$|\varepsilon(z-\mu_{i})^{r_{i,\tau}+1}Y_{3}(z)+\varepsilon^{2}Y_{4}(z,\varepsilon)|\leqslant
C(\tilde{b}^{r_{i,\tau}+1}\varepsilon^{2+\frac{1}{r_{i,\tau}}}+\varepsilon^{2})<C\varepsilon^{2},$
(5.15)

where $C$ is some constant independent of $\varepsilon$ and $\tilde{b}$. Since
$q_{i}-r_{i,\tau}\geqslant r_{i,\tau}+1$, it is also straightforward to check
that for sufficiently small $\varepsilon$ and sufficiently large $\tilde{b}$,
all zeroes of the function
$z\mapsto(z-\mu_{i})^{q_{i}}+\varepsilon(z-\mu_{i})^{r_{i,\tau}}\gamma_{i,\tau}$
belong to the ball $T_{\tilde{b}\varepsilon^{\frac{1}{r_{i,\tau}}}}(\mu_{i})$.
Taking this fact and estimates (5.14), (5.15) into consideration, we apply the
Rouché theorem and arrive at asymptotic expansion (2.18).

Suppose that $q_{i}-2r_{i,\tau}\geqslant 1$. The zeroes of the function
$z\mapsto(z-\mu_{i})^{q_{i}}+\varepsilon(z-\mu_{i})^{r_{i,\tau}}\gamma_{i,\tau}$
can be found explicitly. The first of them is a $r_{i,\tau}$-multiple zero
$z_{0}=\mu_{i}$ and $q_{i}-r_{i,\tau}$ simple zeroes

$z_{j}(\varepsilon):=\mu_{i}-\varepsilon^{\frac{1}{q_{i}-r_{i,\tau}}}(-\gamma_{i,\tau})^{\frac{1}{q_{i}-r_{i,\tau}}}e^{\frac{2\pi\mathrm{i}}{q_{i}-r_{i,\tau}}j},\qquad
j=1,\ldots,q_{i}-r_{i,\tau}.$ (5.16)

First we consider a ball
$T_{\tilde{b}\varepsilon^{\frac{1}{r_{i,\tau}}}}(\mu_{i})$ and we observe that
under our assumption, $q_{i}-r_{i,\tau}\geqslant r_{i,\tau}+1$. Hence, for
sufficiently small $\varepsilon$, the ball
$T_{\tilde{b}\varepsilon^{\frac{1}{r_{i,\tau}}}}(\mu_{i})$ does not contain
the zeroes $z_{j}(\varepsilon)$. Moreover, on the boundary of this ball, the
estimate holds true:

$\displaystyle|(z-\mu_{i})^{q_{i}}+\varepsilon(z-\mu_{i})^{r_{i,\tau}}\gamma_{i,\tau}|\geqslant$
$\displaystyle\varepsilon|\gamma_{i,\tau}||(z-\mu_{i})^{r_{i,\tau}}|-|(z-\mu_{i})^{q_{i}}|$
$\displaystyle=$
$\displaystyle\tilde{b}^{r_{i,\tau}}\varepsilon^{2}|\gamma_{i,\tau}|-\tilde{b}^{q_{i}}\varepsilon^{2+\frac{q_{i}-2r_{i,\tau}}{r_{i,\tau}}}>\frac{|\gamma_{i,\tau}|}{2}\tilde{b}^{r_{i,\tau}}\varepsilon^{2}.$

We also observe that estimate (5.15) remains true in the considered case. Then
we apply Rouché theorem to the ball
$T_{\tilde{b}\varepsilon^{\frac{1}{r_{i,\tau}}}}(\mu_{i})$ and conclude that
this ball contains exactly $r_{i,\tau}$ zeroes of equation (5.13) counting
their orders. This implies asymptotic expansion (2.19).

Now we consider another ball
$T_{\tilde{b}\varepsilon^{\frac{2}{q_{i}-r_{i,\tau}}}}(z_{j}(\varepsilon))$
for $z_{j}$ defined in (5.16) with some fixed $j$. For sufficiently small
$\varepsilon$, this ball does not contain the origin and other points $z_{s}$
with $s\neq j$. On the boundary of this ball, we have the following estimate:

$\displaystyle|(z-\mu_{i})^{q_{i}}+\varepsilon(z-\mu_{i})^{r_{i,\tau}}\gamma_{i,\tau}|\geqslant$
$\displaystyle
C\varepsilon^{\frac{r_{i,\tau}}{q_{i}-r_{i,\tau}}}|(z-\mu_{i})^{q_{i}-r_{i,\tau}}+\varepsilon\gamma_{i,\tau}|$
(5.17) $\displaystyle=$ $\displaystyle
C\varepsilon^{\frac{r_{i,\tau}}{q_{i}-r_{i,\tau}}}\prod\limits_{s=1}^{q_{i}-r_{i,\tau}}|z-z_{s}|\geqslant
C\tilde{b}\varepsilon^{\frac{q_{i}+1}{q_{i}-r_{i,\tau}}},$

where $C$ is some constant independent of $\varepsilon$ and $\tilde{b}$. We
observe that under our assumptions

$\frac{q_{i}+1}{q_{i}-r_{i,\tau}}\leqslant 2.$

In view of the latter inequality, on the boundary of the ball
$T_{\tilde{b}\varepsilon^{\frac{2}{q_{i}-r_{i,\tau}}}}(z_{j}(\varepsilon))$,
the estimates

$|\varepsilon(z-\mu_{i})^{r_{i,\tau}+1}Y_{3}(z)+\varepsilon^{2}Y_{4}(z,\varepsilon)|\leqslant
C(\varepsilon^{\frac{q_{i}+1}{q_{i}-r_{i,\tau}}}+\varepsilon^{2})<C\varepsilon^{\frac{q_{i}+1}{q_{i}-r_{i,\tau}}}$

hold true, where $C$ is some constant independent of $\varepsilon$ and
$\tilde{b}$. This estimate and (5.17) allow us to apply the Rouché theorem and
we conclude that the ball
$T_{\tilde{b}\varepsilon^{\frac{2}{q_{i}-r_{i,\tau}}}}(z_{j}(\varepsilon))$
contains exactly one zero of equation (5.11). This gives asymptotic expansion
(2.20). The proof is complete.

##  6 Emerging resonances and eigenvalues

In this section we determine whether the poles of the operators
$\mathcal{R}_{\varepsilon,\tau}$ described in the previous section correspond
to the eigenvalues or resonances of the operator $\mathcal{H}_{\varepsilon}$.
In this way, we shall prove Theorems 2.4, 2.5.

We fix $p\in\\{1,\ldots,m\\}$ and assume that
$\Lambda_{p}=\ldots=\Lambda_{p+n-1}$, where $n\geqslant 1$ is a multiplicity
of the eigenvalue $\Lambda_{p}$ of the operator $\mathcal{H}^{\prime}$. In
what follows, we analyze the nature of poles $k_{ij}(\varepsilon)$ from
Theorem 2.2 with asymptotic behavior (2.13). In order to do this, we analyze
the behavior of the associated nontrivial solutions to problem (2.3) with
$f=0$.

###  6.1 Bottom of the spectrum

In this subsection we assume that $p=1$ and we prove Theorem 2.4. In the
considered case, the meromorphic continuation of the resolvent given the
operator $\mathcal{R}_{\varepsilon,\tau}$ is independent of $\tau$ and the
same is true for the corresponding poles $k_{ij}(\varepsilon)$ from Theorem
2.2 with asymptotic behavior (2.13). The behavior of the associated nontrivial
solutions to problem (2.3) with $f=0$ is provided by representation (2.7). In
this representation, the first sum over $j=1,\ldots,p-1$ is obviously missing
and the leading term at infinity is the sum over $j=p,\ldots,p+n-1$.

We fix one of the poles $k_{ij}(\varepsilon)$ with some
$i\in\\{1,\ldots,n\\}$, $j=1,\ldots,q_{i}$. It follows from (4.21), (4.22),
(4.23), (4.3), (4.4) that the coefficients $c_{s,\varepsilon}^{\pm}$,
$s=1,\ldots,n$, in representation (2.7) coincide with $l_{s}^{\varepsilon}$
determined by (4.21). Since we deal with a non-trivial solution of system
(4.21), this implies that at least one of the coefficients
$c_{s,\varepsilon}^{\pm}$ is non-zero. Then representation (2.7) for a non-
trivial solution to problem (2.3) defined in (4.22) and formulae (2.43) imply
that if

$\operatorname{Re}k_{ij}(\varepsilon)>0,$ (6.1)

then a non-trivial solution $\psi_{\varepsilon}$ decays exponentially at
infinity and thus, is an eigenfunction, while the opposite inequality

$\operatorname{Re}k_{ij}(\varepsilon)\leqslant 0$ (6.2)

ensures that $\psi_{\varepsilon}$ is not in $W_{2}^{2}(\Omega)$.

Asymptotic expansions for $k_{ij}$ established in Theorems 2.2, 2.3 allow us
to check effectively the above conditions. Namely, if
$\operatorname{Re}\mu_{i}>0$, this ensures inequality (6.1) for sufficiently
small $\varepsilon$. Then the poles $k_{ij}$, $j=1,\ldots,q_{i}$ correspond to
the eigenvalues
$\lambda_{ij}(\varepsilon)=\Lambda_{p}-k_{ij}^{2}(\varepsilon)$ with
asymptotic expansions (2.21), (2.22), (2.23) as described in the formulation
of Theorem 2.4. If $\operatorname{Re}\mu_{i}<0$, this guarantees inequality
(6.2) for sufficiently small $\varepsilon$ and the poles $k_{ij}$ correspond
to the resonances
$\lambda_{ij}(\varepsilon)=\Lambda_{p}-k_{ij}^{2}(\varepsilon)$ with same
asymptotic expansions (2.21), (2.22), (2.23).

If $\operatorname{Re}\mu_{i}=0$, in order to understand which of conditions
(6.1), (6.2) is realized, we need to check the next term in the asymptotic
expansion for $k_{ij}$. According Theorem 2.3, this can be done under an
additional assumption $2r_{i,\tau}\leqslant q_{i}-1$ and only for poles
$k_{ij,\tau}$ with $j=r_{i}+1,\ldots,q_{i}$. Then asymptotic expansion (2.20)
imply that under condition (2.24), the pole $k_{ij}$,
$j=r_{i,\tau}+1,\ldots,q_{i}$ corresponds to an eigenvalue, while under
condition (2.25) the pole $k_{ij}$ corresponds to a resonance. The asymptotic
behavior for this eigenvalue/resonance is given by (2.23) if $\mu_{i}\neq 0$
and it is given by (2.26) if $\mu_{i}=0$.

###  6.2 Internal thresholds in the spectrum

In this subsection we study the nature of the poles emerging from internal
points $\Lambda_{p}$ in the essential spectrum and we prove Theorem 2.5. As in
the previous subsection, here we again analyze the behavior at infinity of
non-trivial solutions $\psi_{\varepsilon}$ associated with poles $k_{ij}$ and
this analysis will be based on representation (2.7). However, there are
important differences. The matter is that now the representation involves also
a sum over $j=1,\ldots,p-1$. Its terms can decay or grow at infinity. As it
has been mentioned in the proof of Theorem 2.1, according identity (2.43), the
functions $e^{-\tau K_{j}(k)|x_{d}|}$, $j=1,\ldots,p-1$, decay exponentially
at infinity if $\tau\operatorname{Im}k^{2}<0$ and grow exponentially or vary
periodically as $\tau\operatorname{Im}k^{2}\geqslant 0$. This means that apart
of the sign of $\operatorname{Re}k_{ij}$, we should also control the sign of
$\tau\operatorname{Im}k_{ij}^{2}$. One more point is that now we have two
meromorphic continuations of the resolvent, the operators
$\mathcal{R}_{\varepsilon,\tau}$, $\tau\in\\{-1,+1\\}$, respectively, two sets
of their poles $k_{ij,\tau}(\varepsilon)$ converging to zero as
$\varepsilon\to+0$. We observe that according Theorems 2.2, 2.3, the first
terms in asymptotic expansions for poles $k_{ij,\tau}$ are the eigenvalues
$\mu_{i}$ of the matrix $\mathrm{M}_{1}$ and they are independent of $\tau$.
In fact, we can see the influence of $\tau$ on the asymptotic behavior of
$k_{ij}$ only in formulae (2.20), for $j=r_{i,\tau}+1,\ldots,q_{i}$ under an
additional assumption $2r_{i,\tau}\leqslant q_{i}-1$.

We choose $i\in\\{1,\ldots,N\\}$, $j\in\\{1,\ldots,q_{i}\\}$,
$\tau\in\\{-1,+1\\}$ and consider the pole $k_{ij,\tau}(\varepsilon)$. As in
the above proof of Theorem 2.4, it is easy to see that at least one of the
coefficients $c_{s,\varepsilon}^{\pm}$, $s=1,\ldots,n$, in representation
(2.7) for the associated non-trivial functions $\psi_{\varepsilon}$ is non-
zero. Then it follows from asymptotic expansions (2.16), (2.18), (2.19),
(2.20) that conditions (2.27), (2.30), (2.31), (2.34) ensure that
$\operatorname{Re}k_{ij,\tau}(\varepsilon)>0$ and
$\tau\operatorname{Im}k_{ij,\tau}(\varepsilon)<0$ and hence, the pole
$k_{ij,\tau}(\varepsilon)$ corresponds to an eigenvalue
$\lambda_{ij,\tau}(\varepsilon)=\Lambda_{p}-k_{ij,\tau}^{2}(\varepsilon)$. The
stated asymptotic behavior for this eigenvalue is implied by (2.16), (2.18),
(2.19), (2.20). In the same way, conditions (2.35), (2.38) ensure that
$\operatorname{Re}k_{ij,\tau}(\varepsilon)<0$ and hence, the pole
$k_{ij,\varepsilon}$ corresponds to a resonance
$\lambda_{ij,\tau}(\varepsilon)=\Lambda_{p}-k_{ij,\tau}^{2}(\varepsilon)$ and
it possesses the stated behavior.

If $\mu_{i}$ is a simple eigenvalue of the matrix $\mathrm{M}_{1}$, then
$q_{i}=1$. In this case, a non-trivial solution $l_{\varepsilon}$ to system
(4.21) associated with $k_{i1,\tau}(\varepsilon)$ converges to the vector
$\mathrm{e}_{i}$. Indeed, it follows from representation (5.4) that the
corresponding zero $z_{ij,\tau}(\varepsilon)$ of equation (5.5) is an
eigenvalue of the matrix
$\mathrm{M}_{1}-\varepsilon\tilde{\mathrm{M}}_{2}(z_{ij,\tau}(\varepsilon),\varepsilon)$
and $l_{\varepsilon}$ is an associated eigenvector. And since
$z_{ij,\tau}(\varepsilon)$ converges to $\mu_{i}$ and both these eigenvalues
are simple, the eigenvector associated with $k_{i1,\tau}(\varepsilon)$
converges to $l_{\varepsilon}$ as well. Now by representation (4.22), (4.23),
definition (2.10) of the operator $\mathcal{G}_{p,\tau}$ and formula (4.11) we
conclude that the coefficients $c_{s,\varepsilon}^{\pm}$ in representation
(2.7) satisfy the identities:

$c_{s,\varepsilon}^{\pm}=\varepsilon\sum\limits_{t=1}^{n}\int\limits_{\Omega}e^{\mp
K_{t}(0)x_{d}}\overline{\psi_{s}(x^{\prime})}\mathcal{L}_{1}\mathrm{e}_{i,t}\psi_{t-p+1}\,dx+o(\varepsilon).$

Hence, by condition (2.42), at least one of the coefficients
$c_{s,\varepsilon}^{\pm}$ is non-zero. As above, it is easy to see that one of
conditions (2.27) or (2.30) and one of conditions (2.39) or (2.41) ensures
that the functions $e^{-K_{s}(k_{ij,\tau}(\varepsilon))|x_{d}|}$ grows
exponentially at infinity and hence, the same is true for the function
$\psi_{\varepsilon}$. Therefore, the pole $k_{ij,\tau}(\varepsilon)$
corresponds to a resonance. The asymptotic expansion for this resonance can be
established as above. The proof of Theorem 2.5 is complete.

## Acknowledgments

The authors thank the referees for useful remarks. The research by D.I.B. and
D.A.Z. is supported by the Russian Science Foundation (Grant No. 20-11-19995).

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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