$,

$\displaystyle\text{\rm
dist}(\kappa^{2},\sigma(N_{0}^{-1}))^{-1}\max\left\\{\lambda_{1}^{-1},{\text{\rm
Re}(\kappa^{2})}\right\\}\ \leq$ $\displaystyle\begin{cases}1&\text{\rm
Re}(\kappa^{2})<0\\\ 2^{-1/2}&0\leq\text{\rm
Re}(\kappa^{2})\leq\lambda_{1}^{-1},\ \text{\rm Re}(\kappa^{2})\leq|\text{\rm
Im}(\kappa^{2})|\\\ 1&\lambda_{1}^{-1}<\text{\rm Re}(\kappa^{2})\leq|\text{\rm
Im}(\kappa^{2})|\end{cases}$ $\displaystyle\leq$ $\displaystyle 1\,.$

∎

Given $r>\frac{1}{\lambda_{1}}$, let us define

(3.10)
$r_{\\!\circ}:=\frac{1}{2}\,\min_{\lambda\in\sigma(N_{0})\cap[1/r,+\infty)}\text{\rm
dist}(\lambda^{-1},\sigma(N_{0}^{-1})\backslash\\{\lambda^{-1}\\})\,,\qquad
r_{\\!+}:=r+r_{\\!\circ}\,.$

Then, for any $r_{\\!*}\in(0,r_{\circ}\,]$,

(3.11) $\\{z\in{\mathbb{C}}:\text{\rm
dist}(z,\sigma(N_{0}^{-1}))<r_{\\!*}\,\\}\cap
D_{r_{\\!+}}=\bigcup_{\lambda\in\sigma(N_{0})\cap[1/r,+\infty)}D_{r_{\\!*}}(\lambda^{-1})$

and the discs on the right are disjoint.

######  Theorem 3.4.

For any $r>\frac{1}{\lambda_{1}}$, let $r_{\\!\circ}$ and $r_{\\!+}$ be
defined as in (3.10). Then, for any $r_{\\!*}\in(0,r_{\\!\circ}\,]$ and for
any $\varepsilon\in(0,1)$ such that

(3.12)
$\varepsilon\,\sqrt{2}\,\left({\frac{|\Omega|}{4\pi}}^{\frac{1}{2}}\,\sqrt{r}_{\\!\\!+}\,e^{\,\varepsilon\sqrt{r}_{\\!\\!+}d_{\Omega}}+\varepsilon\lambda_{1}\right)\frac{r_{\\!+}^{2}}{r_{\\!*}}<1\,,$

one has

(3.13) $\mathsf{e}(H(\varepsilon))\cap D_{r_{\\!+}}\
\subset\bigcup_{\lambda\in\sigma(N_{0})\cap[1/r,+\infty)}D_{r_{\\!*}}(\lambda^{-1})\,.$

###### Proof.

By (3.11), it is enough to show that if
$\kappa\in{\mathbb{C}}\backslash{\mathbb{C}}_{+}$ is such that
$|\kappa|^{2}<r_{\\!+}$ and $\text{\rm
dist}(\kappa^{2},\sigma(N_{0}^{-1}))\geq r_{\\!*}$ then
$M_{\kappa}(\varepsilon)$ has a bounded inverse. By

$M_{\kappa}(\varepsilon)=M_{\kappa}(0)\big{(}1+M_{\kappa}(0)^{-1}(M_{\kappa}(\varepsilon)-M_{\kappa}(0))\big{)}\,,$

$M_{\kappa}(\varepsilon)$ has a bounded inverse whenever

(3.14) $\|M_{\kappa}(0)^{-1}(M_{\kappa}(\varepsilon)-M_{\kappa}(0))\|<1\,.$

The identity

$N_{\varepsilon\kappa}-N_{0}=\kappa\int_{0}^{\varepsilon}N_{s\kappa}^{(1)}\,ds\,,\qquad
N_{\kappa}^{(1)}u(x):=\frac{i}{4\pi}\int_{\Omega}e^{i\kappa|x-y|}\,u(y)\,dy\,,$

yields

$\|N_{\varepsilon\kappa}-N_{0}\|\leq|\kappa|\int_{0}^{\varepsilon}\|N_{s\kappa}^{(1)}\|\,ds\leq\frac{1}{4\pi}\,|\Omega|^{1/2}\,|\kappa|\int_{0}^{\varepsilon}{e^{s\,|\text{\rm
Im}(\kappa)|\,d_{\Omega}}}ds\leq{\frac{|\Omega|}{4\pi}}^{\frac{1}{2}}\,\varepsilon\,|\kappa|\,e^{\varepsilon\,|\text{\rm
Im}(\kappa)|\,d_{\Omega}}\,.$

Hence, by Lemma 3.3,

$\displaystyle\|M_{\kappa}(0)^{-1}(M_{\kappa}(\varepsilon)-M_{\kappa}(0))\|\leq\|M_{\kappa}(0)^{-1}\|\,\|(M_{\kappa}(\varepsilon)-M_{\kappa}(0))\|$
$\displaystyle\leq$
$\displaystyle|\kappa|^{2}\|M_{\kappa}(0)^{-1}\|\,\|(1-\varepsilon^{2})N_{\varepsilon\kappa}-N_{0}\|\leq|\kappa|^{2}\|M_{\kappa}(0)^{-1}\|\,\big{(}\|N_{\varepsilon\kappa}-N_{0}\|+\varepsilon^{2}\|N_{0}\|\big{)}$
$\displaystyle\leq$
$\displaystyle\varepsilon\,\sqrt{2}\,|\kappa|^{2}\left({\frac{|\Omega|}{4\pi}}^{\frac{1}{2}}\,|\kappa|\,e^{\,\varepsilon\,|\text{\rm
Im}(\kappa)|\,d_{\Omega}}+\varepsilon\lambda_{1}\right)\frac{\max\\{\lambda_{1}^{-1},{\text{\rm
Re}(\kappa^{2})}\\}}{\text{\rm dist}(\kappa^{2},\sigma(N_{0}^{-1}))}$
$\displaystyle\leq$
$\displaystyle\varepsilon\,\sqrt{2}\,\left({\frac{|\Omega|}{4\pi}}^{\frac{1}{2}}\,\sqrt{r}_{\\!\\!+}\,e^{\,\varepsilon\sqrt{r}_{\\!\\!+}d_{\Omega}}+\varepsilon\lambda_{1}\right)\frac{r_{\\!+}^{2}}{r_{\\!*}}\,.$

Then, (3.12) gives (3.14). ∎

######  Corollary 3.5.

Given $r>\frac{1}{\lambda_{1}}$, let $r_{\\!\circ}$ and $r_{\\!+}$ be as in
(3.10) and define

$c_{r}:=\sqrt{2}\,\left({\frac{|\Omega|}{4\pi}}^{\frac{1}{2}}\,\sqrt{r}_{\\!\\!+}\,e^{\,\sqrt{r}_{\\!\\!+}d_{\Omega}}+\lambda_{1}\right)r_{\\!+}^{2}\,.$

Then, for any $\varepsilon\in(0,1)$ such that $\varepsilon\leq
r_{\\!\circ}/c_{r}$, one has

(3.15) $\mathsf{e}(H(\varepsilon))\cap D_{r_{\\!+}}\
\subset\bigcup_{\lambda\in\sigma(N_{0})\cap[1/r,+\infty)}D_{\varepsilon_{r}}(\lambda^{-1})\,,\qquad\varepsilon_{r}:=c_{r}\,\varepsilon\,.$

Hence, for any $\varepsilon\leq r_{\\!\circ}/c_{r}$ and for any
$\kappa^{2}_{\circ}(\varepsilon)\in H(\varepsilon)\cap D_{r_{\\!+}}$, there
exists an unique $\lambda_{\circ}\in\sigma_{disc}(N_{0})$ such that

(3.16) $|\kappa^{2}_{\circ}(\varepsilon)-\lambda_{\circ}^{-1}|\leq
c_{r}\,\varepsilon\,.$

###### Proof.

By

$\varepsilon\,\sqrt{2}\,\left({\frac{|\Omega|}{4\pi}}^{\frac{1}{2}}\,\sqrt{r}_{\\!\\!+}\,e^{\,\varepsilon\sqrt{r}_{\\!\\!+}d_{\Omega}}+\varepsilon\lambda_{1}\right)\frac{r_{\\!+}^{2}}{c_{r}\varepsilon}<\sqrt{2}\,\left({\frac{|\Omega|}{4\pi}}^{\frac{1}{2}}\,\sqrt{r}_{\\!\\!+}\,e^{\,\sqrt{r}_{\\!\\!+}d_{\Omega}}+\lambda_{1}\right)\frac{r_{\\!+}^{2}}{c_{r}}=1\,,$

the thesis is consequence of Theorem 3.4 with $r_{\\!*}=c_{r}\varepsilon$. ∎

We are now in the position to prove our main result and provide the
$\varepsilon$-expansion for $\kappa_{\circ}^{2}(\varepsilon)$.

###  3.1. Proof of Theorem 1.1

Let $\lambda_{\circ}\in\sigma_{disc}(N_{0})$ with
$\dim(\ker(\lambda_{\circ}-N_{0}))=m$. By [6, Theorem 1, Section 10.4.3],
there are $m^{\prime}\leq m$ multivalued functions $z\mapsto\zeta_{j}(z)$ (not
necessarily distinct), such that

$\zeta_{j}(0)=\lambda_{\circ}\,,\qquad\ker(N_{z}-\zeta_{j}(z))\not=\\{0\\}\,.$

Such functions have, in a neighborhood of $0\in{\mathbb{C}}$, convergent
Puiseux series expansions

(3.17)
$\zeta_{j}(z)=\lambda_{\circ}+\sum_{n=1}^{+\infty}\zeta_{j}^{(n)}z^{n/p_{j}}\,,\qquad\sum_{j=1}^{m^{\prime}}p_{j}=m\,.$

Since $z\mapsto N_{iz}$ is self-adjoint on the real axis, by [13, Theorem 1,
Section 2, Chapter II] , $z\mapsto\zeta_{j}(iz)$ is holomorphic in a
neighborhood of $0\in{\mathbb{C}}$ with a convergent power series expansion

(3.18)
$\zeta_{j}(iz)=\lambda_{\circ}+\sum_{n=1}^{+\infty}\widetilde{\zeta}_{j}^{\,(n)}(iz)^{n}\,.$

By (3.17) and (3.18), one has, whenever $p_{j}>1$,

$\zeta_{j}^{(1)}=\lim_{z\rightarrow
0}\,(\zeta_{j}(iz)-\lambda_{\circ})(iz)^{-1/p_{j}}=\lim_{z\rightarrow
0}\,(\widetilde{\zeta}_{j}^{\,(1)}(iz)^{1-1/p_{j}}+o(z^{1-1/p_{j}}))=0\,.$

Iterating this argument, one gets $\zeta_{j}^{(n)}=0$ for any $n\geq 1$.
Hence, $m^{\prime}=m$, any $z\mapsto\zeta_{j}(z)$ is single-valued,
holomorphic in a neighborhood of $0\in{\mathbb{C}}$ and (3.17) holds with
$p_{j}=1$.

Let $\kappa_{\circ,j}^{2}(\varepsilon)$ denote a resonance close to
$\lambda_{\circ}^{-1}$ according to the previous Corollary. Since
$\ker(M_{\kappa_{\circ}(\varepsilon)}(\varepsilon))\not=\\{0\\}$,
equivalently,

$\ker(N_{\varepsilon\kappa_{\circ,j}(\varepsilon)}-((1-\varepsilon^{2})\kappa_{\circ,j}^{2}(\varepsilon))^{-1})\not=\\{0\\}\,,$

one gets

$\zeta_{j}(\varepsilon\kappa_{\circ}(\varepsilon))=((1-\varepsilon^{2})\kappa_{\circ,j}^{2}(\varepsilon))^{-1}\,,$

i.e., $\kappa_{\circ,j}(\varepsilon)$ solves the equation

(3.19) $f_{j}(\varepsilon,\kappa_{\circ,j}(\varepsilon))=0\,,$

where

$f_{j}(\varepsilon,z):=1-(1-\varepsilon^{2})z^{2}\zeta_{j}(\varepsilon z)\,.$

Since

$f_{j}(0,\lambda_{\circ}^{-1/2})=1-\lambda_{\circ}^{-1}\zeta(0)=0\,,\qquad\frac{\partial
f_{j}}{\partial
z}(0,\lambda_{\circ}^{-1})=-2\lambda_{\circ}^{-1/2}\zeta(0)=-2\lambda_{\circ}^{1/2}\not=0\,,$

by the implicit function theorem for analytic functions,
$\varepsilon\mapsto\kappa_{\circ,j}(\varepsilon)$ is analytic in a
neighborhood of zero and, writing

$\kappa_{\circ,j}(\varepsilon)=\lambda_{\circ}^{-1/2}+\sum_{n=1}^{+\infty}\kappa_{\circ,j}^{(n)}\,\varepsilon^{n}\,,$

by

$\frac{\partial
f_{j}}{\partial\varepsilon}(0,\lambda_{\circ}^{-1/2})+\frac{\partial
f_{j}}{\partial z}(0,\lambda_{\circ}^{-1/2})\,\kappa_{\circ,j}^{(1)}=0\,,$

one gets

(3.20)
$\kappa_{\circ,j}^{(1)}=-\frac{\zeta_{j}^{(1)}}{2\lambda_{\circ}^{2}}\,.$

Let us now determine $\zeta_{j}^{(1)}$. Denoting by $e_{j}^{\circ}$, $1\leq
j\leq m$, the orthonormal set of eigenvectors corresponding to
$\lambda_{\circ}$, the eigenvector equation

$N_{z}e(z)=\zeta_{j}(z)e(z)\,,\qquad e(0)=e_{j}^{\circ}$

has, by [6, Theorem 2, Section 10.4.3], $m^{\prime}\leq m$ solutions $z\mapsto
e_{j,j^{\prime}}(z)$, $1\leq j^{\prime}\leq m_{j}$,
$\sum_{j=1}^{m^{\prime}}m_{j}=m$, which are continuous at $0$ and have, in a
neighborhood of $0\in{\mathbb{C}}$, convergent Puiseux series expansions

(3.21)
$e_{j,j^{\prime}}(z)=e_{j}^{\circ}+\sum_{n=1}^{+\infty}e_{j,j^{\prime}}^{(n)}\,z^{n/p_{j}^{\prime}}\,.$

Now, we proceed as above. Since $z\mapsto N_{iz}$ is self-adjoint on the real
axis, by [13, Theorem 1, Section 2, Chapter II] , $z\mapsto
e_{j,j^{\prime}}(iz)$ is holomorphic in a neighborhood of $0\in{\mathbb{C}}$
with a convergent power series expansion

(3.22)
$e_{j,j^{\prime}}(iz)=e_{j}^{\circ}+\sum_{n=1}^{+\infty}\widetilde{e}_{j,j^{\prime}}^{\,(n)}\,(iz)^{n}\,.$

By (3.21) and (3.22), one has, whenever $p_{j}^{\prime}>1$,

$e_{j,j^{\prime}}^{(1)}=\lim_{z\rightarrow
0}\,(e_{j,j^{\prime}}(iz)-e_{j}^{\circ})(iz)^{-1/p_{j}^{\prime}}=\lim_{z\rightarrow
0}\,(\widetilde{e}_{j,j^{\prime}}^{\,(1)}\,(iz)^{1-1/p_{j}^{\prime}}+o(z^{1-1/p_{j}^{\prime}}))=0\,.$

Iterating this argument, one gets $e_{j,j^{\prime}}^{(n)}=0$ for any $n\geq
1$. Hence, any $z\mapsto e_{j,j^{\prime}}(z)$ is holomorphic in a neighborhood
of $0\in{\mathbb{C}}$ and (3.21) holds with $p_{j}^{\prime}=1$.

Then, differentiating at $z=0$ the relation

$\langle e^{\circ}_{j},N_{z}e_{j,j^{\prime}}(z)\rangle=\zeta_{j}(z)\langle
e^{\circ}_{j},e_{j,j^{\prime}}(z)\rangle$

one gets

$\langle e^{\circ}_{j},N_{0}^{(1)}e^{\circ}_{j}\rangle+\langle
e^{\circ}_{j},N_{0}e_{j,j^{\prime}}^{(1)}\rangle=\zeta_{j}^{(1)}+\lambda_{\circ}\langle
e^{\circ}_{j},e_{j,j^{\prime}}^{(1)}\rangle\,,$

where

$N_{0}^{(1)}u(x)=\frac{i}{4\pi}\int_{\Omega}u(x)\,dx=\frac{i}{4\pi}\,\langle
1,u\rangle\,.$

Hence,

$\zeta_{j}^{(1)}=\langle
e^{\circ}_{j},N_{0}^{(1)}e^{\circ}_{j}\rangle=\frac{i}{4\pi}\,|\langle
1,e^{\circ}_{j}\rangle|^{2}\,.$

Acknowledgements. The authors gratefully acknowledge the support of the Next
Generation EU-Prin project 2022 ”Singular Interactions and Effective Models in
Mathematical Physics”.

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