# Membranes with thin and heavy inclusions: asymptotics of spectra

Yuriy Golovaty  Dept. of Mechanics and Mathematics, Ivan Franko National
University of Lviv, Universytetska str.1, Lviv, 79000, Ukraine

###### Abstract.

We study the asymptotic behaviour of eigenvalues of 2D vibrating systems with
mass density perturbed in a vicinity of closed curves. The threshold case in
which the resonance frequencies of the membrane and the frequencies of thin
inclusion coincide is investigated. The perturbed eigenvalue problem can be
realized as a family of self-adjoint operators acting on varying Hilbert
spaces. However the so-called limit operator is non-self-adjoint and possesses
the Jordan chains of length $2$. Apart from the lack of self-adjointness, the
operator has non-compact resolvent. As a consequence, its spectrum has a
complicated structure, for instance, the spectrum contains a countable set of
eigenvalues with infinite multiplicity. The complete asymptotic analysis of
eigenvalues has been carried out.

###### Key words and phrases:

Asymptotics of eigenvalues, eigenvalue of infinite multiplicity, quasimode,
non-self-adjoint operator, concentrated mass, singular perturbation

###### 1991 Mathematics Subject Classification:

35B25, 35P05, 74K15

##  1\. Introduction and Statement of Problem

The mechanical systems with strongly inhomogeneous mass distributions have
become the subject of intensive experimental and theoretical studies since the
time of Poisson and Bessel [1, Ch.2], and a lot of research has been devoted
to the analysis of vibrating systems with so-called added masses.
Historically, the ﬁrst relevant mathematical models in classical mechanics go
back to the first half of the 20th century (see e.g. [2] and the references
given there). Many authors have investigated properties of strings and rods
with the mass densities perturbed by finite or infinite sums $\sum
M_{k}\delta(x-x_{k})$, where $\delta$ is Dirac’s delta-function and $M_{k}$ is
an added mass at the point $x_{k}$. Recently, such models in dimensions two
and three with heavy inclusions of a different geometry are widely used not
only in mechanics, but in various fields of science and technology such as
physics of liquid crystals, physical chemistry of polymers, micelles and
microemulsions, molecular theory, cell membrane theory [3, 4, 5]. For
instance, cell membranes are known to contain embedded proteins and various
colloidal particles [6].

In higher dimensions, the perturbation of mass densities by the
$\delta$-functions often leads to incorrect mathematical models, because the
formal differential equations which appear have no mathematical meaning. As an
example of such ill-posed problem we can consider the eigenvalue problem for
the Laplace operator

$-\Delta u=\lambda(1+M\delta(x))u\text{ \ in }\Omega,\qquad u=0\text{ \ on
}\partial\Omega,$

where $\Omega$ is a bounded domain in $\mathbb{R}^{3}$ containing the origin.
The equation has no non-trivial solution, because any such solution $u$ has a
singularity at $x=0$ and therefore the product $\delta(x)u(x)=u(0)\delta(x)$
is not defined. The new and at the same time obvious idea was instead to
replace the $\delta$-function with its regularization
$\varepsilon^{-3}q(x/\varepsilon)$, where $q$ is a function of compact
support, and study the asymptotic behaviour of eigenvalues and eigenfunctions
as $\varepsilon\to 0$. The problem was first investigated by E. Sánchez-
Palencia [7, 8], who proved the existence of the so-called local
eigenvibrations: the eigenfunctions are significant in a small neighborhood of
the origin only.

The model in which the density is perturbed by $\sum M_{k}\delta(x-x_{k})$ is
not adequate even in the one-dimensional case, when dealing with the large
masses $M_{k}$. The very heavy inclusions cause a strong local reaction of
vibrating system, but this phenomenon can not be described on the discrete set
which is a support of the sum of Dirac’s functions. The geometry of small
domains where the large masses are loaded should also have an effect on the
form of eigenvibrations. In [9], asymptotic analysis was applied to a spectral
problem for the Sturm-Liouville operator with weight function of the form
$\rho_{\varepsilon}(x)=\rho(x)+\varepsilon^{-m}q(x/\varepsilon)$, where $q$ is
a function of compact support and $m\in\mathbb{R}$. For the case $m=1$ the
perturbation is a $\delta$-like sequence, but the most interesting cases of
the limit behaviour of eigenfunctions as $\varepsilon\to 0$ are those when the
power $m$ is greater than $1$.

These advanced models have attracted considerable attention in the
mathematical literature over the last three decades (see e.g. [10] for a
review). The spectral properties of differential operators with weight
functions having the form

$\rho_{\varepsilon}(x)=\rho(x)+\sum\varepsilon^{-m_{k}}q_{k,\varepsilon}(x),$

where $q_{k,\varepsilon}$ are compactly supported in vicinity of different
sets, have been investigated in numerous articles. We mention here [11] for
the Laplace operator in dimension $3$, [12, 13] for ordinary differential
operators of the fourth order and the biharmonic operator, [14, 15, 16] for
boundary value problems on junctions of a very complicated geometry, [17, 18]
for the Sturm-Liouville operators on metric graphs. The spectral properties of
strings with rapidly oscillating and periodic densities have been treated in
[19, 20, 21] in the framework of homogenization theory. Another model in which
the heavy inclusions were regarded as rigid ones has been studied in [22].
Since the $90$s of the last century, a series of papers was published
concerning 2D and 3D elastic systems with many concentrated masses near the
boundary [23, 24, 25, 26, 27, 28, 29, 30]. New asymptotic results for the
spectral problems in domains surrounded by thin stiff and heavy bands, when
the mass density and stiffness are simultaneously perturbed in a neighbourhood
of the boundary, were obtained in [31, 32, 33, 34]. The Neumann eigenvalue
problem for a membrane, almost the entire mass of which is concentrated around
the boundary, was studied in [35].

Figure 1. Membrane with heavy and thin inclusion

In this paper we study the eigenvibration characteristics of a membrane with
heavy and thin inclusion inside. Let $\Omega$ be a smooth bounded domain of
$\mathbb{R}^{2}$, and let $\gamma\subset\Omega$ be a smooth closed curve. We
will denote by $\omega_{\varepsilon}$ the $\varepsilon$-neighborhood of
$\gamma$ (i.e., the union of all open balls of radius $\varepsilon$ around a
point on $\gamma$). Consider $\varepsilon>0$ sufficiently small such that
$\omega_{\varepsilon}\subset\Omega$ and the boundary
$\partial\omega_{\varepsilon}$ is smooth. Assume $\rho$ is a smooth uniformly
positive functions in $\Omega$, and

$\rho_{\varepsilon}(x)=\begin{cases}\rho(x),&\text{if
}x\in\Omega\setminus\omega_{\varepsilon},\\\
\varepsilon^{-2}q_{\varepsilon}(x),&\text{if
}x\in\omega_{\varepsilon}.\end{cases}$ (1.1)

To specify explicit dependence of $q_{\varepsilon}$ on $\varepsilon$ we
introduce the Fermi normal coordinates in $\omega_{\varepsilon}$ (see Fig. 1).
Let $\alpha\colon[0,|\gamma|)\to\mathbb{R}^{2}$ be the unit-speed smooth
parametrization of $\gamma$ with the natural parameter $s$, and $|\gamma|$ is
the length of $\gamma$. Then the vector
$\nu=(-\dot{\alpha}_{2},\dot{\alpha}_{1})$ is a unit normal on $\gamma$. Set
$x=\alpha(s)+r\nu(s)$ for
$(s,r)\in[0,|\gamma|)\times(-\varepsilon,\varepsilon)$, where $r$ is the
signed distance from $x$ to $\gamma$. Suppose that

$q_{\varepsilon}(x)=q({\textstyle\frac{r}{\varepsilon}}),$

where $q$ is a smooth positive function in $[-1,1]$. We see that the
perturbation of mass density varies on the normal direction to $\gamma$ only.
Let us consider the spectral problem

$-\Delta
u_{\varepsilon}+au_{\varepsilon}=\lambda^{\varepsilon}\rho_{\varepsilon}u_{\varepsilon}\quad\text{in
}\Omega,\qquad\ell u_{\varepsilon}=0\quad\text{on }\partial\Omega,$ (1.2)

where $a\in C^{\infty}(\overline{\Omega})$ and $\ell v=0$ denotes the
Dirichlet, Neumann or Robin boundary conditions on $\partial\Omega$. Our goal
is to describe the asymptotic behaviour of the eigenvalues
$\lambda^{\varepsilon}$ of (1.2) as $\varepsilon\to 0$.

This paper is a continuation of [36, 37], where the problem with the Dirichlet
boundary condition and more general perturbation
$\varepsilon^{-m}q_{\varepsilon}$ of mass density has been treated using the
variational approach and the case $m=3$ has been completely investigated by
asymptotic methods. Also, it has been shown that there exist five different
limit behaviours for the spectrum and eigenspaces depending on $m$: $m<1$,
$m=1$, $1<m<2$, $m=2$ and $m>2$. These cases differ by the form of
eigenvibrations and place on the membrane where the main part of their
“energy” is concentrated in the limit. From the mathematical viewpoint, the
difference between the cases is that the spectral parameter in the limit
eigenvalue problems appears alternately both in a differential equation on
$\Omega$ and in an equation on the strip $\gamma\times(-1,1)$, which is a
dilatation of $\omega_{\varepsilon}$, and even in coupling conditions on
$\gamma$.

The threshold case $m=2$ is the most difficult and interesting one to analyse,
because the spectral parameter is included simultaneously in two differential
equations, and the limit spectral problem is associated with a non-self-
adjoint operator. The main insight of the present work is to exhibit this non-
self-adjoint operator, its spectrum and generalized eigenspaces.

To better understand what happens in dimension $2$, we briefly discuss the
similar model for a string. Recently, we have revised some results from [9]
concerning the case $m=2$. In [38], we have studied the limiting behavior, as
$\varepsilon\to 0$, of eigenvalues and eigenfunctions of the problem

$\displaystyle-
y_{\varepsilon}^{\prime\prime}+Qy_{\varepsilon}=\lambda^{\varepsilon}\rho_{\varepsilon}y_{\varepsilon},\quad
x\in(a,b),$ (1.3) $\displaystyle
y_{\varepsilon}(a)\cos\alpha+y_{\varepsilon}^{\prime}(a)\sin\alpha=0,\quad
y_{\varepsilon}(b)\cos\beta+y_{\varepsilon}^{\prime}(b)\sin\beta=0$ (1.4)

with the weight function $\rho_{\varepsilon}$ given by (1.1), where
$\Omega=(a,b)$ is an interval containing the origin, and
$\omega_{\varepsilon}=(-\varepsilon,\varepsilon)$. For each real $\alpha$ and
$\beta$ the problem can be associated with a family of self-adjoint operators
$S_{\varepsilon}$ in the weighted space $L_{2}(\rho_{\varepsilon},\Omega)$.
The operators $S_{\varepsilon}$ are defined by
$S_{\varepsilon}\phi=\rho_{\varepsilon}^{-1}(-\phi^{\prime\prime}+Q\phi)$ on
functions $\phi\in W_{2}^{2}(\Omega)$ obeying boundary conditions (1.4). The
spectra of $S_{\varepsilon}$ are real, discrete and simple. It is worth noting
that the study of operator families acting on varying spaces entails some
mathematical difficulties. First of all, the question arises how to understand
the convergence of such families. Next, if these operators do converge in some
sense, does this convergence implies the convergence of their spectra (see,
e.g., [19, III.1], [39, 40]). By abandoning the self-adjointness, we have
realized (1.3), (1.4) as a family of non-self-adjoint matrix operators
$\mathcal{A}_{\varepsilon}$ acting on the fixed Hilbert space
$L_{2}(\rho,(a,b))\times L_{2}(q,(-1,1))$ (see details in [38]). The operators
are certainly similar to self-adjoint ones and their resolvents are compact.
Moreover the spectra of $\mathcal{A}_{\varepsilon}$ and $S_{\varepsilon}$
coincide and the corresponding eigenspaces are isomorphic. It has been proved
that $\mathcal{A}_{\varepsilon}$ converge in the norm resolvent sense (the
resolvents of $\mathcal{A}_{\varepsilon}$ converge in the uniform norm as
$\varepsilon\to 0$) to the matrix operator $\mathcal{A}$ associated with the
eigenvalue problem

$\displaystyle-v^{\prime\prime}+Qv=\lambda\rho v\;\;\text{in
}(a,0)\cup(0,b),\quad-w^{\prime\prime}=\lambda qw\;\;\text{in }(-1,1),$
$\displaystyle\quad w^{\prime}(-1)=0,\quad w^{\prime}(1)=0,\quad
v(-0)=w(-1),\quad v(+0)=w(1),$ $\displaystyle\quad
v(a)\cos\alpha+v^{\prime}(a)\sin\alpha=0,\quad
v(b)\cos\beta+v^{\prime}(b)\sin\beta=0.$

Surprisingly enough, $\mathcal{A}$ is not similar to a self-adjoint one,
because it possesses multiple eigenvalues with the Jordan chains of length
$2$. In view of [41, Ch.1],[42], the norm resolvent convergence
$\mathcal{A}_{\varepsilon}\to\mathcal{A}$ implies the “number-by-number”
convergence of the corresponding eigenvalues and some results on the
convergence of eigenspaces.

Problem (1.2) can also be associated with a family of matrix operators acting
on the same Hilbert space. However, in two dimensions there is no uniform
convergence of resolvents, because the limit operator $\mathcal{P}$ which can
be obtained in the strong resolvent topology has non-compact resolvent [43,
Theorem IV.2.26]. It is known that the strong resolvent convergence of
operators does not imply the convergence of their spectra. For this reason, we
study the convergence of eigenvalues of (1.2) without touching on the
convergence of corresponding operators. The operator $\mathcal{P}$ is non-
self-adjoint and possesses nontrivial Jordan chains. In Sec. 3, we describe
the spectrum of $\mathcal{P}$ and the generalized eigenspaces. The spectrum is
discrete and real; it consists of eigenvalues of three types: (a) eigenvalues
of finite multiplicity with generalized eigenspaces generated by eigenvectors
only; (b) infinite-fold eigenvalues with generalized eigenspaces generated by
eigenvectors only; (c) infinite-fold eigenvalues with generalized eigenspaces
containing both eigenvectors and generalized eigenvectors. In fact, this
spectrum is a union of spectra of two operators that correspond to two
different parts of the vibrating system, namely, the membrane clamped along
the curve $\gamma$ and the thin heavy inclusion. Using the method of
quasimodes we describe the asymptotic behaviour of eigenvalues of (1.2) as
perturbation of $\sigma(\mathcal{P})$. In Sec. 4 and 5, we prove the existence
of a countable number of eigenvalues with the asymptotics
$\lambda_{\varepsilon,k}=\lambda+\varepsilon\lambda_{1}^{(k)}+O(\varepsilon^{2})$
as $\varepsilon\to 0$, $k\in\mathbb{N}$, where $\lambda$ is an eigenvalue of
type (b) or (c). The set of correctors
$\\{\lambda_{1}^{(k)}\\}_{k=1}^{\infty}$ is a spectrum of some
pseudodifferential operator $N_{\lambda}$ acting in $L_{2}(\gamma)$. In
addition, if $\lambda$ is an eigenvalue of type (c), there also exists $2d$
eigenvalues with the half-integer power asymptotics
$\mu_{\varepsilon,j}^{\pm}=\lambda\pm\varepsilon^{1/2}\lambda_{1/2,j}+\varepsilon\lambda_{1,j}^{\pm}+O(\varepsilon^{3/2})$,
as $\varepsilon\to 0$, where $d$ is a number of Jordan chains of length $2$
corresponding to $\lambda$. In Sec. 6 we prove that there exist $K$
eigenvalues of (1.2) with the asymptotics
$\lambda_{\varepsilon,k}=\lambda+\varepsilon\lambda_{1}^{(k)}+O(\varepsilon^{2})$,
where $\lambda$ is an eigenvalue of type (a) with multiplicity $K$.

##  2\. Formal construction of the limit operator

It will be convenient to parameterize the curve $\gamma$ by points of a
circle. It will allow us not to indicate every time that functions on $\gamma$
are periodic on $s$. Let $S$ be the circle of the length $|\gamma|$. Then
$\omega_{\varepsilon}$ is diffeomorphic to the cylinder
$S\times(-\varepsilon,\varepsilon)$. We also set $\omega=S\times(-1,1)$. Here
and subsequently, writing “in $\omega$” after an equation, we mean that the
equation is considered in the rectangle $(0,|\gamma|)\times(-1,1)$ and the
corresponding solution is a $|\gamma|$-periodic function on $s$.

Let us denote by $\gamma_{t}$ the curve that is obtained from $\gamma$ by
flowing for “time” $t$ along the normal vector field, i.e.,
$\gamma_{t}=\\{x\in\mathbb{R}^{2}\colon\;x=\alpha(s)+t\nu(s),\;s\in S\\}$.
Then the boundary of $\omega_{\varepsilon}$ consists of two curves
$\gamma_{-\varepsilon}$ and $\gamma_{\varepsilon}$. The domain $\Omega$ is
divided by $\gamma$ into subdomains $\Omega_{-}$ and $\Omega_{+}$. Suppose
that $\partial\Omega_{-}=\partial\Omega\cup\gamma_{-}$ and
$\partial\Omega_{+}=\gamma_{+}$, where $\gamma_{-}$ and $\gamma_{+}$ are two
edges of the cut $\gamma$. In the sequel, the coordinate $r$ increases in the
direction from $\Omega^{-}$ to $\Omega^{+}$.

For any $\varepsilon>0$, problem (1.2) admits a self-adjoint operator
realization in the weighted Lebesgue space $L^{2}(\rho_{\varepsilon},\Omega)$.
We introduce operator $T_{\varepsilon}$ defined by

$T_{\varepsilon}\phi=\rho_{\varepsilon}^{-1}(-\Delta\phi+a\phi)$

on functions $\phi\in W_{2}^{2}(\Omega)$ obeying the boundary condition
$\ell\phi=0$ on $\partial\Omega$. Obviously, the spectrum of $T_{\varepsilon}$
is real and discrete.

We look for the approximation to the eigenvalue $\lambda^{\varepsilon}$ and
the corresponding eigenfunction $u_{\varepsilon}$ of (1.2) in the form

$\lambda^{\varepsilon}\sim\lambda+\cdots,\qquad
u_{\varepsilon}(x)\sim\begin{cases}v(x)+\cdots&\text{if
}x\in\Omega\setminus\omega_{\varepsilon},\\\
w(s,\textstyle\frac{r}{\varepsilon})+\cdots&\text{if
}x\in\omega_{\varepsilon}.\end{cases}$ (2.1)

Since $u_{\varepsilon}$ solves (1.2) and the domain $\omega_{\varepsilon}$
shrinks to $\gamma$ as $\varepsilon\to 0$, the function $v$ must be a solution
of the equation $-\Delta v+av=\lambda\rho v$ in $\Omega\setminus\gamma$ that
satisfies the boundary condition $\ell v=0$ on $\partial\Omega$. Of course,
$v$ must also satisfy appropriate transmission conditions on $\gamma$. To find
these conditions, we must examine more closely the equation in (1.2) in a
vicinity of $\gamma$.

Returning to the local coordinates $(s,r)$, we see that the vectors
$\alpha=(\dot{\alpha}_{1},\dot{\alpha}_{2})$,
$\nu=(-\dot{\alpha}_{2},\dot{\alpha}_{1})$ give the Frenet frame for $\gamma$.
The Jacobian of transformation

$x_{1}=\dot{\alpha}_{1}(s)-r\dot{\alpha}_{2}(s),\quad
x_{2}=\dot{\alpha}_{2}(s)+r\dot{\alpha}_{1}(s)$

has the form $J(s,r)=1-r\varkappa(s)$, where
$\varkappa=\det(\dot{\alpha},\ddot{\alpha})$ is the signed curvature of
$\gamma$. We see that $J$ is positive for sufficiently small $r$, because the
curvature $\varkappa$ is bounded on $\gamma$. In addition, the Laplace-
Beltrami operator becomes

$\Delta\phi=J^{-1}\left(\partial_{s}(J^{-1}\partial_{s}\phi)+\partial_{r}(J\partial_{r}\phi)\right).$

In the local coordinates $(s,n)$, where $n=r/\varepsilon$, the Laplacian can
be written as

$\Delta=(1-\varepsilon
n\varkappa)^{-1}\left(\varepsilon^{-2}\partial_{n}(1-\varepsilon
n\varkappa)\partial_{n}+\partial_{s}\big{(}(1-\varepsilon
n\varkappa)^{-1}\partial_{s}\big{)}\right).$

From this we readily deduce the representation

$\Delta=\varepsilon^{-2}\partial^{2}_{n}-\varepsilon^{-1}\varkappa\partial_{n}-n\varkappa^{2}\partial_{n}+\partial^{2}_{s}+\varepsilon(2n\varkappa\partial^{2}_{s}+n\varkappa^{\prime}\partial_{s}-n^{2}\varkappa^{3}\partial_{n})+\varepsilon^{2}P_{\varepsilon},$
(2.2)

where $P_{\varepsilon}$ is a PDE of the second order on $s$ and the first one
on $n$ whose coefficients are uniformly bounded in $\omega$ with respect to
$\varepsilon$. Then using this representation and (2.1) we obtain the equation
$-\partial^{2}_{n}w=\lambda qw$ in $\omega$. Next, the eigenfunction
$u_{\varepsilon}$ as an element of $\mathop{\rm dom}T_{\varepsilon}$ satisfies
the conditions

$[u_{\varepsilon}]_{-\varepsilon}=0,\quad[u_{\varepsilon}]_{\varepsilon}=0,\quad[\partial_{r}u_{\varepsilon}]_{-\varepsilon}=0,\quad[\partial_{r}u_{\varepsilon}]_{\varepsilon}=0,$
(2.3)

where $[\,\cdot\,]_{t}$ stands for the jump of a function across $\gamma_{t}$.
These conditions imply $v^{\pm}=w(\,\cdot\,,\pm 1)$ and
$\partial_{n}w(\,\cdot\,,\pm 1)=0$, where $v^{\pm}$ denote the one-side traces
of $v$ on $\gamma$, i.e., $v^{\pm}=v|_{\gamma_{\pm}}$. Combining the latter
equalities, we can formally deduce that the pair $(v,w)$ must be an
eigenvector of the spectral problem

$\displaystyle-$ $\displaystyle\Delta v+av=\lambda\rho v\quad\text{in
}\Omega\setminus\gamma,\qquad\ell v=0\quad\text{on }\partial\Omega,$ (2.4)
$\displaystyle-$ $\displaystyle\partial^{2}_{n}w=\lambda qw\>\text{ in
}\omega,\quad\partial_{n}w(s,-1)=0,\;\>\partial_{n}w(s,1)=0,$ (2.5)
$\displaystyle v^{-}=w(s,-1),\quad v^{+}=w(s,1),\quad s\in S$ (2.6)

with the spectral parameter $\lambda$; (2.4)-(2.6) will be regarded as the
limit problem.

We will show that the problem can be associated with a non-self-adjoint matrix
operator. Moreover this operator is not similar to a self-adjoint one, because
it possesses generalized eigenvectors. It means that the limit problem admits
no self-adjoint realization. It it worth mentioning that the similar non-self-
adjoint problem appeared in [31], where the spectral problem for a membrane
surrounded by a thin stiff band was studied. Actually perturbed spectral
problem (1.2) and the problem in [31] give us nontrivial examples of self-
adjoint operators acting on varying Hilbert spaces that converge in some sense
to a non-self-adjoint operator.

##  3\. Properties of the limit operator

We use the following notation. The spectrum and resolvent set of a linear
operator $T$ are denoted by $\sigma(T)$ and $\varrho(T)$, respectively. Let
$T^{*}$ denote the adjoint operator of $T$. For any $z\in\varrho(T)$,
$\mathrm{R}_{z}(T)=(T-zI)^{-1}$ is the resolvent operator. Here and
subsequently, $I$ is an identity operator.

###  3.1. Spectrum of the limit operator

We introduce the operators

$\displaystyle\mathring{A}=\rho^{-1}(-\Delta+a)\quad\text{in
}L_{2}(\rho,\Omega),\quad\mathop{\rm dom}\mathring{A}=\\{f\in
W_{2}^{2}(\Omega\setminus\gamma)\colon\ell f=0\;\;\text{on
}\partial\Omega\\},$ $\displaystyle B=-q^{-1}\partial^{2}_{n}\quad\text{in
}L_{2}(q,\omega),\quad\mathop{\rm dom}B=\left\\{g\in
W_{2}^{2,0}(\omega)\colon\partial_{n}g(\,\cdot\,,-1)=\partial_{n}g(\,\cdot\,,1)=0\right\\},$

where $W_{2}^{2,0}(\omega)$ is the anisotropic Sobolev space

$W_{2}^{2,0}(\omega)=\left\\{g\in L_{2}(\omega)\colon\partial^{k}_{n}g\in
L_{2}(\omega)\text{ for }k=1,2\right\\}.$

In the space $\mathcal{L}=L_{2}(\rho,\Omega)\times L_{2}(q,\omega)$ we
consider the matrix operator

$\mathcal{P}=\begin{pmatrix}\mathring{A}&0\\\
0&B\end{pmatrix},\quad\mathop{\rm
dom}\mathcal{P}=\bigl{\\{}(f,g)\in\mathop{\rm
dom}\mathring{A}\times\mathop{\rm dom}B\colon\\\
f^{-}=g(\,\cdot\,,-1),\>\;f^{+}=g(\,\cdot\,,1)\bigr{\\}}.$

Now (2.4)–(2.6) can be written as $\mathcal{P}u=\lambda u$ with the notation
$u=(v,w)^{\top}$. For simplicity of notation we often write $(v,w)$ instead of
$(v,w)^{\top}$. Note that the operator $\mathcal{P}$ is non-self-adjoint.
Direct computations show that

$\mathcal{P}^{*}=\begin{pmatrix}A&0\\\
0&\mathring{B}\end{pmatrix},\quad\mathop{\rm
dom}\mathcal{P}^{*}=\bigl{\\{}(f,g)\in\mathop{\rm dom}A\times
W_{2}^{2,0}(\omega)\colon\\\
\partial_{r}f^{-}=-\partial_{n}g(\,\cdot\,,-1),\>\;\partial_{r}f^{+}=\partial_{n}g(\,\cdot\,,1)\bigr{\\}},$

where $A$ is the restrictions of $\mathring{A}$ to $\mathop{\rm
dom}A=\\{f\in\mathop{\rm dom}\mathring{A}\colon f=0\;\text{ on }\gamma\\}$ and
$\mathring{B}$ is the extension of $B$ to the whole space
$W_{2}^{2,0}(\omega)$. Here $\partial_{r}f^{\pm}$ stand for the one-side
traces of the normal derivative of $f$ on $\gamma$.

######  Lemma 3.1.

The spectrum of $B$ consists of a countable set of real eigenvalues of
infinite multiplicity. Moreover, $\lambda$ belongs to $\sigma(B)$ if and only
if $\lambda$ is an eigenvalue of the Sturm-Liouville problem

$y^{\prime\prime}+\lambda q(n)y=0,\;\;n\in(-1,1),\quad
y^{\prime}(-1)=0,\;\;y^{\prime}(1)=0.$ (3.1)

###### Proof.

Obviously, the operator $B$ is self-adjoint. For given $\lambda\in\mathbb{C}$
and $g\in L_{2}(q,\omega)$, the equation $(B-\lambda I)\phi=g$ can be treated
as the boundary value problem for the ordinary differential equation

$-(\partial^{2}_{n}+\lambda q(n))\phi=q(n)g(s,n)\;\text{ in
}\omega,\quad\partial_{n}\phi(s,-1)=0,\;\>\partial_{n}\phi(s,1)=0$

with the parameter $s\in S$. If $\lambda$ is not an eigenvalue of (3.1), then
the problem has a unique solution $\phi(s,\,\cdot\,)$ for each
$g(s,\,\cdot\,)\in L_{2}(-1,1)$ and almost all $s\in S$. Moreover $\phi$
belongs to $L_{2}(q,\omega)$ in view of its integral representation via the
Green function. Otherwise, if $\lambda$ is an eigenvalue of (3.1) with
eigenfunction $y$, the problem is unsolvable for some right-hand sides $g$,
because the corresponding homogeneous problem has infinitely many linearly
independent solutions of the form $b(s)y(n)$, where $b\in L_{2}(\gamma)$.
Therefore the spectrum of $B$ consists of all eigenvalues of (3.1), and the
corresponding eigenspaces are infinite-dimensional. ∎

######  Theorem 3.2.

The operator $\mathcal{P}$ has real discrete spectrum. Moreover

$\sigma(\mathcal{P})=\sigma(A)\cup\sigma(B).$

###### Proof.

Let us construct the resolvent of $\mathcal{P}$ in an explicit form. Given
$\mu\in\mathbb{C}$, $f\in L_{2}(\rho,\Omega)$ and $g\in L_{2}(q,\omega)$, we
write $(\mathring{A}-\mu I)v=f$, $(B-\mu I)w=g$. The second equation admits a
unique solution $w=R_{\mu}(B)g$ if $\mu\in\varrho(B)$, and then $v$ is a
solution of the problem

$\displaystyle-$ $\displaystyle\Delta v+av-\mu\rho v=\rho f\quad\text{in
}\Omega\setminus\gamma,\quad\ell v=0\quad\text{on }\partial\Omega,$ (3.2)
$\displaystyle v^{-}=w(\,\cdot\,,-1),\quad v^{+}=w(\,\cdot\,,1).$

Suppose the operator $T(\mu)\colon W_{2}^{2}(\omega)\to L_{2}(\rho,\Omega)$
solves the problem

$\displaystyle-$ $\displaystyle\Delta\phi+a\phi-\mu\rho\phi=0\quad\text{in
}\Omega\setminus\gamma,\quad\ell\phi=0\text{ on }\partial\Omega,$ (3.3)
$\displaystyle\phi^{-}=\psi(\,\cdot\,,-1),\quad\phi^{+}=\psi(\,\cdot\,,1)$

for a given function $\psi\in W_{2}^{2}(\omega)$. If $\mu\in\varrho(A)$, then
the problem has a unique solution $\phi=T(\mu)\psi$ and hence $T(\mu)$ is
bounded. Next, $v$ can be represented as the sum of a solution of (3.2)
subject to the homogeneous boundary conditions on $\gamma_{\pm}$ and a
solution of (3.3) with $\psi=w$. Therefore

$v=R_{\mu}(A)f+T(\mu)w=R_{\mu}(A)f+T(\mu)R_{\mu}(B)g,$

provided $\mu\in\varrho(A)\cap\varrho(B)$. Then the resolvent of $\mathcal{P}$
can be written in the form

$R_{\mu}(\mathcal{P})=\begin{pmatrix}R_{\mu}(A)&T(\mu)R_{\mu}(B)\\\
0&R_{\mu}(B)\end{pmatrix}.$

The equality $\sigma(\mathcal{P})=\sigma(A)\cup\sigma(B)$ follows directly
from this representation and the fact that $T(\mu)$ is bounded for
$\mu\in\varrho(A)$. Obviously,
$\sigma(A)\cup\sigma(B)\subset\sigma(\mathcal{P})$. Suppose, contrary to our
claim, that $\sigma(\mathcal{P})$ is not contained in
$\sigma(A)\cup\sigma(B)$. Then there exists $\mu\in\sigma(\mathcal{P})$ such
that $\mu\in\rho(A)\cap\rho(B)$. Hence, the operators $R_{\mu}(A)$,
$R_{\mu}(B)$ and $T(\mu)$ are bounded, and therefore, the operator
$R_{\mu}(\mathcal{P})$ is also bounded, a contradiction. Clearly,
$\sigma(\mathcal{P})$ is real and discrete, because $A$ and $B$ are self-
adjoint operators with discrete spectra. ∎

######  Corollary 3.3.

The operator $\mathcal{P}$ has non-compact resolvent.

###### Proof.

The entry $R_{\mu}(B)$ of the matrix $R_{\mu}(\mathcal{P})$ is a non-compact
operator, which is clear from Lemma 3.1. The main reason for which
$R_{\mu}(B)$ is not compact is that the second derivative $\partial^{2}_{n}$
is a non-elliptic PDE in the two-dimensional domain $\omega$. ∎

######  Remark 3.4.

The operator $A$ can be represent as the direct sum $A_{-}\oplus A_{+}$, where

$\displaystyle A_{-}=\rho^{-1}(-\Delta+a),\quad\mathop{\rm dom}A_{-}=\\{f\in
W_{2}^{2}(\Omega_{-})\colon\ell f=0\;\text{ on }\partial\Omega,\;f=0\;\text{
on }\gamma\\},$ $\displaystyle A_{+}=\rho^{-1}(-\Delta+a),\quad\mathop{\rm
dom}A_{+}=\\{f\in W_{2}^{2}(\Omega_{+})\colon f=0\text{ on }\gamma\\}.$

Hence $\sigma(\mathcal{P})=\sigma(A_{-})\cup\sigma(A_{+})\cup\sigma(B)$.

###  3.2. Structure of generalized eigenspaces

Let $X_{\mu}$ be the generalized eigenspace corresponding to
$\mu\in\sigma(\mathcal{P})$, i.e., $X_{\mu}=\\{h\in\mathcal{L}\colon
h\in\ker(\mathcal{P}-\mu I)^{k}\text{ for some }k\in\mathbb{N}\\}$. If a
vector $h$ belongs to $\ker(\mathcal{P}-\mu I)^{j}$ and $(\mathcal{P}-\mu
I)^{j-1}h\neq 0$, one says that $h$ is a generalized eigenvector of rank $j$.
The eigenspace $X_{\mu}^{0}=\ker(\mathcal{P}-\mu I)$ is a subspace of
$X_{\mu}$; eigenvectors are precisely the generalized eigenvectors of rank
$1$.

####  3.2.1. Case $\lambda\in\sigma(A)\setminus\sigma(B)$.

We look first for non-trivial solutions $u=(v,w)$ of $\mathcal{P}u=\lambda u$.
Since $\lambda$ does not belong to $\sigma(B)$, we have $w=0$. This in turn
implies $v|_{\gamma}=0$, by (2.6). Therefore $v$ must be a eigenvector of $A$
corresponding to $\lambda$. All eigenvalues of $A$ have finite multiplicities.
Assume that $\lambda$ is an eigenvalue of multiplicity $K$ and
$V_{1}$,…,$V_{K}$ are the eigenfunctions of $A$ such that

$\int_{\Omega}\rho V_{i}V_{j}\,dx=\delta_{i,j},\qquad i,j=1,\ldots,K.$ (3.4)

Here $\delta_{i,j}$ is the Kronecker symbol. Then $X_{\lambda}^{0}$ is spanned
by

$u_{1}=(V_{1},0),\;\;u_{2}=(V_{2},0),\;\;\dots\;\;,u_{K}=(V_{K},0).$ (3.5)

The generalized eigenvectors of rank $2$ satisfy the equation
$(\mathcal{P}-\lambda I)u_{*}=u$, where $u$ is an eigenvector of
$\mathcal{P}$. There is no such vectors $u_{*}=(v_{*},w_{*})$ in this case,
because the second components of all the eigenvectors $u_{k}$ are zero. In
fact, $Bw_{*}=\lambda w_{*}$ implies $w_{*}=0$. Therefore $(A-\lambda
I)v_{*}=v$, where $v$ is a linear combination of $V_{j}$. The last equation is
unsolvable, since $A$ is self-adjoint.

####  3.2.2. Case $\lambda\in\sigma(B)\setminus\sigma(A)$.

Suppose that $y$ is an eigenfunction of (3.1) corresponding to $\lambda$. Then
$B$ has a countable collection of linearly independent eigenfunctions
$\big{\\{}b_{j}(s)y(n)\big{\\}}_{j=1}^{\infty}$, where
$\\{b_{j}\\}_{j=1}^{\infty}$ is a basis in $L_{2}(\gamma)$ consisting of
smooth functions. On the other hand, the problem

$-\Delta v+av=\lambda\rho v\>\text{ in }\Omega\setminus\gamma,\quad\ell
v=0\>\text{ on }\partial\Omega,\quad v^{-}=y(-1)b,\quad v^{+}=y(1)b$ (3.6)

is uniquely solvable for any $b\in C^{\infty}(\gamma)$, since $\lambda$ does
not belong to $\sigma(A)$. Therefore $\mathcal{P}$ possesses the countable set
of linearly independent eigenvectors

$u_{1}=(v_{1},b_{1}y),\;u_{2}=(v_{2},b_{2}y),\;\dots\;,u_{j}=(v_{j},b_{j}y),\;\dots,$
(3.7)

where $v_{j}$ is a solution of (3.6) with $b_{j}$ in place of $b$. Note that
the values $y(\pm 1)$ are different from zero and hence all $v_{j}$ are non-
trivial solutions. In this case, finding generalized eigenvectors of rank $2$
leads to the equation $(B-\lambda I)w_{*}=by$, which is unsolvable for $b\neq
0$. Hence $X_{\lambda}=X^{0}_{\lambda}$ and $\dim X_{\lambda}=\infty$.

####  3.2.3. Case $\lambda\in\sigma(A)\cap\sigma(B)$.

Since $\lambda$ belongs to $\sigma(B)$, any eigenvector of $\mathcal{P}$ has
the form $(v(x),b(s)y(n))$, where $v$ is a solution of (3.6). But now we can
not solve problem (3.6) for any function $b$, since $\lambda$ is a point of
$\sigma(A)$.

######  Proposition 3.5.

Let $f\in L_{2}(\Omega)$ and $b_{\pm}\in W_{2}^{3/2}(\gamma)$. Assume that
$\lambda$ is an eigenvalue of $A$ of multiplicity $K$ and the corresponding
eigenfunctions $V_{1},\dots,V_{K}$ satisfy (3.4). Then the problem

$-\Delta v+av=\lambda\rho v+f\>\text{ in }\Omega\setminus\gamma,\quad\ell
v=0\>\text{ on }\partial\Omega,\quad v^{-}=b_{-},\quad v^{+}=b_{+}$ (3.8)

admits a solution $v\in W_{2}^{2}(\Omega\setminus\gamma)$ if and only if

$\int_{\gamma}\big{(}b_{+}\partial_{r}V_{k}^{+}-b_{-}\partial_{r}V_{k}^{-}\big{)}\,d\gamma+\int_{\Omega}fV_{k}\,dx=0$
(3.9)

for all $k=1,\dots,K$.

###### Proof.

This statement is a simple consequence of the Fredholm alternative for the
self-adjoint operator $A$ with compact resolvent. Conditions (3.9) can be
obtained by multiplying the equation in (3.8) by $V_{1},\dots,V_{K}$ in turn
and then integrating by parts twice in view of the boundary conditions. ∎

In view of Proposition 3.5, problem (3.6) is solvable if and only if
$(b,\Psi_{k})_{L_{2}(\gamma)}=0$ for all $k=1,\dots,K$, where

$\Psi_{k}=y(1)\,\partial_{r}V_{k}^{+}-y(-1)\,\partial_{r}V_{k}^{-}.$ (3.10)

Let $H_{\lambda}$ be the subspace in $L_{2}(\gamma)$ spanned by
$\Psi_{1},\dots,\Psi_{K}$. Hence the solvability of (3.6) is equivalent to the
orthogonality of $b$ to $H_{\lambda}$.

######  Proposition 3.6.

Assume $\lambda$ is an eigenvalue of $A=A_{-}\oplus A_{+}$ and $k_{\pm}$ is
the multiplicity of $\lambda$ in the spectrum of $A_{\pm}$. Then $\dim
H_{\lambda}\geq\max(k_{+},k_{-})$.

###### Proof.

Since the operator $A$ is a direct sum of $A_{+}$ and $A_{-}$, we can choose a
basis in the eigenspace of $A$ such that $V_{1},\dots,V_{k_{+}}$ are
identically equal to zero in $\Omega_{-}$ and the rest of $k_{-}$
eigenfunctions $V_{k_{+}+1},\dots,V_{K}$ are identically equal to zero in
$\Omega_{+}$. Note that $k_{+}+k_{-}=K$. Then $H_{\lambda}$ is the linear span
of

$\partial_{r}V_{1}^{+},\,\ldots\,,\partial_{r}V_{k_{+}}^{+},\,\partial_{r}V_{k_{+}+1}^{-},\,\ldots\,,\partial_{r}V_{K}^{-}.$
(3.11)

In general, these normal derivatives are linearly dependent in
$L_{2}(\gamma)$, but the first of $k_{+}$ derivatives are always linearly
independent as well as the last of $k_{-}$ ones. Suppose, contrary to our
claim, that the functions
$\partial_{r}V_{1}^{+},\,\ldots\,,\partial_{r}V_{k_{+}}^{+}$ are linearly
dependent in $L_{2}(\gamma)$. Then there exists an eigenfunction $v$ of the
Dirichlet type problem $-\Delta v+av=\lambda\rho v$ in $\Omega_{+}$, $v=0$ on
$\partial\Omega_{+}$ such that $\partial_{r}v=0$ on the whole boundary of
$\Omega_{+}$, but this is impossible. The same conclusion can be drawn for the
second part of the normal derivatives. ∎

Assume $\dim H_{\lambda}=d$ and choose a basis $\\{b_{j}\\}_{j=1}^{\infty}$ in
$L_{2}(\gamma)$ such that $b_{1}$,…,$b_{d}$ belong to $H_{\lambda}$, while
$b_{k}$ for $j>d$ are elements of $H^{\perp}_{\lambda}$, and $b_{j}\in
C^{\infty}(\gamma)$. Then

$(V_{1},0),\,\dots,\,(V_{K},0),\,(v_{d+1},b_{d+1}y),\,(v_{d+2},b_{d+2}y),\dots$
(3.12)

is a countable set of linearly independent eigenvectors of $\mathcal{P}$. Here
$v_{j}$ is a solution of (3.6) for $b=b_{j}$ which is orthogonal to the span
of $V_{1},\dots,V_{K}$ in $L_{2}(\rho,\Omega)$. So $X_{\lambda}^{0}$ is
infinite-dimensional.

In this case, we can also find the generalized eigenvectors
$u_{*}=(v_{*},w_{*})$ of rank $2$. They satisfy the equation
$(\mathcal{P}-\lambda)u_{*}=u$, where $u=(v,w)$ is an eigenvector of
$\mathcal{P}$. Hence $v_{*}$ and $w_{*}$ solve the problem

$\displaystyle-$ $\displaystyle\Delta v_{*}+av_{*}=\lambda\rho v_{*}+\rho
v\quad\text{in }\Omega\setminus\gamma,\quad\ell v_{*}=0\quad\text{on
}\partial\Omega,$ (3.13) $\displaystyle-$
$\displaystyle\partial^{2}_{n}w_{*}=\lambda qw_{*}+qw\quad\text{in
}\omega,\qquad\partial_{n}w_{*}(\,\cdot\,,-1)=0,\quad\partial_{n}w_{*}(\,\cdot\,,1)=0,$
(3.14) $\displaystyle v_{*}^{-}=w_{*}(\,\cdot\,,-1),\quad
v_{*}^{+}=w_{*}(\,\cdot\,,1).$ (3.15)

It is easily seen that (3.13)–(3.15) is unsolvable if $w\neq 0$. Therefore the
generalized eigenvectors can be associated with some non-trivial linear
combinations of the eigenvectors $(V_{1},0),\dots,(V_{K},0)$ only. We set
$w=0$ and $v=c_{1}V_{1}+\cdots+c_{K}V_{K}$. Then $w_{*}(s,n)=b_{*}(s)y(n)$ and

$\displaystyle-$ $\displaystyle\Delta v_{*}+av_{*}=\lambda\rho
v_{*}+\rho\sum_{j=1}^{K}c_{j}V_{j}\quad\text{in
}\Omega\setminus\gamma,\quad\ell v_{*}=0\quad\text{on }\partial\Omega,$ (3.16)
$\displaystyle v_{*}^{-}=y(-1)\,b_{*},\quad v_{*}^{+}=y(1)\,b_{*}.$ (3.17)

By Proposition 3.5 and (3.4), the problem admits a solution if and only if

$c_{1}=-(b_{*},\Psi_{1})_{L_{2}(\gamma)},\,\dots\,,c_{K}=-(b_{*},\Psi_{K})_{L_{2}(\gamma)}.$

Since $v$ is non-zero, $b_{*}$ must have a non-trivial projection onto the
subspace $H_{\lambda}$. But $b_{1}$,…,$b_{d}$ have this property. Putting
$b_{*}=b_{j}$ for $j=1,\dots,d$ into (3.16), (3.17), we obtain the Jordan
chains of length $2$

$(V^{(j)},0)\;\;\mapsto\;\;(v^{(j)}_{*},b_{j}y),\quad j=1,\ldots,d,$

where $V^{(j)}=c_{1}^{(j)}V_{1}+\cdots+c_{K}^{(j)}V_{j}$ with
$c_{i}^{(j)}=-(b_{j},\Psi_{i})_{L_{2}(\gamma)}$.

There are no generalized eigenvectors of rank $3$, because all the
eigenvectors of rank $2$ have nonzero second components and the equation
$(B-\lambda I)y_{*}=b_{j}y$ is unsolvable for $b_{j}\neq 0$. Hence the space
$X_{\lambda}$ has the basis consisting of:

  * $\circ$

$d$ Jordan chains of length $2$

$(V^{(1)},0)\longmapsto(v^{(1)}_{*},b_{1}y),\,\ldots\,,(V^{(d)},0)\longmapsto(v^{(d)}_{*},b_{d}y),$
(3.18)

  * $\circ$

the series of eigenvectors of the form

$(v_{d+1},b_{d+1}y),\,(v_{d+2},b_{d+2}y),\,\dots\,,(v_{j},b_{j}y),\,\dots\,,$
(3.19)

  * $\circ$

$K-d$ eigenvectors of the form

$(V^{(d+1)},0),\,\dots\,,(V^{(K)},0).$ (3.20)

Here $V^{(1)},\dots,V^{(K)}$ are linearly independent eigenfunctions of $A$
corresponding to eigenvalue $\lambda$.

We summarize the information about the spectrum and generalized eigenspaces of
the limit operator $\mathcal{P}$ that we have obtained.

######  Theorem 3.7.

Let $X_{\lambda}$ be the generalized eigenspace corresponding to
$\lambda\in\sigma(\mathcal{P})$ and $X_{\lambda}^{0}$ be the eigenspace.

(i) If $\lambda\in\sigma(A)\setminus\sigma(B)$, then $X_{\lambda}$ is finite
dimensional, $X_{\lambda}=X_{\lambda}^{0}$, and the basis in $X_{\lambda}$ is
given by (3.5).

(ii) The part $\sigma(B)\setminus\sigma(A)$ of $\sigma(\mathcal{P})$ consists
of eigenvalues $\lambda$ of infinite multiplicity with eigenspaces
$X_{\lambda}=X_{\lambda}^{0}$ generated by vectors (3.7).

(iii) The part $\sigma(A)\cap\sigma(B)$ is also consists of eigenvalues
$\lambda$ of infinite multiplicity, but $X_{\lambda}\neq X_{\lambda}^{0}$.
Apart from the eigenvectors, there exist the generalized eigenvectors of rank
$2$, and the basis in $X_{\lambda}$ is given by (3.18)–(3.20). The dimension
of factor space $X_{\lambda}/X_{\lambda}^{0}$ does not exceed the multiplicity
of $\lambda$ in the spectrum of $A$, namely

$\max(k_{-},k_{+})\leq\dim X_{\lambda}/X_{\lambda}^{0}\leq k_{-}+k_{+},$

where $k_{-}$ and $k_{+}$ are the multiplicity of $\lambda$ in the spectra of
$A_{-}$ and $A_{+}$ respectively.

######  Remark 3.8.

The operator $\mathcal{P}$ has always the eigenvalue $\lambda=0$ of infinite
multiplicity, since $0\in\sigma(B)$. The zero eigenvalue is the smallest
infinite-fold one, because $B$ is non-negative. All negative eigenvalues, if
they exist, have finite multiplicities.

##  4\. Asymptotics of eigenvalues in the case
$\lambda_{0}\in\sigma(B)\setminus\sigma(A)$

We will first focus our attention on perturbations of infinite-fold
eigenvalues. In this section, we construct the asymptotics of countable set of
eigenvalues $\lambda_{\varepsilon}$ of (1.2) that converge to an infinite-fold
eigenvalue $\lambda_{0}$ of $\mathcal{P}$ when $\lambda_{0}\not\in\sigma(A)$.
We look for the asymptotics in the form

$\lambda^{\varepsilon}\sim\lambda_{0}+\lambda_{1}\varepsilon+\cdots,\quad
u_{\varepsilon}(x)\sim\begin{cases}v_{0}(x)+\varepsilon
v_{1}(x)+\cdots&\text{if }x\in\Omega\setminus\omega_{\varepsilon},\\\
w_{0}(s,\textstyle\frac{r}{\varepsilon})+\varepsilon
w_{1}(s,\textstyle\frac{r}{\varepsilon})+\cdots&\text{if
}x\in\omega_{\varepsilon},\end{cases}$ (4.1)

where $(v_{0},w_{0})$ is an non-zero element of $X_{\lambda_{0}}$. To match
the expansions on $\gamma_{\pm\varepsilon}$, we write $v_{k}$ in the local
coordinates $(s,n)$. Then (2.3) becomes

$\displaystyle w_{0}(s,\pm 1)+\varepsilon w_{1}(s,\pm
1)-v_{0}(s,\pm\varepsilon)-\varepsilon v_{1}(s,\pm\varepsilon)+\cdots\sim 0,$
$\displaystyle\varepsilon^{-1}\partial_{n}w_{0}(s,\pm
1)+\partial_{n}w_{1}(s,\pm
1)-\partial_{r}v_{0}(s,\pm\varepsilon)-\varepsilon\partial_{r}v_{1}(s,\pm\varepsilon)+\cdots\sim
0.$

After expanding $v_{k}(s,\pm\varepsilon)$ and their derivatives into the
Taylor series about $n=\pm 0$ for fixed $s$, we in particular derive
$w_{1}(\,\cdot\,,\pm 1)=v_{1}^{\pm}\pm\partial_{r}v_{0}^{\pm}$ and
$\partial_{n}w_{1}(\,\cdot\,,\pm 1)=\partial_{r}v_{0}^{\pm}$. Of course,
$w_{0}(\,\cdot\,,\pm 1)=v_{0}^{\pm}$ and $\partial_{n}w_{0}(\,\cdot\,,\pm
1)=0$, since $(v_{0},w_{0})\in\mathop{\rm dom}\mathcal{P}$.

Substituting (4.1) into (1.2), and taking into account representation (2.2),
we obtain that the pair $(v_{1},w_{1})$ solves the problem

$\displaystyle-\Delta v_{1}+av_{1}=\lambda_{0}\rho v_{1}+\lambda_{1}\rho
v_{0}\quad\text{in }\Omega\setminus\gamma,\qquad\ell v_{1}=0\quad\text{on
}\partial\Omega,$ (4.2)
$\displaystyle-\partial^{2}_{n}w_{1}=\lambda_{0}qw_{1}-\varkappa\partial_{n}w_{0}+\lambda_{1}qw_{0}\quad\text{in
}\omega,$ (4.3)
$\displaystyle\phantom{-}\partial_{n}w_{1}(\,\cdot\,,-1)=\partial_{r}v_{0}^{-},\quad\partial_{n}w_{1}(\,\cdot\,,1)=\partial_{r}v_{0}^{+},$
(4.4)
$\displaystyle\phantom{-}v_{1}^{-}=w_{1}(\,\cdot\,,-1)+\partial_{r}v_{0}^{-},\quad
v_{1}^{+}=w_{1}(\,\cdot\,,1)-\partial_{r}v_{0}^{+}.$ (4.5)

The compatibility condition for (4.3), (4.4) reads

$y(-1)\,\partial_{r}v_{0}^{-}-y(1)\,\partial_{r}v_{0}^{+}+\varkappa\int_{-1}^{1}\partial_{n}w_{0}\,y\,dn=\lambda_{1}\int_{-1}^{1}qw_{0}y\,dn\quad\text{on
}\gamma,$ (4.6)

where $y$ is an eigenfunction of (3.1) corresponding to $\lambda_{0}$. We have
applied the Fredholm alternative for the operator with compact resolvent
associated with the Sturm-Liouville problem (3.1). The relation (4.6) can be
treated as a spectral equation on $\lambda_{1}$ for a pseudodifferential
operator on $\gamma$. We introduce the Dirichlet-to-Neumann maps
$N^{\pm}_{\lambda}$ in $L_{2}(\gamma)$ as follows. Let $z$ be the solution of
problem

$-\Delta z+az=\lambda\rho z\;\;\text{in }\Omega_{-},\quad
z=\varphi\quad\text{on }\gamma,\;\;\ell z=0\quad\text{on }\partial\Omega$
(4.7)

for given $\varphi$. Set $N^{-}_{\lambda}\varphi=\partial_{r}z|_{\gamma}$. We
follow [44] in assuming that

$\mathop{\rm dom}N^{-}_{\lambda}=\big{\\{}\varphi\in L_{2}(\gamma)\colon z\in
W_{2}^{1}(\Omega_{-})\text{ and }\partial_{r}z|_{\gamma}\in
L_{2}(\gamma)\big{\\}}.$

Likewise, we set $N^{+}_{\lambda}\varphi=-\partial_{r}z|_{\gamma}$, where $z$
is a solution of the problem

$-\Delta z+az=\lambda\rho z\;\;\text{in }\Omega_{+},\quad
z=\varphi\quad\text{on }\gamma,$ (4.8)

and $\mathop{\rm dom}N^{+}_{\lambda}=\big{\\{}\varphi\in L_{2}(\gamma)\colon
z\in W_{2}^{1}(\Omega_{+})\text{ and }\partial_{r}z|_{\gamma}\in
L_{2}(\gamma)\big{\\}}$. The operators $N^{\pm}_{\lambda}$ transform the
Dirichlet data on $\gamma$ for solutions of the corresponding boundary value
problems into the Neumann ones. Both operators are well-defined if
$\lambda\not\in\sigma(A)$. The minus sing in definition of $N^{+}_{\lambda}$
indicates that the direction of axis $r$ coincides with the inward normal on
$\gamma=\partial\Omega_{+}$.

It follows from (3.7) that $w_{0}(s,n)=b_{0}(s)y(n)$ and $v_{0}$ is a solution
of (3.6) for $b=b_{0}$. Then
$\partial_{r}v_{0}^{-}=y(-1)\,N^{-}_{\lambda_{0}}b_{0}$ and
$\partial_{r}v_{0}^{+}=-y(1)\,N^{+}_{\lambda_{0}}b_{0}$. Consequently,
condition (4.6) reads

$y^{2}(-1)\,N^{-}_{\lambda_{0}}b_{0}+y^{2}(1)\,N^{+}_{\lambda_{0}}b_{0}+\varkappa
b_{0}\int_{-1}^{1}yy^{\prime}\,dn=\lambda_{1}b_{0}\int_{-1}^{1}qy^{2}\,dn.$

Suppose $\|y\|_{L_{2}(q,(-1,1))}=1$ and write
$\theta_{\lambda_{0}}^{\pm}=y^{2}(\pm 1)$. Since

$\int_{-1}^{1}yy^{\prime}\,dn=\tfrac{1}{2}\int_{-1}^{1}(y^{2})^{\prime}\,dn=\tfrac{1}{2}\left(\theta_{\lambda_{0}}^{+}-\theta_{\lambda_{0}}^{-}\right),$

we can finally rewrite (4.6) in the form
$N_{\lambda_{0}}b_{0}=\lambda_{1}b_{0}$, where

$N_{\lambda}=\theta_{\lambda}^{-}N^{-}_{\lambda}+\theta_{\lambda}^{+}N^{+}_{\lambda}+\tfrac{1}{2}\,(\theta_{\lambda}^{+}-\theta_{\lambda}^{-})\varkappa.$
(4.9)

Note that the values $\theta_{\lambda_{0}}^{-}$ and $\theta_{\lambda_{0}}^{+}$
do not depend on our choice of $y$, because all eigenvalues of the Sturm-
Liouville problem are simple.

######  Proposition 4.1.

The operator $N_{\lambda}$ is self-adjoint, bounded below and has compact
resolvent for all $\lambda\not\in\sigma(A)$.

###### Proof.

The operators $N^{-}_{\lambda}$ and $N^{+}_{\lambda}$ are self-adjoint in
$L_{2}(\gamma)$, bounded below and have compact resolvents [44, Th.3.1]. The
linear combination
$\theta_{\lambda}^{-}N^{-}_{\lambda}+\theta_{\lambda}^{+}N^{+}_{\lambda}$ has
the same properties, since $\theta_{\lambda}^{-}$ and $\theta_{\lambda}^{+}$
are positive. Finally, $N_{\lambda}$ is a perturbation of this linear
combination by the operator of multiplication by the bounded function
$\tfrac{1}{2}\,(\theta_{\lambda}^{+}-\theta_{\lambda}^{-})\varkappa(s)$, which
completes the proof. ∎

Denote by $\\{\lambda_{1}^{(k)}\\}_{k=1}^{\infty}$ the eigenvalues of
$N_{\lambda_{0}}$. So we have calculated the countable set of correctors
$\lambda_{1}^{(k)}$ in asymptotics (4.1). To keep the mathematics rather
simple we suppose that the spectrum of $N_{\lambda_{0}}$ is simple. It means
that the infinite-fold eigenvalue $\lambda_{0}$ of $\mathcal{P}$
asymptotically splits into an infinite number of simple eigenvalues of
$T_{\varepsilon}$ under perturbation. Let $\\{b^{(k)}\\}_{k=1}^{\infty}$ be
the collection of orthonormal eigenfunctions of $N_{\lambda_{0}}$.

Let us fix the corrector $\lambda_{1}^{(k)}$ and set $w_{0}=w^{(k)}=b^{(k)}y$.
Then $v_{0}=v^{(k)}$ is a unique solution of (3.6) with $\lambda_{0}$ and
$b^{(k)}$ in place of $\lambda$ and $b$ respectively. Such a choice of
$(v^{(k)},w^{(k)})$ ensures that problem (4.3), (4.4) is solvable, since the
compatibility condition $N_{\lambda_{0}}b^{(k)}=\lambda_{1}^{(k)}b^{(k)}$
holds. The problem admits the family of solutions

$w_{1}(s,n)=\hat{w}_{1}^{(k)}(s,n)+b_{1}(s)y(n),$

where $b_{1}$ is an arbitrary $L_{2}(\gamma)$-function and $\hat{w}_{1}^{(k)}$
is the partial solution subject to the condition

$\int_{-1}^{1}qy\hat{w}_{1}^{(k)}\,dn=0\quad\text{ for all }s\in S.$ (4.10)

According to (4.2) and (4.5), with this choice of $w_{1}$ the function $v_{1}$
can be written as $v_{1}=\hat{v}_{1}^{(k)}+\tilde{v}_{1}$, where
$\hat{v}_{1}^{(k)}$ (resp. $\tilde{v}_{1}$) is a solution of the problem

$-\Delta\phi+a\phi=\lambda\rho\phi+\lambda_{1}^{(k)}\rho v^{(k)}\text{ in
}\Omega\setminus\gamma,\quad\ell\phi=0\text{ on
}\partial\Omega,\quad\phi^{-}=g_{-},\;\;\phi^{+}=g_{+}$ (4.11)

with the Dirichlet data $g_{\pm}=\hat{w}_{1}^{(k)}(\,\cdot\,,\pm
1)\mp\partial_{n}v^{(k)}|_{\gamma_{\pm}}$ (resp. $g_{\pm}=y(\pm 1)b_{1}$). The
term $\hat{v}_{1}^{(k)}$ is uniquely defined, while $\tilde{v}_{1}$ along with
$b_{1}$ will be fixed below.

Set $a_{0}(s)=a(s,0)$. The pair $(v_{2},w_{2})$ in the asymptotics of
$u_{\varepsilon}$ solves the problem

$\displaystyle-\Delta v_{2}+av_{2}=\lambda_{0}\rho v_{2}+\lambda_{1}\rho
v_{1}+\lambda_{2}\rho v_{0}\quad\text{in }\Omega\setminus\gamma,\quad\ell
v_{2}=0\quad\text{on }\partial\Omega,$ (4.12)
$\displaystyle-\partial^{2}_{n}w_{2}=\lambda_{0}qw_{2}-(\varkappa\partial_{n}-\lambda_{1}q)w_{1}-(\varkappa^{2}n\partial_{n}-\partial_{s}^{2}+a_{0}-\lambda_{2}q)w_{0}\quad\text{in
}\omega,$ (4.13)
$\displaystyle\phantom{-}\partial_{n}w_{2}(\,\cdot\,,-1)=\partial_{r}v_{1}^{-}-\partial^{2}_{r}v_{0}^{-},\quad\partial_{n}w_{2}(\,\cdot\,,1)=\partial_{r}v_{1}^{+}+\partial^{2}_{r}v_{0}^{+},$
(4.14)
$\displaystyle\phantom{-}v_{2}^{-}=w_{2}(\,\cdot\,,-1)+\partial_{r}v_{1}^{-}-\partial^{2}_{r}v_{0}^{-},\quad
v_{2}^{+}=w_{2}(\,\cdot\,,1)-\partial_{r}v_{1}^{+}-\partial^{2}_{r}v_{0}^{+}.$
(4.15)

By reasoning as above, we obtain that the solution $w_{2}$ exists if and only
if

$(N_{\lambda_{0}}-\lambda_{1}^{(k)}I)b_{1}=\lambda_{2}b^{(k)}+h_{1}^{(k)},$
(4.16)

where

$h_{1}^{(k)}=y(1)\big{(}\partial_{r}\hat{v}_{1}^{(k)}|_{\gamma_{+}}+\partial^{2}_{r}v^{(k)}|_{\gamma_{+}}\big{)}-y(-1)\big{(}\partial_{r}\hat{v}_{1}^{(k)}|_{\gamma_{-}}-\partial^{2}_{r}v^{(k)}|_{\gamma_{-}}\big{)}\\\
-\int_{-1}^{1}(\varkappa\partial_{n}-\lambda_{1}^{(k)}q)y\hat{w}_{1}^{(k)}\,dn-\int_{-1}^{1}\big{(}\varkappa^{2}nyy^{\prime}\,b^{(k)}-y^{2}(\partial_{s}^{2}b^{(k)}-a_{0}b^{(k)})\big{)}\,dn.$
(4.17)

Since $\lambda_{1}^{(k)}$ is a simple eigenvalue of $N_{\lambda_{0}}$, the
second corrector $\lambda_{2}=\lambda_{2}^{(k)}$ can be uniquely calculated
from the solvability condition for equation (4.16)

$\lambda_{2}^{(k)}=-(h_{1}^{(k)},b^{(k)})_{L_{2}(\gamma)}.$

Moreover, there exists a unique solution $b_{1}^{(k)}$ of (4.16) satisfying
the condition $(b_{1}^{(k)},b^{(k)})_{L_{2}(\gamma)}=0$. Hence $v_{1}^{(k)}$
is now uniquely defined by choosing $\tilde{v}_{1}$ to be a solution of (4.11)
with the Dirichlet data $g_{\pm}=y(\pm 1)b_{1}^{(k)}$. The compatibility
condition (4.16) allows us to solve problems (4.13), (4.14) and (4.12), (4.15)
one after another and to find solutions $w_{2}^{(k)}$ and $v_{2}^{(k)}$.

The process used to ﬁnd the leading terms of asymptotic expansions (4.1) can
be continued to systematically construct all other terms. For fixed
$\lambda_{0}\in\sigma(B)\setminus\sigma(A)$ we consider the countable
collection of formal approximations to eigenvalues and eigenfunctions of the
perturbed problem

$\displaystyle\Lambda_{\varepsilon}^{(k)}=\lambda_{0}+\lambda_{1}^{(k)}\varepsilon+\lambda_{2}^{(k)}\varepsilon^{2}+\lambda_{3}^{(k)}\varepsilon^{3},$
(4.18)
$\displaystyle\hat{u}_{\varepsilon}^{(k)}(x)=\begin{cases}v_{0}^{(k)}(x)+\varepsilon
v_{1}^{(k)}(x)+\varepsilon^{2}v_{2}^{(k)}(x)+\varepsilon^{3}v_{3}^{(k)}(x)&\text{if
}x\in\Omega\setminus\omega_{\varepsilon},\\\
w_{0}^{(k)}(s,\textstyle\frac{r}{\varepsilon})+\varepsilon
w_{1}^{(k)}(s,\textstyle\frac{r}{\varepsilon})+\varepsilon^{2}w_{2}^{(k)}(s,\textstyle\frac{r}{\varepsilon})+\varepsilon^{3}w_{3}^{(k)}(s,\textstyle\frac{r}{\varepsilon})&\text{if
}x\in\omega_{\varepsilon}.\end{cases}$ (4.19)

Let $H$ be a Hilbert space with norm $\|\cdot\|$ and let $T$ be a self-adjoint
operator in $H$ with a domain $\mathop{\rm dom}T$. We say a pair
$(\mu,u)\in\mathbb{R}\times\mathop{\rm dom}T$ is a quasimode of $T$ with the
accuracy $\delta$, if $\|u\|=1$ and $\|(T-\mu I)u\|\leq\delta$. Of course, if
$\delta=0,$ then $\mu$ is an eigenvalue of $T$ with the eigenvector $u$.

######  Proposition 4.2 ([45, p.139]).

Assume $(\mu,u)$ is a quasimode of $T$ with accuracy $\delta>0$ and the
spectrum of $T$ is discrete in the interval $[\mu-\delta,\mu+\delta]$. Then
there exists an eigenvalue $\mu_{*}$ of $T$ such that
$|\mu_{*}-\mu|\leq\delta$.

In order to construct the quasimodes of $T_{\varepsilon}$, we must modify the
approximations of eigenfunctions, because they do not belong to $\mathop{\rm
dom}T_{\varepsilon}$. By construction, the functions $v_{j}^{(k)}$ and
$w_{j}^{(k)}$ are smooth, since the coefficients $a$, $\rho$ and $q$ are
smooth. But, in general, $\hat{u}_{\varepsilon}^{(k)}$ have jump
discontinuities on $\partial\omega_{\varepsilon}$.

Figure 2. Plot of the function $\zeta$.

Let us define the function $\zeta$ plotted in Fig. 2. This function is smooth
outside the origin, $\zeta(r)=1$ for $r\in[0,\beta/2]$ and $\zeta(r)=0$ in the
set $\mathbb{R}\setminus[0,\beta)$. We can choose $\beta$ small enough such
that the local coordinates $(s,r)$ are well defined in $\omega_{2\beta}$. Set

$\eta_{\varepsilon}^{(k)}=\big{(}[\hat{u}_{\varepsilon}^{(k)}]_{\varepsilon}+[\partial_{r}\hat{u}_{\varepsilon}^{(k)}]_{\varepsilon}\,(r-\varepsilon)\big{)}\,\zeta(r-\varepsilon)\\\
+\big{(}[\hat{u}_{\varepsilon}^{(k)}]_{-\varepsilon}+[\partial_{r}\hat{u}_{\varepsilon}^{(k)}]_{-\varepsilon}\,(r+\varepsilon)\big{)}\,\zeta(-r-\varepsilon).$
(4.20)

The function is different from zero in the set
$\omega_{\beta+\varepsilon}\setminus\omega_{\varepsilon}$ only. And it is easy
to check that $\eta_{\varepsilon}^{(k)}$ and
$\partial_{r}\eta_{\varepsilon}^{(k)}$ have the same jumps across the boundary
of $\omega_{\varepsilon}$ as $\hat{u}_{\varepsilon}^{(k)}$ and
$\partial_{r}\hat{u}_{\varepsilon}^{(k)}$ respectively. Therefore the function
$U_{\varepsilon}^{(k)}=\hat{u}_{\varepsilon}^{(k)}-\eta_{\varepsilon}^{(k)}$
belongs to the domain of $T_{\varepsilon}$. Moreover we have not changed
$\hat{u}_{\varepsilon}^{(k)}$ too much, since

$\sup_{x\in\Omega\setminus\overline{\omega}_{\varepsilon}}\big{(}|\eta_{\varepsilon}^{(k)}(x)|+|\Delta\eta_{\varepsilon}^{(k)}(x)|\big{)}\leq
c\varepsilon^{3}.$ (4.21)

It follows from the explicit formula for $\eta_{\varepsilon}^{(k)}$ and
smallness of jumps of $\hat{u}_{\varepsilon}^{(k)}$,
$\partial_{r}\hat{u}_{\varepsilon}^{(k)}$ across
$\partial\omega_{\varepsilon}$. All the jumps are of order
$O(\varepsilon^{3})$, as $\varepsilon\to 0$, by construction.

We will henceforth write
$n_{\varepsilon}^{(k)}=\|U_{\varepsilon}^{(k)}\|^{-1}_{L^{2}(\rho_{\varepsilon},\Omega)}$.

######  Lemma 4.3.

For each $k\in\mathbf{N}$, the pair
$(\Lambda_{\varepsilon}^{(k)},n_{\varepsilon}^{(k)}U_{\varepsilon}^{(k)})$
constructed above is a quasimode of the operator $T_{\varepsilon}$ with the
accuracy $O(\varepsilon^{2})$ as $\varepsilon\to 0$.

###### Proof.

For simplicity we shall drop the index $k$ in the sequel and write
$(\Lambda_{\varepsilon},U_{\varepsilon})$, $v_{j}$, $w_{j}$ and $\lambda_{j}$
instead of $(\Lambda_{\varepsilon}^{(k)},U_{\varepsilon}^{(k)})$,
$v_{j}^{(k)}$, $w_{j}^{(k)}$ and $\lambda_{j}^{(k)}$. Set
$F_{\varepsilon}=(T_{\varepsilon}-\Lambda_{\varepsilon})U_{\varepsilon}$. Then
we have

$F_{\varepsilon}=(-\Delta+a-\Lambda_{\varepsilon}\rho)U_{\varepsilon}=\sum_{j=0}^{3}\varepsilon^{j}(-\Delta
v_{j}+av_{j}-\rho\sum_{i=0}^{j}\lambda_{i}v_{j-i})+\Delta\eta_{\varepsilon}-a\eta_{\varepsilon}+\Lambda_{\varepsilon}\rho\eta_{\varepsilon}$

outside $\omega_{\varepsilon}$. From our choice of $v_{j}$, we derive that the
first sum in the right-hand side vanishes. Therefore
$\sup_{x\in\Omega\setminus\omega_{\varepsilon}}|F_{\varepsilon}(x)|\leq
c_{1}\varepsilon^{3}$, because of (4.21). Applying representation (2.2) of the
Laplace operator in $\omega$, we have

$\Delta-a+\varepsilon^{-2}\Lambda_{\varepsilon}q=\sum_{j=0}^{3}\varepsilon^{j-2}p_{j}+\varepsilon^{2}P_{\varepsilon}-a_{\varepsilon},$

where $p_{0}=\partial^{2}_{n}+\lambda_{0}q$,
$p_{1}=-\varkappa\partial_{n}+\lambda_{1}q$,
$p_{2}=-n\varkappa^{2}\partial_{n}+\partial_{s}^{2}-a(s,0)+\lambda_{2}q$,

$p_{3}=2n\varkappa\partial^{2}_{s}+n\varkappa^{\prime}\partial_{s}-n^{2}\varkappa^{3}\partial_{n}-na(s,0)\partial_{n}+\lambda_{3}q,$

and $a_{\varepsilon}(s,n)=a(s,\varepsilon n)-a(s,0)-\varepsilon
n\partial_{n}a(s,0)$. Then, in the domain $\omega_{\varepsilon}$, we obtain

$F_{\varepsilon}=(-\Delta+a-\varepsilon^{-2}\Lambda_{\varepsilon}q)U_{\varepsilon}=\sum_{j=0}^{3}\varepsilon^{j-2}\sum_{i=0}^{j-i}p_{i}w_{j-i}-\varepsilon^{2}P_{\varepsilon}U_{\varepsilon}+a_{\varepsilon}=-\varepsilon^{2}P_{\varepsilon}U_{\varepsilon}+a_{\varepsilon},$

since the functions $w_{0},\ldots,w_{3}$ solve the equations
$\sum_{i=0}^{j-i}p_{i}w_{j-i}=0$, by construction. Consequently
$\sup_{x\in\omega_{\varepsilon}}|F_{\varepsilon}(x)|=\varepsilon^{2}\sup_{x\in\omega_{\varepsilon}}|P_{\varepsilon}U_{\varepsilon}(x)|+\sup_{x\in\omega_{\varepsilon}}|a_{\varepsilon}(x)|\leq
c_{2}\varepsilon^{2}$. Hence

$\|F_{\varepsilon}\|^{2}_{L^{2}(\rho_{\varepsilon},\Omega)}\leq\int_{\Omega\setminus\omega_{\varepsilon}}\rho|F_{\varepsilon}|^{2}\,dx+\varepsilon^{-2}\int_{\omega_{\varepsilon}}q|F_{\varepsilon}|^{2}\,dx\leq
c_{3}\varepsilon^{6}+c_{4}|\omega_{\varepsilon}|\varepsilon^{2}\leq
c_{5}\varepsilon^{3}.$

The main contribution to the $L^{2}(\rho_{\varepsilon},\Omega)$-norm of
$U_{\varepsilon}$ is given by the integral
$\varepsilon^{-2}\int_{\omega_{\varepsilon}}q|w_{0}|^{2}\,dx$ which is of
order $O(\varepsilon^{-1})$, as $\varepsilon\to 0$. Hence
$\|U_{\varepsilon}\|_{L^{2}(\rho_{\varepsilon},\Omega)}\geq
c_{6}\varepsilon^{-1/2}$ for $\varepsilon$ small enough, i.e.,
$n_{\varepsilon}\leq c_{7}\varepsilon^{1/2}$. This bound along with
$\|F_{\varepsilon}\|_{L^{2}(\rho_{\varepsilon},\Omega)}\leq
c_{8}\varepsilon^{3/2}$ yields

$\|(T_{\varepsilon}-\Lambda_{\varepsilon}I)(n_{\varepsilon}U_{\varepsilon})\|_{L^{2}(\rho_{\varepsilon},\Omega)}=n_{\varepsilon}\|F_{\varepsilon}\|_{L^{2}(\rho_{\varepsilon},\Omega)}\leq
c_{9}\varepsilon^{2},$ (4.22)

and this is precisely the assertion of the lemma. ∎

######  Theorem 4.4.

Suppose that $\lambda_{0}$ is an eigenvalue of the limit operator
$\mathcal{P}$ such that $\lambda_{0}\in\sigma(B)\setminus\sigma(A)$. Assume
the spectrum $\\{\lambda_{1}^{(k)}\\}_{k\in\mathbb{N}}$ of the operator
$N_{\lambda_{0}}$ is simple. Then there exists a countable set of eigenvalues
$\lambda_{\varepsilon,k}$, $k\in\mathbb{N}$, in the spectrum of
$T_{\varepsilon}$ that converge to $\lambda_{0}$ and admit the asymptotics

$\lambda_{\varepsilon,k}=\lambda_{0}+\varepsilon\lambda_{1}^{(k)}+O(\varepsilon^{2}),\quad\text{as
}\varepsilon\to 0.$ (4.23)

###### Proof.

Fix $I\in\mathbb{N}$. In view of Proposition 4.2 and Lemma 4.3, there exist
eigenvalues $\lambda_{\varepsilon,1},\dots,\lambda_{\varepsilon,I}$ of
$T_{\varepsilon}$ and a constant $c_{I}$ such that

$|\lambda_{\varepsilon,k}-\lambda_{0}-\varepsilon\lambda_{1}^{(k)}|\leq
c_{I}\varepsilon^{2}$ (4.24)

for $k=1,2,\dots,I$ and $\varepsilon$ small enough. Moreover these eigenvalues
are pairwise different. Suppose, contrary to our claim, that some eigenvalue
$\lambda_{\varepsilon}$ of $T_{\varepsilon}$ simultaneously satisfies two
estimates (4.24), for example when $k=1$ and $k=2$. Then
$\lambda_{\varepsilon}-\lambda_{0}-\varepsilon\lambda_{1}^{(1)}\leq
c_{I}\varepsilon^{2}$ and
$\lambda_{0}+\varepsilon\lambda_{1}^{(2)}-\lambda_{\varepsilon}\leq
c_{I}\varepsilon^{2}$. Adding these inequalities yields
$\lambda_{1}^{(2)}-\lambda_{1}^{(1)}\leq 2c_{I}\varepsilon$ for all
$\varepsilon$ small enough. But this is impossible, because the spectrum of
$N_{\lambda_{0}}$ is simple and therefore
$\lambda_{1}^{(1)}<\lambda_{1}^{(2)}$. Write

$\delta_{I}^{\varepsilon}=\varepsilon\max_{k=1,\dots,I}|\lambda_{1}^{(k)}|+c_{I}\varepsilon^{2}.$

Hence the interval
$[\lambda_{0}-\delta_{I}^{\varepsilon},\lambda_{0}+\delta_{I}^{\varepsilon}]$
contains at least $I$ eigenvalues of $T_{\varepsilon}$ that possess the
asymptotics (4.23). ∎

Figure 3. Bifurcation of the eigenvalue
$\lambda_{0}\in\sigma(B)\setminus\sigma(A)$.

##  5\. Asymptotics of eigenvalues in the case
$\lambda_{0}\in\sigma(A)\cap\sigma(B)$

In view of Theorem 3.7, if the set $\sigma(A)\cap\sigma(B)$ is non-empty, the
operator $\mathcal{P}$ possesses generalized eigenvectors of rank $2$. This
requires changing the structure of asymptotics

$\displaystyle\lambda^{\varepsilon}$
$\displaystyle\sim\lambda_{0}+\lambda_{1/2}\varepsilon^{1/2}+\lambda_{1}\varepsilon+\cdots,$
(5.1) $\displaystyle u_{\varepsilon}(x)$
$\displaystyle\sim\begin{cases}v_{0}(x)+\varepsilon^{1/2}v_{1/2}(x)+\varepsilon
v_{1}(x)+\cdots&\text{for }x\in\Omega\setminus\omega_{\varepsilon},\\\
w_{0}(s,\textstyle\frac{r}{\varepsilon})+\varepsilon^{1/2}w_{1/2}(s,\textstyle\frac{r}{\varepsilon})+\varepsilon
w_{1}(s,\textstyle\frac{r}{\varepsilon})+\cdots&\text{for
}x\in\omega_{\varepsilon}.\end{cases}$ (5.2)

Here $(v_{0},w_{0})$ is an non-zero element of the eigenspace spanned by
vectors (3.12). If $y$ is a normalized eigenfunction of (3.1) corresponding to
$\lambda_{0}$, then $w_{0}(s,n)=b_{0}(s)y(n)$ for some $b_{0}\in
L_{2}(\gamma)$. As above, substituting the series in (1.2) in particular
yields

$\displaystyle-\Delta v_{1/2}+av_{1/2}=\lambda_{0}\rho
v_{1/2}+\lambda_{1/2}\rho v_{0}\text{ in }\Omega\setminus\gamma,\quad\ell
v_{1/2}=0\text{ on }\partial\Omega,$ (5.3)
$\displaystyle-\partial^{2}_{s}w_{1/2}=\lambda_{0}qw_{1/2}+\lambda_{1/2}qw_{0}\quad\text{in
}\omega,$ (5.4)
$\displaystyle\phantom{-}\partial_{n}w_{1/2}(\,\cdot\,,-1)=0,\quad\partial_{n}w_{1/2}(\,\cdot\,,1)=0,$
(5.5) $\displaystyle\phantom{-}v_{1/2}^{-}=w_{1/2}(\,\cdot\,,-1),\quad
v_{1/2}^{+}=w_{1/2}(\,\cdot\,,1).$ (5.6)

Since (5.4), (5.5) can be written as
$(B-\lambda_{0})w_{1/2}=\lambda_{1/2}b_{0}y$ and $B$ is self-adjoint, a
solution $w_{1/2}$ exists if only if

$\lambda_{1/2}b_{0}=0.$ (5.7)

This condition is a branching point of our algorithm.

### Integer power asymptotics: case $w_{0}\neq 0$

If $b_{0}$ is a non-zero function, then $\lambda_{1/2}=0$ by necessity. Then
problem (5.3)–(5.6) turns into the limit problem (2.4)–(2.6) and
$(v_{1/2},w_{1/2})\in X^{0}_{\lambda_{0}}$. Without loss of generality we
assume that $(v_{1/2},w_{1/2})=0$, i.e., this vector is absorbed by the
leading term of the asymptotics. Moreover, a trivial verification shows that
all terms $\lambda_{j/2}$, $(v_{j/2},w_{j/2})$ with half-integer indexes in
(5.1), (5.2) can be treated as equal to zero. In this case, we come back to
the integer power asymptotics (4.1), but the construction of quasimodes needs
a slight modification. The next terms $\lambda_{1}$, $(v_{1},w_{1})$ solve
problem (4.2)–(4.5), and therefore condition (4.6) must hold. But now we
cannot rewrite (4.6) in the form of the spectral equation for
$N_{\lambda_{0}}$, because this operator is not defined for
$\lambda_{0}\in\sigma(A)$.

We will “extend” $N_{\lambda}$ to $\sigma(A)$ by means of a restriction of the
space in which it acts. A slight change in the proof of Proposition 3.5
actually shows that both problems (4.7) and (4.8) are solvable for
$\lambda\in\sigma(A)$ if the function $\varphi$ in the boundary condition on
$\gamma$ is orthogonal to the subspace $H_{\lambda}\subset L_{2}(\gamma)$
spanned by functions (3.10). Although solutions of the problems, in this case,
are ambiguously determined, we can subject them to some additional condition.
Namely, there exists a unique solution $z$ of (4.7) (or (4.8)) satisfying the
condition $\partial_{r}z|_{\gamma}\perp H_{\lambda}$. So we can define the
Dirichlet-to-Neumann map $M^{-}_{\lambda}\varphi=\partial_{r}z|_{\gamma}$ on
$H_{\lambda}^{\perp}$, where $z$ is a solution of (4.7) belonging to
$H_{\lambda}^{\perp}$. Similarly, we define
$M^{+}_{\lambda}\varphi=-\partial_{r}z|_{\gamma}$, where $z\in
H_{\lambda}^{\perp}$ is a solution of (4.8). Both operators are well defined
for $\lambda\in\mathbb{C}$. In fact, we have

$M^{\pm}_{\lambda}=(I-P_{\lambda})N^{\pm}_{\lambda}(I-P_{\lambda}),$

where $P_{\lambda}$ is the orthogonal projector onto the subspace
$H_{\lambda}$. Moreover, $M^{\pm}_{\lambda}=N^{\pm}_{\lambda}$ for
$\lambda\in\mathbb{C}\setminus\sigma(A)$, since the subspace $H_{\lambda}$ is
trivial in this case.

Now solvability condition (4.6) becomes
$M_{\lambda_{0}}b_{0}=\lambda_{1}b_{0}$, where

$M_{\lambda}=\theta_{\lambda}^{-}M^{-}_{\lambda}+\theta_{\lambda}^{+}M^{+}_{\lambda}+\tfrac{1}{2}(\theta_{\lambda}^{+}-\theta_{\lambda}^{-})\varkappa.$

The pseudodifferential operator $M_{\lambda}$ has the same properties as
$N_{\lambda}$. Suppose that the spectrum of $M_{\lambda_{0}}$ is simple and
$\lambda_{1}^{(k)}$ is the $k$-th eigenvalue of $M_{\lambda}$ with the
normalized eigenfunction $b^{(k)}$. We set $w_{0}=b^{(k)}y$.

Assume $\lambda_{0}$ is an $K$-fold eigenvalue of $A$ with the eigenspace
$\mathcal{V}_{\lambda_{0}}$. In order to shorten notation, we introduce the
vector $\mathbf{V}=(V_{1},\dots,V_{K})$, where the eigenfunctions $V_{k}$ are
subject to condition (3.4). Then the leading term $v_{0}=v_{0}^{(k)}$ in (5.2)
solves (3.6) for $\lambda=\lambda_{0}$ and $b=b^{(k)}$ and has the form

$v_{0}^{(k)}(x)=\hat{v}_{0}^{(k)}(x)+\alpha_{0}\cdot\mathbf{V}(x),$

where $\alpha_{0}$ is an arbitrary vector in $\mathbb{R}^{K}$ and
$\hat{v}_{0}^{(k)}$ is a partial solution of (3.6) such that
$\hat{v}_{0}^{(k)}\perp\mathcal{V}_{\lambda_{0}}$. The dot denotes the scalar
product in $\mathbb{R}^{K}$. To determine $v_{0}^{(k)}$ uniquely, we should
calculate $\alpha_{0}$. Next, we have

$w_{1}(s,n)=\hat{w}_{1}^{(k)}(s,n)+b_{1}(s)y(n),$ (5.8)

where $\hat{w}_{1}^{(k)}$ is a partial solution of (4.3), (4.4) subject to
condition (4.10), and $b_{1}$ is an arbitrary $L_{2}(\gamma)$-function. Assume
that $b_{1}=g_{0}+g$, where $g_{0}\in H_{\lambda_{0}}$ and $g\in
H_{\lambda_{0}}^{\perp}$. Then problem (4.2), (4.5) for $v_{1}$ admits
solutions

$v_{1}=\phi+\psi+\alpha_{1}\cdot\mathbf{V},\quad\alpha_{1}\in\mathbb{R}^{K},$
(5.9)

where $\phi$ and $\psi$ solve the problems

$\displaystyle-\Delta\phi+a\phi=\lambda_{0}\rho\phi\text{ \ in
}\Omega\setminus\gamma,\quad\ell\phi=0\text{ \ on
}\partial\Omega,\quad\phi=y(\pm 1)\,g\text{ \ on }\gamma_{\pm};$
$\displaystyle\begin{aligned}
&-\Delta\psi+a\psi=\lambda_{0}\rho\psi+\lambda_{1}^{(k)}\rho\left(\hat{v}_{0}^{(k)}+\alpha_{0}\cdot\mathbf{V}\right)\text{
\ in }\Omega\setminus\gamma,\quad\ell\psi=0\text{ \ on }\partial\Omega,\\\
&\phantom{-}\psi^{\pm}=\hat{w}^{(k)}_{1}(\,\cdot\,,\pm
1)\mp\partial_{r}\hat{v}_{0}^{(k)}+y(\pm
1)\,g_{0}\mp\alpha_{0}\cdot\partial_{r}\mathbf{V}\text{ \ on
}\gamma\end{aligned}$

respectively. Since $g$ is orthogonal to $H_{\lambda_{0}}$, the first problem
is solvable and admits a solution belonging to
$\mathcal{V}_{\lambda_{0}}^{\perp}$. As for the second one, its solvability
conditions can be written in the form
$\big{(}C_{\lambda_{0}}-\lambda_{1}^{(k)}I\big{)}\alpha_{0}=f$, where
$C_{\lambda_{0}}$ is a matrix with the entries

$c_{ij}=\int_{\gamma_{+}}\partial_{r}V_{i}\>\partial_{r}V_{j}\,d\gamma+\int_{\gamma_{-}}\partial_{r}V_{i}\>\partial_{r}V_{j}\,d\gamma,\quad
i,j=1,\dots,K,$

and $f$ is a vector with the components

$f_{i}=\lambda_{1}^{(k)}\int_{\Omega}\rho\hat{v}_{0}^{(k)}V_{i}\,dx+\int_{\gamma_{+}}\big{(}\hat{w}_{1}^{(k)}-\partial_{r}\hat{v}_{0}^{(k)}+y(1)\,g_{0}\big{)}\,\partial_{r}V_{i}\,d\gamma\\\
-\int_{\gamma_{-}}\big{(}\hat{w}_{1}^{(k)}+\partial_{r}\hat{v}_{0}^{(k)}+y(-1)\,g_{0}\big{)}\,\partial_{r}V_{i}\,d\gamma,\quad
i=1,\dots,K.$

If we suppose that $\lambda_{1}^{(k)}$ is not an eigenvalue of
$C_{\lambda_{0}}$, then the solvability conditions for $\psi$ can be fulfilled
for any $g\in H_{\lambda_{0}}$. It is enough to set
$\alpha_{0}=\big{(}C_{\lambda_{0}}-\lambda_{1}^{(k)}I\big{)}^{-1}f$. Then the
problem has a solution $\phi\in\mathcal{V}_{\lambda_{0}}^{\perp}$.

Note that the vector $\alpha_{0}$ has not yet been defined, because $f$
depends on the unknown function $g_{0}$. Using representations (5.8) and (5.9)
along with the fact that $g\in H_{\lambda_{0}}^{\perp}$, we can write the
solvability condition for (4.13), (4.14) in the form

$(M_{\lambda_{0}}-\lambda_{1}^{(k)}I)g=\lambda_{2}^{(k)}b^{(k)}+\lambda_{1}^{(k)}g_{0}+h_{1}^{(k)},$
(5.10)

where $h_{1}^{(k)}$ is given by (4.17). For the equation (5.10) to be
meaningful, we need to ensure that the right hand side is orthogonal to
$H_{\lambda_{0}}$. Obviously, $b^{(k)}$ belongs to $H_{\lambda_{0}}^{\perp}$.
Next, if $\lambda_{1}^{(k)}$ is different from zero, there exists a unique
vector $g_{0}\in H_{\lambda_{0}}$ such that
$\lambda_{1}^{(k)}g_{0}+h_{1}^{(k)}\in H_{\lambda_{0}}^{\perp}$. With $g_{0}$
in hand, we can uniquely defined $f$, $\alpha_{0}$, and finally the leading
term $(v_{0}^{(k)},w_{0}^{(k)})$. The solvability condition for (5.10) has the
form
$\lambda_{2}^{(k)}=-(h_{1}^{(k)}+\lambda_{1}^{(k)}g_{0},b^{(k)})_{L_{2}(\gamma)}$.
Then there exists a unique solution $g$ satisfying the condition
$(g,b^{(k)})_{L_{2}(\gamma)}=0$. And finally, we can uniquely define
$w_{1}^{(k)}$ and $v_{1}^{(k)}$ (up to the vector $\alpha_{1}$). This process
can be continued to systematically construct all other terms
$\lambda_{j}^{(k)}$, $v_{j}^{(k)}$ and $w_{j}^{(k)}$ in the approximation
$\Lambda_{\varepsilon}^{(k)}$, $\hat{u}_{\varepsilon}^{(k)}$ of the form
(4.18), (4.19). As in the previous section, $\hat{u}_{\varepsilon}^{(k)}$ can
be improved to the element
$U_{\varepsilon}^{(k)}=\hat{u}_{\varepsilon}^{(k)}-\eta_{\varepsilon}^{(k)}$
from the domain of operator $T_{\varepsilon}$, where
$\eta_{\varepsilon}^{(k)}$ is given by (4.21).

Summarizing results of the above calculations, we obtain the following
statement.

######  Lemma 5.1.

The pairs
$(\Lambda_{\varepsilon}^{(k)},\|U_{\varepsilon}^{(k)}\|^{-1}_{L^{2}(\rho_{\varepsilon},\Omega)}U_{\varepsilon}^{(k)})$,
$k\in\mathbb{N}$, are quasimodes of $T_{\varepsilon}$ with the accuracy
$O(\varepsilon^{2})$ as $\varepsilon\to 0$ that approximate the part of
spectrum lying in a vicinity of $\lambda_{0}\in\sigma(A)\cap\sigma(B)$.

The lemma can be proved similarly to Lemma 4.3.

### Integer power asymptotics: case $w_{0}=0$

Suppose now that $\lambda_{1/2}$ and $b_{0}$ are equal to zero simultaneously.
In this case, we also construct the integer power asymptotics (4.1). Since
$w_{0}=0$, problem (4.2)–(4.5) becomes

$\displaystyle-\Delta v_{1}+av_{1}=\lambda_{0}\rho v_{1}+\lambda_{1}\rho
v_{0}\quad\text{in }\Omega\setminus\gamma,\qquad\ell v_{1}=0\quad\text{on
}\partial\Omega,$ (5.11)
$\displaystyle-\partial^{2}_{n}w_{1}=\lambda_{0}qw_{1}\text{ \ in
}\omega,\quad\partial_{n}w_{1}(\,\cdot\,,-1)=\partial_{r}v_{0}^{-},\quad\partial_{n}w_{1}(\,\cdot\,,1)=\partial_{r}v_{0}^{+},$
(5.12)
$\displaystyle\phantom{-}v_{1}^{-}=w_{1}(\,\cdot\,,-1)+\partial_{r}v_{0}^{-},\quad
v_{1}^{+}=w_{1}(\,\cdot\,,1)-\partial_{r}v_{0}^{+}.$ (5.13)

Compatibility condition (4.6) for problem (5.12) takes the form

$y(-1)\,\partial_{r}v_{0}^{-}-y(1)\,\partial_{r}v_{0}^{+}=0.$ (5.14)

If we set $v_{0}=\alpha\cdot\mathbf{V}$, then the last condition can be
written as $\alpha\cdot\Psi=0$, where
$\alpha\in\mathbb{R}^{K}\setminus\\{0\\}$, $\Psi=(\Psi_{1},\dots,\Psi_{K})$
and $\Psi_{k}$ are given by (3.10). Let $y_{*}$ be the solution of the Cauchy
problem $-y_{*}^{\prime\prime}=\lambda_{0}qy_{*}$ in $(-1,1)$, $y_{*}(-1)=0$,
$y_{*}^{\prime}(-1)=1$. Problem (5.12) admits the solution
$w_{1}(s,n)=y_{*}(n)\,\alpha\cdot\partial_{r}\mathbf{V}^{-}(s)+y(n)b_{1}(s)$,
where $b_{1}$ is an arbitrary $L_{2}(\gamma)$-function. Obviously,
$\partial_{n}w_{1}(\,\cdot\,,-1)=\partial_{r}v_{0}^{-}$. The Lagrange identity
yields $(yy_{*}^{\prime}-y^{\prime}y_{*})|_{-1}^{1}=0$, and hence,
$y_{*}^{\prime}(1)=y(-1)/y(1)$. Then condition (5.14) becomes
$\alpha\cdot\partial_{r}\mathbf{V}^{+}=y_{*}^{\prime}(1)\,\alpha\cdot\partial_{r}\mathbf{V}^{-}$,
and we have

$\partial_{n}w_{1}(\,\cdot\,,1)=y_{*}^{\prime}(1)\,\alpha\cdot\partial_{r}\mathbf{V}^{-}=\alpha\cdot\partial_{r}\mathbf{V}^{+}=\partial_{r}v_{0}^{+}.$

Next, we can rewrite (5.11) and (5.13) in the form

$\displaystyle-\Delta v_{1}+av_{1}=\lambda_{0}\rho
v_{1}+\lambda_{1}\rho\,\alpha\cdot\mathbf{V}\;\;\text{in
}\Omega\setminus\gamma,\quad\ell v_{1}=0\;\;\text{on }\partial\Omega,$ (5.15)
$\displaystyle\quad
v_{1}^{-}=\alpha\cdot\partial_{r}\mathbf{V}^{-}+y(-1)b_{1},\quad
v_{1}^{+}=(y_{*}(1)-y_{*}^{\prime}(1))\,\alpha\cdot\partial_{r}\mathbf{V}^{+}+y(1)b_{1}.$
(5.16)

To achieve solvability of the problem one needs to choose proper vectors
$\alpha$ along with the parameter $\lambda_{1}$. Multiplying the equation in
(5.15) by $V_{1},\dots,V_{K}$ in turn and integrating by parts twice in view
of the boundary conditions (5.16) yield the equation
$L\alpha=\lambda_{1}\alpha$, where the symmetric matrix
$L=(l_{ij})_{i,j=1}^{K}$ has the entries

$l_{ij}=\int_{\gamma}\big{(}(y_{*}(1)-y_{*}^{\prime}(1))\,\partial_{r}V_{i}^{+}\partial_{r}V_{j}^{+}+\partial_{r}V_{i}^{-}\partial_{r}V_{j}^{-}\big{)}\,d\gamma.$
(5.17)

Note that $L$ does not depend on $b_{1}$, because of (5.14).

Hence, the number $\lambda_{1}$ and non-zero vector $\alpha$ satisfy two
conditions $L\alpha=\lambda_{1}\alpha$ and $\alpha\cdot\Psi=0$. We must find
the eigenvectors of $L$ which are “orthogonal” to $\Psi$. The dimension of the
space $h_{\lambda_{0}}=\\{\alpha\in\mathbb{C}^{K}\colon\alpha\cdot\Psi=0\\}$
is $K-d$, since $\dim H_{\lambda_{0}}=d$. Assume that $L$ has the simple
eigenvalues $\nu_{1},\dots,\nu_{p}$ with the eigenvectors
$\alpha_{1},\dots,\alpha_{p}$ belonging to $h_{\lambda_{0}}$. Obviously,
$p\leq K-d$. For any pair $(\nu_{k},\alpha_{k})$, we can solve (5.15), (5.16)
and find

$v_{1}^{(k)}=\hat{v}_{1}^{(k)}+\alpha_{1}^{(k)}\cdot\mathbf{V}$

up to the vector $\alpha_{1}^{(k)}\in\mathbb{R}^{K}$. Also, it follows from
the first condition in (5.16) that

$b_{1}^{(k)}(s)=y(-1)^{-1}\hat{v}_{1}^{(k)}(s,-1)-\alpha_{k}\cdot\partial_{r}\mathbf{V}(s,-1),$

and now the function
$w_{1}^{(k)}=y_{*}\;\alpha_{k}\cdot\partial_{r}\mathbf{V}^{-}+y\;b_{1}^{(k)}$
is uniquely defined. We continue in this fashion obtaining $p$ quasimodes of
$T_{\varepsilon}$ of the form

$\displaystyle\Lambda_{\varepsilon}^{(k)}=\lambda_{0}+\nu_{k}\varepsilon+\lambda_{2,k}\varepsilon^{2}+\lambda_{3,k}\varepsilon^{3},$
(5.18) $\displaystyle
W_{\varepsilon}^{(k)}(x)=\begin{cases}\alpha_{k}\cdot\mathbf{V}(x)+\varepsilon
v_{1}^{(k)}(x)+\varepsilon^{2}v_{2}^{(k)}(x)+\varepsilon^{3}v_{3}^{(k)}(x)-\eta_{\varepsilon}^{(k)}(x)&\text{if
}x\in\Omega\setminus\omega_{\varepsilon},\\\
\phantom{\alpha_{k}\cdot\mathbf{V}(x)+\;}\varepsilon
w_{1}^{(k)}(s,\textstyle\frac{r}{\varepsilon})+\varepsilon^{2}w_{2}^{(k)}(s,\textstyle\frac{r}{\varepsilon})+\varepsilon^{3}w_{3}^{(k)}(s,\textstyle\frac{r}{\varepsilon})&\text{if
}x\in\omega_{\varepsilon}.\end{cases}$ (5.19)

Now we summarize the results of our calculations. The next lemma can be proved
similarly to Lemma 4.3.

######  Lemma 5.2.

The pairs
$(\Lambda_{\varepsilon}^{(k)},\|W_{\varepsilon}^{(k)}\|^{-1}_{L^{2}(\rho_{\varepsilon},\Omega)}W_{\varepsilon}^{(k)})$,
$k=1,\dots,p$, are quasimodes of $T_{\varepsilon}$ with the accuracy
$O(\varepsilon^{2})$ as $\varepsilon\to 0$ that approximate a part of spectrum
lying in a vicinity of $\lambda_{0}\in\sigma(A)\cap\sigma(B)$.

### Half-integer power asymptotics

The set of quasimodes in Lemma 5.1 does not approximate all eigenvalues of
$T_{\varepsilon}$ that converge to $\lambda_{0}$. We assume that
$\lambda_{1/2}$ in (5.2) is different from zero, and then $b_{0}=0$, by (5.7).
Recalling now (3.12), we have $v_{0}=\beta_{0}\cdot\mathbf{V}$, where
$\beta_{0}$ is an arbitrary vector in $\mathbb{R}^{K}$ such that
$\|\beta_{0}\|=1$. In this case, we will use some finite-dimensional operator
instead of $M_{\lambda_{0}}$ to split the limit multiple eigenvalue
$\lambda_{0}$. Reasoning as above we deduce that the problem (5.3)-(5.6) has a
solution of the form $w_{1/2}(s,n)=b_{1/2}(s)y(n)$,
$v_{1/2}=\hat{v}_{1/2}+\beta_{1/2}\cdot\mathbf{V}$ where
$\beta_{1/2}\in\mathbb{R}^{K}$ and $\hat{v}_{1/2}$ is a partial solution of
the problem

$\displaystyle-\Delta v_{1/2}+av_{1/2}=\lambda_{0}\rho
v_{1/2}+\lambda_{1/2}\rho v_{0}\text{ \ in }\Omega\setminus\gamma,\quad\ell
v_{1/2}=0\text{ \ on }\partial\Omega,$ (5.20)
$\displaystyle\phantom{-}v_{1/2}^{-}=y(-1)\,b_{1/2},\qquad
v_{1/2}^{+}=y(1)\,b_{1/2}$ (5.21)

that is orthogonal to $\mathcal{V}_{\lambda_{0}}$ in $L_{2}(\rho,\Omega)$. By
Proposition 3.5, the solvability conditions for (5.20), (5.21) can be written
in the vector form

$\int_{\gamma}b_{1/2}\Psi\,d\gamma+\lambda_{1/2}\beta_{0}=0.$ (5.22)

For the next terms we have

$\displaystyle-\Delta
v_{1}+av_{1}=\rho(\lambda_{0}v_{1}+\lambda_{1/2}v_{1/2}+\lambda_{1}v_{0})\text{
\ in }\Omega\setminus\gamma,\quad\ell v_{1}=0\text{ \ on }\partial\Omega,$
(5.23)
$\displaystyle-\partial^{2}_{n}w_{1}=\lambda_{0}qw_{1}+\lambda_{1/2}qw_{1/2}\quad\text{in
}\omega,$ (5.24)
$\displaystyle\phantom{-}\partial_{n}w_{1}(\,\cdot\,,-1)=\partial_{r}v_{0}^{-},\qquad\partial_{n}w_{1}(\,\cdot\,,1)=\partial_{r}v_{0}^{+},$
(5.25)
$\displaystyle\phantom{-}v_{1}^{-}=w_{1}(\,\cdot\,,-1)+\partial_{r}v_{0}^{-},\qquad
v_{1}^{+}=w_{1}(\,\cdot\,,1)-\partial_{r}v_{0}^{+}.$ (5.26)

Problem (5.24), (5.25) is solvable if
$y(1)\,\partial_{r}v_{0}^{+}-y(-1)\,\partial_{r}v_{0}^{-}+\lambda_{1/2}b_{1/2}=0.$
Since $\partial_{r}v_{0}^{\pm}=\beta_{0}\cdot\partial_{r}\mathbf{V}^{\pm}$, it
can be written in the form

$\beta_{0}\cdot\big{(}y(1)\,\partial_{r}\mathbf{V}^{+}-y(-1)\,\partial_{r}\mathbf{V}^{-}\big{)}+\lambda_{1/2}b_{1/2}=0.$

Using notation (3.10), we have

$\beta_{0}\cdot\Psi+\lambda_{1/2}b_{1/2}=0.$ (5.27)

Multiplying this equality by $\Psi^{\top}$, integrating over $\gamma$ and
recalling (5.22), we finally discover
$\bigl{(}G_{\lambda_{0}}-\lambda_{1/2}^{2}\bigr{)}\,\beta_{0}=0$, where
$G_{\lambda_{0}}$ is the Gram matrix of $\Psi_{1}$,…,$\Psi_{K}$. This matrix
is semi-positive and its rank is equal to the dimension of $H_{\lambda_{0}}$.

Suppose $\omega^{2}$ is a positive simple eigenvalue of $G_{\lambda_{0}}$ with
the eigenvector $\beta_{0}$. So there exist two different correctors
$\lambda_{1/2}=\omega$ and $\lambda_{1/2}=-\omega$ in asymptotics (5.1) with
the same leading term $v_{0}=\beta_{0}\cdot\mathbf{V}$ in approximation (5.2).
First assume that $\lambda_{1/2}=\omega$. From (5.27), we have
$b_{1/2}=-\omega^{-1}\beta_{0}\cdot\Psi$. Up to a function $b_{1}$, we can
find $w_{1}(s,n)=\hat{w}_{1}(s,n)+b_{1}(s)y(n)$, where $\hat{w}_{1}$ is a
partial solution of (5.24), (5.25) subject to the condition
$(\hat{w}_{1},y)_{L_{2}(q,(-1,1))}=0$. Next, problem (5.23), (5.26) is
solvable if and only if
$\int_{\gamma}b_{1}\Psi\,d\gamma+\omega\beta_{1/2}+\lambda_{1}\beta_{0}=h$,
where

$h=\int_{\gamma_{+}}\big{(}\hat{w}_{1}-\partial_{r}v_{0}\big{)}\,\partial_{r}\mathbf{V}_{k}\,\,d\gamma-\int_{\gamma_{-}}\big{(}\hat{w}_{1}+\partial_{r}v_{0}\big{)}\,\partial_{r}\mathbf{V}\,d\gamma-\omega\int_{\Omega}\rho\hat{v}_{1/2}\mathbf{V}_{k}\,dx.$

Also, a solution of the problem

$\displaystyle-\partial^{2}_{n}w_{3/2}=q(\lambda_{0}w_{3/2}+\omega
w_{1}+\lambda_{1}w_{1/2})\quad\text{in }\omega,$
$\displaystyle\quad\partial_{n}w_{3/2}(\,\cdot\,,-1)=\partial_{r}v_{1/2}^{-},\quad\partial_{n}w_{3/2}(\,\cdot\,,1)=\partial_{r}v_{1/2}^{+}.$

exists if and only if

$y(-1)\,\partial_{r}v_{1/2}^{-}-y(1)\,\partial_{r}v_{1/2}^{+}=\omega
b_{1}+\lambda_{1}b_{1/2}.$ (5.28)

Reasoning as in the previous step, we can rewrite this condition in the form

$\bigl{(}G_{\lambda_{0}}-\omega^{2}I\bigr{)}\beta_{1/2}=2\omega\lambda_{1}\beta_{0}-\omega
h+f,$

where

$f=y(-1)\,\int_{\gamma_{-}}\partial_{r}\hat{v}_{1/2}\Psi\,d\gamma-y(1)\,\int_{\gamma_{+}}\partial_{r}\hat{v}_{1/2}\Psi\,d\gamma.$

Since $\omega^{2}$ is an eigenvalue of $G_{\lambda_{0}}$, the system admits a
solution $\beta_{1/2}$ if and only if

$\lambda_{1}=\tfrac{1}{2\omega}\,\beta_{0}\cdot(\omega h-f).$ (5.29)

Although the unit vector $\beta_{0}$ is defined up to the change of sign,
$\lambda_{1}$ is uniquely defined by (5.29), because the transformation
$\beta_{0}\mapsto-\beta_{0}$ implies that the vectors $h$ and $f$ also change
their sings. We fix this solution such that $\beta_{1/2}\cdot\beta_{0}=0$.
Then

$b_{1}=\tfrac{1}{\omega}\,\Big{(}y(-1)\,\partial_{r}\hat{v}_{1/2}^{-}-y(1)\,\partial_{r}\hat{v}_{1/2}^{+}-\beta_{1/2}\cdot\Psi-\lambda_{1}b_{1/2}\Big{)},$

by (5.28). Assuming that $\lambda_{1/2}^{+}=\omega$ we have calculated
$\lambda_{1}^{+}$, $w_{1/2}^{(+)}$, $w_{1}^{(+)}$ and $v_{1/2}^{(+)}$ in
(5.1), (5.2). We can continue in this fashion obtaining the next terms of the
asymptotics. Taking $\lambda_{1/2}^{-}=-\omega$ we can compute analogously the
terms $\lambda_{k}^{-}$, $w_{k}^{(-)}$ and $v_{k}^{(-)}$ in the asymptotics of
other eigenvalue and eigenfunction. A simple analysis of the foregoing
formulas shows that $w_{1/2}^{(-)}=-w_{1/2}^{(+)}$ and
$v_{1/2}^{(-)}=-v_{1/2}^{(+)}$. Moreover, by construction, we have

$w_{1/2}^{(\pm)}(s,n)=\mp\omega^{-1}\,y(n)\,\beta_{0}\cdot\Psi(s),\qquad
v_{1/2}^{(\pm)}(x)=\pm v_{*}(x),$

where $v_{*}$ is a solution of the problem

$-\Delta v+av=\lambda_{0}\rho v+\omega\rho\,\beta_{0}\cdot\mathbf{V}\text{ in
}\Omega\setminus\gamma,\quad\ell v=0\text{ on }\partial\Omega,\quad
v=-\omega^{-1}y(\pm 1)\,\beta_{0}\cdot\Psi\text{ \ on }\gamma_{\pm}$

that is orthogonal to $\mathcal{V}_{\lambda_{0}}$.

Let us summarize the above considerations in the following lemma.

######  Lemma 5.3.

Let $d$ be the dimension of $H_{\lambda_{0}}$, where
$\lambda_{0}\in\sigma(A)\cap\sigma(B)$. Suppose that all non-zero eigenvalues
$\omega_{1}^{2},\dots,\omega_{d}^{2}$ of the matrix $G_{\lambda_{0}}$ are
simple and $\beta_{0,1},\dots,\beta_{0,d}$ are the corresponding normalized
eigenvectors. Then the operator $T_{\varepsilon}$ possesses $d$ pairs of the
quasimodes
$(\hat{\mu}_{\varepsilon,j}^{\pm},n^{\pm}_{\varepsilon,j}V^{(\pm)}_{\varepsilon,j})$,
$j=1,\ldots,d$, with the accuracy $O(\varepsilon^{3/2})$ as $\varepsilon\to
0$, where $n^{\pm}_{\varepsilon,j}$ is a normalizing factor and

$\displaystyle\hat{\mu}_{\varepsilon,j}^{\pm}=\lambda_{0}\pm\varepsilon^{1/2}\omega_{j}+\sum_{k=2}^{6}\varepsilon^{k/2}\lambda_{k/2,j}^{\pm},$
$\displaystyle
V^{(\pm)}_{\varepsilon,j}=\beta_{0,j}\cdot\mathbf{V}\pm\varepsilon^{1/2}v_{*,j}+\sum_{k=2}^{6}\varepsilon^{k/2}v_{k/2,j}^{(\pm)}-\eta^{(\pm)}_{\varepsilon,j}\text{
\ in }\Omega\setminus\omega_{\varepsilon},$ $\displaystyle
V^{(\pm)}_{\varepsilon,j}=\mp\varepsilon^{1/2}\omega_{j}^{-1}y(\tfrac{r}{\varepsilon})\,\beta_{0,j}\cdot\Psi(s)+\sum_{k=2}^{6}\varepsilon^{k/2}w_{k/2,j}^{(\pm)}(s,\tfrac{r}{\varepsilon})\text{
\ in }\omega_{\varepsilon}.$

The small correctors $\eta^{(\pm)}_{\varepsilon,j}$ are defined as in (4.20)
with $V^{(\pm)}_{\varepsilon,j}$ in place of $\hat{u}_{\varepsilon}^{(k)}$.

###### Proof.

The proof differs from the proof of Lemma 4.3 only by estimate (4.22). The
approximations to eigenvalues and eigenfunctions have been constructed up to
order $O(\varepsilon^{3})$ and therefore the remainder term $F_{\varepsilon}$
(with the notation of Lemma 4.3) can be also estimated by
$\|F_{\varepsilon}\|_{L^{2}(\rho_{\varepsilon},\Omega)}\leq
c_{1}\varepsilon^{3/2}$. But the leading term $w_{0}$ in this asymptotics is
equal to zero and hence the $L^{2}(\rho_{\varepsilon},\Omega)$-norm of
$V^{(\pm)}_{\varepsilon,j}$ is bounded uniformly with respect to
$\varepsilon$. Then the normalizing factor $n^{\pm}_{\varepsilon,j}$ tends to
some positive number as $\varepsilon\to 0$ and

$\|(T_{\varepsilon}-\hat{\mu}_{\varepsilon,j}^{\pm}I)(n^{\pm}_{\varepsilon,j}V^{(\pm)}_{\varepsilon,j})\|_{L^{2}(\rho_{\varepsilon},\Omega)}=n^{\pm}_{\varepsilon,j}\|F_{\varepsilon}\|_{L^{2}(\rho_{\varepsilon},\Omega)}\leq
c_{2}\varepsilon^{3/2},$

which is the desired conclusion. ∎

In view of Lemmas 5.1–5.3, we can prove the following result in the same way
as Theorem 4.4.

######  Theorem 5.4.

Suppose that the set $\sigma(A)\cap\sigma(B)$ is non empty and $\lambda_{0}$
is an eigenvalue of the limit operator $\mathcal{P}$ belonging to this
intersection.

  * (i)

Assume the spectrum $\\{\lambda_{1}^{(k)}\\}_{k\in\mathbb{N}}$ of the operator
$M_{\lambda_{0}}$ is simple. Then there exists a countable set of eigenvalues
$\lambda_{\varepsilon,k}$, $k\in\mathbb{N}$, in the spectrum of
$T_{\varepsilon}$ that admit the asymptotics

$\lambda_{\varepsilon,k}=\lambda_{0}+\varepsilon\lambda_{1}^{(k)}+O(\varepsilon^{2}),\quad\text{as
}\varepsilon\to 0.$

  * (ii)

Assume $\dim H_{\lambda_{0}}=d$ and all positive eigenvalues $\omega_{j}^{2}$
of the matrix $G_{\lambda_{0}}$ are simple. Then operator $T_{\varepsilon}$
possesses $2d$ eigenvalues with the asymptotics

$\displaystyle\mu_{\varepsilon,j}^{-}=\lambda_{0}-\varepsilon^{1/2}\omega_{j}+\varepsilon\lambda_{1,j}^{-}+O(\varepsilon^{3/2}),$
$\displaystyle\mu_{\varepsilon,j}^{+}=\lambda_{0}+\varepsilon^{1/2}\omega_{j}+\varepsilon\lambda_{1,j}^{+}+O(\varepsilon^{3/2}),\quad
j=1,\dots,d,$

as $\varepsilon\to 0$, where $\lambda_{1,j}^{\pm}$ can be calculated from
(5.29).

  * (iii)

Also, the operator $T_{\varepsilon}$ can have no more than $K-d$ eigenvalues
of possessing the asymptotics

$\lambda^{\varepsilon}_{i}=\lambda_{0}+\nu_{i}\varepsilon+O(\varepsilon^{2}),\quad\text{as
}\varepsilon\to 0,$

where $\nu_{i}$ are simple non-zero eigenvalues of the matrix in (5.17).

In this case, the generalized eigenspace $X_{\lambda_{0}}$ contains the Jordan
chains. Note that problem (5.3)–(5.6) for $(v_{1/2},w_{1/2})$ coincides up to
the multiplier $\lambda_{1/2}$ in the right-hand side with problem
(3.13)–(3.15) for generalized eigenvectors. Inspecting the structure of
$V^{(\pm)}_{\varepsilon,j}$ more closely we see that

$V^{(\pm)}_{\varepsilon,j}=U_{j}\pm\varepsilon^{1/2}U^{*}_{j}+O(\varepsilon),\text{
as }\varepsilon\to 0,$

where the vectors $U_{j}=(\beta_{0,j}\cdot\mathbf{V}(x),0)$,
$U^{*}_{j}=\big{(}v_{*,j}(x),-\omega_{j}^{-1}y(n)\,\beta_{0,j}\cdot\Psi(s)\big{)}$
form a Jordan chain of $\mathcal{P}$ corresponding to $\lambda_{0}$. This
observation has the following geometric interpretation. The operator
$T_{\varepsilon}$ possesses pairs of eigenvalues with the asymptotics
$\mu_{\varepsilon,j}^{\pm}=\lambda_{0}\pm\omega_{j}\varepsilon^{1/2}+O(\varepsilon)$
for which the corresponding normalized eigenfunctions
$u_{\varepsilon,j}^{(-)}$ and $u_{\varepsilon,j}^{(+)}$ converge in
$L_{2}(\Omega)$ to the same function $v_{0,j}=\beta_{0,j}\cdot\mathbf{V}$, as
$\varepsilon\to 0$. Although these eigenfunctions remain orthogonal for all
$\varepsilon$ in the weighted space $L_{2}(\rho_{\varepsilon},\Omega)$, they
make an infinitely small angle between them in $L_{2}(\Omega)$ with the
standard norm, and stick together at the limit. In particular, it leads to the
loss of completeness in $L_{2}(\Omega)$ for the limit eigenfunction
collection. Interestingly enough, however, the plane $\pi_{\varepsilon,j}$
that is the span of $u_{\varepsilon,j}^{(-)}$ and $u_{\varepsilon,j}^{(+)}\\}$
has regular behaviour as $\varepsilon\to 0$. The limit position of
$\pi_{\varepsilon,j}$ is the 2-dimensional space $\pi_{j}$ spanned by the
functions $\beta_{0,j}\cdot\mathbf{V}$ and $v_{*,j}$.

We actually have an example of singular perturbations in which the
completeness property of perturbed eigenfunction collection passes in some
sense into the completeness of generalized eigenfunctions of the limit non-
self-adjoint operator. Although we didn’t justify the asymptotics of
eigenfunctions of (1.2), we can formally state that the non-self-adjoint
operator $\mathcal{P}$ contains all the information about the asymptotic
behaviour of eigenvalues of the perturbed problem:

  * $\circ$

the spectrum of $\mathcal{P}$ is a limit set for the spectra
$\sigma(T_{\varepsilon})$ as $\varepsilon\to 0$;

  * $\circ$

knowing the multiplicity of eigenvalues of $\mathcal{P}$, we can divide
$\sigma(T_{\varepsilon})$ into finite or infinite subsets of eigenvalues with
the same limits as $\varepsilon\to 0$;

  * $\circ$

in the case $\lambda\in\sigma(A)\cap\sigma(B)$, the dimension of space
$H_{\lambda}$ indicates how many eigenvalues of $T_{\varepsilon}$ possess the
half-integer power asymptotics;

  * $\circ$

the Jordan chains of $\mathcal{P}$ are involved in the quasimodes of
$T_{\varepsilon}$ (in the formal asymptotics of eigenfunctions of
$T_{\varepsilon}$).

Figure 4. Bifurcation of the eigenvalue
$\lambda_{0}\in\sigma(A)\cap\sigma(B)$.

As we pointed out in the introduction, the problem

$-\Delta
u_{\varepsilon}=\lambda^{\varepsilon}\rho_{\varepsilon}u_{\varepsilon}\quad\text{in
}\Omega,\qquad u_{\varepsilon}=0\quad\text{on }\partial\Omega$ (5.30)

with the perturbed density $\rho_{\varepsilon}(x;m)=\rho(x)$ for
$x\in\Omega\setminus\omega_{\varepsilon}$ and
$\rho_{\varepsilon}(x;m)=\varepsilon^{-m}q_{\varepsilon}(x)$ for
$x\in\omega_{\varepsilon}$, the Dirichlet boundary condition and without the
potential $a$ in the differential equation has been considered in [37]. The
variational methods have been applied to study the asymptotic behaviour of
eigenvalues and eigenfunctions. In this case the problem can be realized as
the family of bounded self-adjoint operators $A_{\varepsilon}$ in
$\mathring{W}^{1}_{2}(\Omega)=\\{v\in W^{1}_{2}(\Omega)\colon
u_{\varepsilon}=0\text{ on }\partial\Omega\\}$. Due to the second
representation theorem [43, Theorem VI.2.23], $A_{\varepsilon}$ is defined by
the identity
$\mathfrak{a}(A_{\varepsilon}u,\phi)=(\rho_{\varepsilon}u,\phi)_{L_{2}(\Omega)}$
for all $\phi\in\mathring{W}^{1}_{2}(\Omega)$, because the Dirichlet form
$\mathfrak{a}(u,\phi)=\int_{\Omega}\nabla u\cdot\nabla\phi\,dx$ is a scalar
product in $\mathring{W}^{1}_{2}(\Omega)$. Problem (5.30) can be written in
the form
$A_{\varepsilon}u_{\varepsilon}=\lambda_{\varepsilon}^{-1}u_{\varepsilon}$.
For $m>1$ the operators $A_{\varepsilon}$ diverge as $\varepsilon\to 0$, and
it has been shown that $\|A_{\varepsilon}\|\leq c\varepsilon^{1-m}$ with a
constant $c$ being independent of $\varepsilon$.

The complete asymptotic analysis of eigenvalues $\lambda_{\varepsilon}$ of
(5.30) has been carried out for $m=3$. We have proved that the spectrum
consists of a countable number of infinite series of eigenvalues with the
asymptotics

$\lambda^{\varepsilon}_{kj}=\mu_{k}\varepsilon+\nu_{kj}\varepsilon^{2}+o(\varepsilon^{2})\quad\text{as
}\varepsilon\to 0,\quad k,j\in\mathbb{N},$ (5.31)

where $\mu_{k}$ is an infinite-fold eigenvalue of $B$ and
$\\{\nu_{kj}\\}_{j=1}^{\infty}$ is a spectrum of pseudodifferential operator
$N_{\mu_{k}}$ acting in $L_{2}(\gamma)$ (see formula (4.9)). The operator
$N_{\mu_{k}}$ is a self-adjoint realization of a boundary value problem for
the Laplace operator in $\Omega$ with coupling conditions on $\gamma$.

Since the first eigenvalue $\mu_{1}$ of $B$ is equal to zero, the low-lying
eigenvalues admit the asymptotics
$\lambda^{\varepsilon}_{1j}=\varepsilon^{2}(\nu_{1j}+o(1))$, $\varepsilon\to
0$. In this case, $\\{\nu_{1j}\\}_{j=1}^{\infty}$ are eigenvalues of the
spectral problem $-\Delta u=\nu q_{0}\delta_{\gamma}u$ in $\Omega$, $u=0$ on
$\partial\Omega$, where $q_{0}=\int_{-1}^{1}q(r)\,dr$ and $\delta_{\gamma}$ is
a distribution in $\mathcal{D}^{\prime}(\Omega)$ acting as
$\langle\delta_{\gamma},\phi\rangle=\int_{\gamma}\phi\,d\gamma$ for any
$\phi\in C^{\infty}(\Omega)$. The problem describes the eigenvibrations of a
membrane with the mass density $\rho_{0}(x)=q_{0}\delta_{\gamma}(x)$, i.e.,
the whole mass of the membrane is supported on $\gamma$ and the rest part of
vibrating system is massless. Such series of infinitesimal eigenvalues with
the asymptotics $\lambda^{\varepsilon}_{1j}=\varepsilon^{m-1}(\nu_{1j}+o(1))$,
$\varepsilon\to 0$, exists for any $m>1$. Hence, the estimate
$\|A_{\varepsilon}\|\leq c\varepsilon^{1-m}$ is sharp. The similar result is
obtained in Theorems 4.4 and 5.4 for problem (1.2) when $m=2$ (see Corollary
5.5).

In [37] some results on the convergence of eigenfunctions have been also
obtained. Interestingly enough, the complete asymptotic description of
$\lambda^{\varepsilon}$ and $u_{\varepsilon}$ in (5.30) includes not only the
operator $B$ but also the operator $A$ which is associated with the problem
$-\Delta u=\lambda u$ in $\Omega\setminus\gamma$, $u=0$ on $\partial\Omega$
and $\gamma$ in this case. In view of (5.31), any point of the positive
spectral half-line is an accumulation point for eigenvalues
$\lambda^{\varepsilon}$, i.e., for each $\lambda>0$ there exists a sequence
$\\{\lambda^{\varepsilon_{i}}\\}_{i=1}^{\infty}$ of eigenvalues such that
$\lambda^{\varepsilon_{i}}\to\lambda$ as $\varepsilon_{i}\to 0$. We have
proved that only the points of $\sigma(A)$ can be approximated by eigenvalues
$\lambda^{\varepsilon_{i}}$ so that the corresponding eigenfunctions
$u_{\varepsilon_{i}}$ converge to nontrivial limits in $L^{2}(\Omega)$. These
limits are eigenfunctions of $A$ corresponding to $\lambda$.

######  Corollary 5.5.

Problem (1.2) has a series of small eigenvalues that admit the asymptotics
$\lambda_{\varepsilon,k}=\varepsilon\nu_{k}+O(\varepsilon^{2})$ as
$\varepsilon\to 0$, where $\nu_{k}$ are eigenvalues of the problem

$-\Delta v+av=0\text{ in }\Omega\setminus\gamma,\quad\ell v=0\text{ on
}\partial\Omega,\quad[v]_{\gamma}=0,\;\;[\partial_{r}v]_{\gamma}+\nu
q_{0}v=0\text{ on }\gamma$ (5.32)

with the spectral parameter $\nu$ in the coupling conditions. In addition, if
$0\in\sigma(A)$, then (1.2) has also a finite number of small eigenvalues with
asymptotics

$\mu_{\varepsilon,j}^{\pm}=\pm\omega_{j}\sqrt{\varepsilon}+O(\varepsilon)\text{
as }\varepsilon\to 0,\quad j=1,\dots,d.$

###### Proof.

The existence of such eigenvalues follows from Theorems 4.4 and 5.4 along with
the observation that $\lambda_{0}=0$ belongs to $\sigma(B)$ (see Remark 3.8).
It remains to derive the eigenvalue problem for $\nu_{k}$. Let us look at
(2.4)–(2.6) and put $\lambda=0$:

$\displaystyle-\Delta v+av=0\;\;\text{in }\Omega\setminus\gamma,\quad\ell
v=0\;\;\text{on }\partial\Omega,\quad v^{-}=w(\,\cdot\,,-1),\quad
v^{+}=w(\,\cdot\,,1),\;\;\text{on }\gamma,$
$\displaystyle\partial^{2}_{n}w=0\>\text{ in
}\omega,\quad\partial_{n}w(\,\cdot\,,-1)=0,\;\>\partial_{n}w(\,\cdot\,,1)=0,$

We see that $w=b(s)$ and hence $v^{-}=v^{+}$, i.e., $v$ is continuous on
$\gamma$. Since $y=1$ in this case, condition (4.6) for $v_{0}=v$ reads as
$\partial_{r}v^{+}-\partial_{r}v^{-}+\lambda_{1}q_{0}b=0$ on $\gamma$. To
complete the proof we replace $\lambda_{1}$ with $\nu$ and recall that
$b=v|_{\gamma}$. Note that problem (5.32) can be written in the form

$-\Delta v+av=\nu q_{0}\delta_{\gamma}v\text{ in }\Omega,\quad\ell v=0\text{
on }\partial\Omega.$

Hence, $\nu_{k}$ are eigenvalues of a membrane with the mass density
$\rho_{0}(x)=q_{0}\delta_{\gamma}(x)$ supported by the curve $\gamma$. ∎

##  6\. Asymptotics of eigenvalues in the case
$\lambda_{0}\in\sigma(A)\setminus\sigma(B)$

For the sake of completeness, we briefly discuss the perturbation of
eigenvalue $\lambda_{0}$ of $\mathcal{P}$ which does not belong to
$\sigma(B)$. In view of part (i) of Theorem 3.7, $\lambda_{0}$ is an
eigenvalue of finite multiplicity. Also, we have
$v_{0}=\alpha_{0}\cdot\mathbf{V}$ and $w_{0}=0$ in asymptotics (4.1), where
$\alpha_{0}\in\mathbb{R}^{K}\setminus\\{0\\}$. Then (4.3) and (4.4) imply

$-\partial^{2}_{n}w_{1}=\lambda_{0}qw_{1}\;\;\text{in
}\omega,\quad\partial_{n}w_{1}(\,\cdot\,,-1)=\alpha_{0}\cdot\partial_{r}\mathbf{V}^{-},\;\;\partial_{n}w_{1}(\,\cdot\,,1)=\alpha_{0}\cdot\partial_{r}\mathbf{V}^{+}.$

Since $\lambda_{0}\not\in\sigma(B)$, there exists a unique solution of the
problem. Suppose the functions $W_{k}$, $k=1,\dots,K$, solve the problems

$-\partial^{2}_{n}w=\lambda_{0}qw\;\;\text{in
}\omega,\quad\partial_{n}w(\,\cdot\,,-1)=\partial_{r}V_{k}^{-},\;\;\partial_{n}w_{1}(\,\cdot\,,1)=\partial_{r}V_{k}^{+},$

and $\mathbf{W}=(W_{1},\dots,W_{K})$. Thus we have
$w_{1}=\alpha_{0}\cdot\mathbf{W}$. Next, we can rewrite (4.2) and (4.5) in the
form

$\displaystyle-\Delta v_{1}+av_{1}=\lambda_{0}\rho
v_{1}+\lambda_{1}\rho\,\alpha_{0}\cdot\mathbf{V}\;\;\text{in
}\Omega\setminus\gamma,\quad\ell v_{1}=0\;\;\text{on }\partial\Omega,$ (6.1)
$\displaystyle\quad
v_{1}^{-}=\alpha_{0}\cdot(\mathbf{W}(\,\cdot\,,-1)+\partial_{r}\mathbf{V}^{-}),\quad
v_{1}^{+}=\alpha_{0}\cdot(\mathbf{W}(\,\cdot\,,1)-\partial_{r}\mathbf{V}^{+}).$
(6.2)

In general, the problem is unsolvable, because $\lambda_{0}$ belongs to the
spectrum of $A$ which is the direct sum of $A_{-}$ and $A_{+}$. To achieve the
solvability one needs to choose proper vectors $\alpha_{0}$ along with the
parameter $\lambda_{1}$. Multiplying the equation in (6.1) by
$V_{1},\dots,V_{K}$ in turn and integrating by parts twice in view of the
boundary conditions (6.2) yield the spectral matrix equation
$R\alpha_{0}=\lambda_{1}\alpha_{0}$, where the matrix $R=(r_{ij})_{i,j=1}^{K}$
has the entries

$r_{ij}=\int_{\gamma}\left((W_{i}(\,\cdot\,,1)-\partial_{r}V_{i}^{+})\partial_{r}V_{j}^{+}+(W_{i}(\,\cdot\,,-1)+\partial_{r}V_{i}^{-})\partial_{r}V_{j}^{-}\right)\,d\gamma.$

The matrix $R$ is symmetric, because it is easy to check that

$\int_{\gamma}W_{i}(\,\cdot\,,\pm
1)\partial_{r}V_{j}^{\pm}\,d\gamma=\int_{\gamma}W_{j}(\,\cdot\,,\pm
1)\partial_{r}V_{i}^{\pm}\,d\gamma.$

Assume that $R$ has $K$ simple eigenvalues
$\lambda_{1}^{(1)},\dots,\lambda_{1}^{(K)}$ with the corresponding eigenvalues
$\alpha_{0}^{(1)},\dots,\alpha_{0}^{(K)}$. For any pair
$(\lambda_{1}^{(k)},\alpha_{0}^{(k)})$, we can solve (6.1), (6.2) and find
$v_{1}^{(k)}=\hat{v}_{1}^{(k)}+\alpha_{1}^{(k)}\cdot\mathbf{V}$ up to the
vector $\alpha_{1}^{(k)}\in\mathbb{R}^{K}$. We continue in this fashion
obtaining $K$ different quasimodes with high enough accuracy of the operator
$T_{\varepsilon}$ that approximate the part of spectrum lying in a vicinity of
$\lambda_{0}$.

######  Theorem 6.1.

Suppose that $\lambda_{0}$ is an eigenvalue of the limit operator
$\mathcal{P}$ such that $\lambda_{0}\in\sigma(A)\setminus\sigma(B)$. Assume
$\lambda_{0}$ has multiplicity $K$ and the matrix $R$ possesses the simple
eigenvalues $\lambda_{1}^{(1)},\dots,\lambda_{1}^{(K)}$. Then there exist $K$
eigenvalues in the spectrum of $T_{\varepsilon}$ that converge to
$\lambda_{0}$ and admit the asymptotics
$\lambda_{\varepsilon,k}=\lambda_{0}+\varepsilon\lambda_{1}^{(k)}+O(\varepsilon^{2})$
as $\varepsilon\to 0$, $k=1,\dots,K$.

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