# Resonances near spectral thresholds for multichannel discrete Schrödinger
operators

M. Assal, O. Bourget, P. Miranda and D. Sambou  Marouane Assal, Pablo Miranda,
Departamento de Matemática y Ciencia de la Computación, Universidad de
Santiago de Chile, Las sophoras 173, Santiago, Chile<EMAIL_ADDRESS><EMAIL_ADDRESS>Olivier Bourget, Facultad de Matemáticas, Pontificia
Universidad Católica de Chile, Vicuna Mackenna 4860, Santiago, Chile.
<EMAIL_ADDRESS>Diomba Sambou, Institut Denis Poisson, Université d’Orléans,
CNRS-UMR 7013, 45067 Orléans Cedex 2, France<EMAIL_ADDRESS>
###### Abstract.

We study the distribution of resonances for discrete Hamiltonians of the form
$H_{0}+V$ near the thresholds of the spectrum of $H_{0}$. Here, the
unperturbed operator $H_{0}$ is a multichannel Laplace type operator on
$\ell^{2}(\mathbb{Z};\mathfrak{G})\cong\ell^{2}(\mathbb{Z})\otimes\mathfrak{G}$
where ${\mathfrak{G}}$ is an abstract separable Hilbert space, and $V$ is a
suitable non-selfadjoint compact perturbation. We distinguish two cases. If
${\mathfrak{G}}$ is of finite dimension, we prove that resonances exist and do
not accumulate at the thresholds in the spectrum of $H_{0}$. Furthermore, we
compute exactly their number and give a precise description on their location
in clusters in the complex plane. If ${\mathfrak{G}}$ is of infinite
dimension, an accumulation phenomenon occurs at some thresholds. We describe
it by means of an asymptotical analysis of the counting function of
resonances. Consequences on the distribution of the complex and the embedded
eigenvalues are also given.

Mathematics subject classification 2020: 47A10, 81Q10, 81U24.

##  1\. Introduction

###  1.1. General setting

Let $\Delta$ be the positive one-dimensional discrete Laplacian defined on the
Hilbert space

$\ell^{2}(\mathbb{Z}):=\big{\\{}u:\mathbb{Z}\to\mathbb{C};\,\|u\|^{2}=\sum_{n\in\mathbb{Z}}|u(n)|^{2}<\infty\big{\\}},$

by

$(\Delta u)(n):=2u(n)-u(n+1)-u(n-1),\;\;\;u\in\ell^{2}(\mathbb{Z}).$

Consider a complex separable Hilbert space $\mathfrak{G}$ and let $M$ be a
compact operator on $\mathfrak{G}$. On the Hilbert space
$\ell^{2}(\mathbb{Z};\mathfrak{G})\cong\ell^{2}(\mathbb{Z})\otimes\mathfrak{G}$,
we introduce the operator $H_{0}$ defined by

(1.1) $H_{0}:=\Delta\otimes I_{\mathfrak{G}}+I_{\ell^{2}(\mathbb{Z})}\otimes
M,$

where $I_{\mathfrak{G}}$ and $I_{\ell^{2}(\mathbb{Z})}$ denote the identity
operators on $\mathfrak{G}$ and $\ell^{2}(\mathbb{Z})$ respectively.

The operator $\Delta$ is bounded, selfadjoint in $\ell^{2}(\mathbb{Z})$ and
its spectrum is absolutely continuous given by $\sigma(\Delta)=\sigma_{{\rm
ac}}(\Delta)=[0,4]$. Then, the spectrum of $H_{0}$ is absolutely continuous
(in particular there are not embedded eigenvalues) and it has the following
band structure in the complex plane

$\sigma(H_{0})=\sigma_{ac}(H_{0})=\bigcup_{\lambda\in\sigma(M)}[\lambda,\lambda+4],$

where the edge points {$\lambda,\lambda+4\\}_{\lambda\in\sigma(M)}$ play the
role of thresholds. These bands are parallel to the real axis, and may be
pairwise disjoint, overlapping or intersecting (see Fig. 1).

${{\rm Re}\,z}$${{\rm
Im}\,z}$$\lambda_{2}$$\tau_{2}$$\bullet$$\bullet$$\lambda_{5}$$\tau_{5}$$\bullet$$\bullet$$0$$4$$\bullet$$\bullet$${\tau_{4}=\lambda_{3}}$$\tau_{3}$$\bullet$$\bullet$$\lambda_{4}$$\bullet$${\lambda_{6}}$$\tau_{6}$$\bullet$$\bullet$${\lambda_{7}}$$\tau_{7}$$\bullet$$\bullet$${\lambda_{8}}$$\tau_{8}$$\bullet$$\bullet$$\lambda_{9}$$\tau_{9}$$\bullet$$\bullet$
Figure 1. Schematic example of $\sigma(H_{0})$. ($\tau_{j}=\lambda_{j}+4$).

Our main purpose in this paper is to study the existence and the distribution
of resonances for operators of the form $H=H_{0}+V$ near the spectral
thresholds of $\sigma(H_{0})$. The specific class of perturbations
$V:\ell^{2}(\mathbb{Z};\mathfrak{G})\to\ell^{2}(\mathbb{Z};\mathfrak{G})$ to
be considered will be introduced later. The resonances will be defined as the
poles of the meromorphic extension of the resolvent of $H$ in some weighted
spaces (see Section 2).

We will study two different cases: the first one is when $\mathfrak{G}$ is
finite-dimensional, and the second one is when $\mathfrak{G}$ is infinite-
dimensional with $M$ a finite-rank operator on $\mathfrak{G}$. Although both
cases may look similar, they present an important difference in terms of the
distribution of resonances. In fact, since in the first case the spectrum of
$M$ consists only on eigenvalues with finite multiplicities, and in the second
one, $0$ is an eigenvalue of $M$ with infinite multiplicity, the set of
spectral thresholds splits into two classes.

  * •

The first class consists on the thresholds
$\\{\lambda,\lambda+4\\}_{\lambda\in\sigma(M)}$ such that $\lambda$ is an
eigenvalue of $M$ with finite multiplicity. Near such points, we prove the
existence of resonances of $H$ and precise their location. More specifically,
under suitable assumptions, we obtain the exact number of resonances near the
thresholds, and we prove that they are distributed in clusters around some
specific points, see Theorems 3.1 and 3.2 (this improves some of the results
in [9]).

  * •

The second class consists of the set of thresholds $\\{0,4\\}$ in the infinite
dimensional case. In contrast to the first class, we show that there is an
infinite number of resonances that accumulate near these points, see Theorem
3.4.

###  1.2. Comments on the literature

The resonance phenomena in quantum mechanics has been mathematically tackled
from various perspectives. For instance, resonances of a stationary quantum
system can be defined as the poles of a suitable meromorphic extension of
either the Green function, the resolvent of the Hamiltonian or the scattering
matrix. The imaginary part of such poles is sometimes interpreted as the
inverse of the lifetime of some associated quasi-eigenstate. A related idea is
to identify the resonances of a quantum Hamiltonian with the discrete
eigenvalues of some non-selfadjoint operator obtained from the original one by
the methods of spectral deformations. Another point of view consists in
defining the resonances dynamically, i.e., in terms of quasi-exponential decay
for the time evolution of the system. This property is somewhat encoded in the
concept of sojourn time. The equivalence between these different perspectives
and formalisms is also an issue.

The study of the existence and the asymptotic behavior of resonances in
different asymptotic regimes has witnessed a lot of progress during the last
thirty years, mainly in the context of continuous configuration spaces. This
has been achieved thanks to the development of many mathematical approaches
such as scattering methods, spectral and variational techniques, semiclassical
and microlocal analysis (many references to this vast literature can be found
in the monographs [17, 18, 11]).

On the other side, the qualitative spectral properties of the discrete Laplace
operator and some selfadjoint generalizations exhibiting dispersive
properties, have been extensively investigated. We primarily refer to [10, 30]
for the multidimensional lattice case ${\mathbb{Z}}^{d}$, [12] and references
therein for trees, [2, 26] for periodic graphs and perturbed graphs,
respectively. Analyses of the continuum limit are performed in [24] and
references therein. The role of the thresholds of the discrete Laplace
operator are specifically studied in [19] (see also references therein). We
also refer the reader to [28] and references therein for studies concerning
Jacobi matrices and block Jacobi matrices.

The study of resonances for quantum Hamiltonians on discrete structures has
been mainly performed on quasi 1D models (see [21] and references therein).
However, in these approaches, perturbations are assumed to be diagonal and
essentially compactly supported. One may quote [7] where results on the
distribution of resonances of compact perturbations of the 1D discrete Laplace
operator were obtained. This suggests that a more systematic analysis of
resonances in the spirit of the works of Bony, Bruneau and Raikov [6, 5]
should be performed in this context. The present paper is the first of a
sequence of studies about resonances for operators on various graph structures
where the perturbation is not necessarily diagonal. In what follows, we focus
on some generalizations of the 1D discrete Laplace operator and study the
distribution of resonances that appear in the neighborhoods of the thresholds,
in perturbative regimes.

The asymptotic behavior of resonances near thresholds has been studied in an
abstract setting in [15]. However, this study does not include the models of
the present work. On the other side, in some continuous waveguides models, the
singularities at the thresholds are similar in structure to the ones that
appear in our case. Thus, related results to ours are obtained in [9, 8].
Nevertheless, using a modified approach, the conclusions that we obtain here
are sharper. More precisely, as we said before, we provide the exact number of
resonances near each threshold and we show that they are distributed in small
clusters whose radii depend on the perturbation parameter. Moreover, we treat
also a case of accumulation of resonances, which do not appear in the above
works.

###  1.3. Plan of the article and notations:

The article is organized as follows: the resonances are defined in Section 2.
The main results, namely Theorems 3.1, 3.2 and 3.4, are stated in Section 3.
In Section 4 we briefly describe some models where our results can be applied.
The proofs of the main theorems are postponed to Sections 5-8. Finally, in the
appendix we prove a result on the multiplicity of resonances needed in our
study. We present this result in an abstract form since it may be of
independent interest.

Notations. Let $\mathcal{K}$ be a separable Hilbert space. We denote by
$\mathcal{B}(\mathcal{K})$ the algebra of bounded linear operators acting on
$\mathcal{K}$. $\mathfrak{S}_{\infty}(\mathcal{K})$ and
$\mathfrak{S}_{p}(\mathcal{K})$, $p\geq 1$, stand for the ideal of compact
operators and the Schatten classes respectively. In particular,
$\mathfrak{S}_{1}(\mathcal{K})$ and $\mathfrak{S}_{2}(\mathcal{K})$ are the
ideals of trace class operators and Hilbert-Schmidt operators on
$\mathcal{K}$, endowed with the norms $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$
respectively. If $U\subset\mathbb{C}$ is an open set, we denote
$\text{Hol}(U;\mathcal{K})$ the set of holomorphic functions on $U$ with
values in $\mathcal{K}$. We denote by $(\delta_{n})_{n\in\mathbb{Z}}$ the
canonical basis of $\ell^{2}(\mathbb{Z})$. Throughout the paper, $\mathcal{H}$
stands for the Hilbert space $\ell^{2}(\mathbb{Z})\otimes\mathfrak{G}$
identified with $\ell^{2}(\mathbb{Z};\mathfrak{G})$,
$(\delta_{n})_{n\in\mathbb{Z}}$ the canonical orthonormal basis of
$\ell^{2}({\mathbb{Z}})$, and $(e_{j})_{j\in I}$ stands for an orthonormal
basis of $\mathfrak{G}$, for some index set $I\subseteq\mathbb{Z}_{+}$. For
$s>0$, let $W_{s}$ be the multiplication operator by the function
$\mathbb{Z}\ni
n\mapsto\|e^{-\frac{s}{2}|\cdot|}\|_{\ell^{2}(\mathbb{Z})}e^{\frac{s}{2}|n|}$
acting on $e^{-\frac{s}{2}|\cdot|}\ell^{2}(\mathbb{Z})$ with values in
$\ell^{2}(\mathbb{Z})$. $W_{-s}$ stands for the multiplication operator by the
function $\mathbb{Z}\ni
n\mapsto\|e^{-\frac{s}{2}|\cdot|}\|^{-1}_{\ell^{2}(\mathbb{Z})}e^{-\frac{s}{2}|n|}$
acting on $\ell^{2}(\mathbb{Z})$ with values in $\ell^{2}(\mathbb{Z})$. We set
$\bm{W}_{\pm s}:=W_{\pm s}\otimes I_{\mathfrak{G}}$. The first quadrant of the
complex plane will be denoted $\mathbb{C}_{1}$, i.e.,
$\mathbb{C}_{1}:=\\{z\in\mathbb{C};\,{\rm Re}\,z>0,{\rm Im}\,z>0\\}$. For
$\varepsilon>0$ and $z_{0}\in\mathbb{C}$, we set
$D_{\varepsilon}(z_{0}):=\\{z\in\mathbb{C};|z-z_{0}|<\varepsilon\\}$ and
$D^{*}_{\varepsilon}(0):=D_{\varepsilon}(z_{0})\setminus\\{z_{0}\\}$.

##  2\. Resonances

In this section, we define the resonances of $H_{\omega}=H_{0}+\omega V$ near
the thresholds of $\sigma(H_{0})$ for a class of potentials $V$ satisfying
Assumption 2.1 below. We consider the following two cases:

Case (A). $\mathfrak{G}$ is finite-dimensional.

Case (B). $\mathfrak{G}$ is infinite-dimensional and $M$ is finite-rank.

In both cases, we assume that $M$ is diagonalizable. From now on, we denote by
$H_{0,A}$ (respectively $H_{0,B}$) the operator $H_{0}$ defined by (1.1) in
case (A) (respectively case (B)). The same notation is used for the perturbed
operators, i.e.,

$H_{\omega,A}:=H_{0,A}+\omega V,\quad H_{\omega,B}:=H_{0,B}+\omega
V,\quad\omega\in\mathbb{C}.$

In the following $N$ stands for the dimension of $\mathfrak{G}$ in case (A)
and for the rank of $M$ in case (B). Let us denote by
$\\{\lambda_{q}\\}_{q=1}^{N}$ the set of eigenvalues of $M$ in case (A). In
case (B) we still denote by $\\{\lambda_{q}\\}_{q=1}^{N}$ the non-zero
eigenvalues of $M$ and set $\lambda_{0}=0$.

The spectra of the operators $H_{0,A}$ and $H_{0,B}$ are given by

$\sigma(H_{0,A})=\bigcup_{j=1}^{N}\Lambda_{j},\quad\sigma(H_{0,B})=\bigcup_{j=1}^{N}\Lambda_{j}\cup[0,4],\qquad\Lambda_{j}:=[\lambda_{j},\lambda_{j}+4].$

Let $\\{\lambda_{q}\\}_{q=1}^{d}$ be the subset of
$\\{\lambda_{q}\\}_{q=1}^{N}$ consisting of its distinct elements, $1\leq
d\leq N$. The sets of the spectral thresholds of $\sigma(H_{0,A})$ and
$\sigma(H_{0,B})$ are denoted by

$\mathcal{T}_{A}:=\\{\lambda_{q},\lambda_{q}+4\\}_{q=1}^{d},\quad\mathcal{T}_{B}:=\\{\lambda_{q},\lambda_{q}+4\\}_{q=0}^{d},$

respectively. In the sequel, we shall use the notation $\bullet$ to refer
either to $A$ or $B$.

######  Definition 2.1.

A threshold $\zeta\in\mathcal{T}_{\bullet}$ is degenerate if there exist
$p\neq q\in\\{1,...,d\\}$ such that
$\zeta=\lambda_{q}=\lambda_{p}+4\in\mathcal{T}_{\bullet}$. Otherwise, $\zeta$
is non-degenerate.

For instance, in the example shown in Figure 1, $\lambda_{3}$ is a degenerate
threshold.

As noticed in [19, Appendix A] in the case of the free discrete Laplacian,
there is a simple relation between the right thresholds and the left ones that
makes possible to reduce the study near the threshold $\lambda_{q}+4$ to that
near $\lambda_{q}$. In order to keep the paper at a reasonable length, in the
following we will state our results only for the left thresholds
$\lambda_{q}$. Analogous results for the right thresholds $\lambda_{q}+4$ hold
with natural modifications.

For $q\in\\{0,1,...,d\\}$, denote by $\nu_{q}$ the dimension of ${\rm
Ker}(M-\lambda_{q})$. Of course $\nu_{0}=\infty$ in case (B). Let us denote by
$\pi_{q}$ the projection onto ${\rm Ker}(M-\lambda_{q})$ defined by

$\pi_{q}:=\frac{1}{2\pi
i}\oint_{|z-\lambda_{q}|=\varepsilon}(z-M)^{-1}dz,\quad 0<\varepsilon\ll 1.$

Given a threshold $\lambda_{q}\in\mathcal{T}_{\bullet}$, one introduces the
parametrization

(2.1) $k\mapsto z_{q}(k)=\lambda_{q}+k^{2},$

where $k$ is a complex variable in a small neighborhood of $0$.

Let $Y=(Y(n,m))_{(n,m)\in\mathbb{Z}^{2}}$ be a matrix with components $Y(n,m)$
acting on $\mathfrak{G}$. Under suitable conditions on the $Y(n,m)$ (see
Assumption 2.1 below), $Y$ acts on $\mathcal{H}$ as

$(Yx)(n)=\sum_{m}Y(n,m)x(m)$, $x\in\mathcal{H}$.

Actually, $Y$ can be interpreted as an operator with summation kernel given by
the function $(n,m)\in\mathbb{Z}^{2}\mapsto Y(n,m)$ with values in a suitable
space. Canonically, $Y$ can be written as

(2.2) $Y=\sum_{n,m}|\delta_{n}\rangle\langle\delta_{m}|\otimes Y(n,m).$

Throughout this paper, we will suppose the following

######  Assumption 2.1.

$M$ is diagonalizable and the perturbation satisfies:

  * -

Case (A). $V$ is of the form (2.2) and $\|V(n,m)\|_{\mathfrak{G}}\leq
Ce^{-\rho(|n|+|m|)}$, $\forall\>(n,m)\in\mathbb{Z}^{2}$, for some constants
$\rho,C>0$.

  * -

Case (B). $V$ is of definite sign ($\pm V\geq 0)$ of the form $V=(1\otimes
K^{\ast})U(1\otimes K)$, where $K\in{\mathfrak{S}}_{p}(\mathfrak{G})$ for some
$p\in[1,+\infty)$, and $U$ is of the form (2.2) with
$\|U(n,m)\|_{\mathfrak{G}}\leq Ce^{-\rho(|n|+|m|)}$,
$\forall\>(n,m)\in\mathbb{Z}^{2}$, for some constants $\rho,C>0$.

To fix ideas, let us give examples of potentials satisfying Assumption 2.1.

  * 1)

In the basis $(e_{j})_{j\geq 0}$ of $\mathfrak{G}$, for each
$(n,m)\in\mathbb{Z}^{2}$ the operator $Y(n,m)$ has the matrix representation

$Y(n,m)=\\{Y_{jk}(n,m)\\}_{j,k\geq 0},\qquad Y_{jk}(n,m):=\langle
e_{j},Y(n,m)e_{k}\rangle_{\mathfrak{G}}.$

In particular, if $f$ is a positive bounded function defined in
$\mathbb{Z}_{+}^{2}$, then typical examples of potentials satisfying
Assumption 2.1 (A) are $V$ such that

$|V_{jk}(n,m)|\leq
f(j,k)e^{-\rho(|n|+|m|)},(j,k)\in\mathbb{Z}_{+}^{2},(n,m)\in\mathbb{Z}^{2},\rho>0.$

If $\beta>2$, Assumption 2.1 (B) holds for $V$ such that $\pm U\geq 0$ and

(2.3)
$|U_{jk}(n,m)|\leq{\rm}\langle(j,k)\rangle^{-\beta}e^{-\rho(|n|+|m|)},(j,k)\in\mathbb{Z}_{+}^{2},(n,m)\in\mathbb{Z}^{2},\rho>0,$

where $\langle(j,k)\rangle:=(1+j^{2}+k^{2})^{1/2}$. For instance, (2.3) is
satisfied if

$U(n,m)=e^{-\rho(|n|+|m|)}\sum_{j,k}\langle(j,k)\rangle^{-\beta}|e_{j}\rangle\langle
e_{k}|,\quad(n,m)\in\mathbb{Z}^{2},\rho>0.$

  * 2)

Under Assumption 2.1 (A) or (B), one has
$V_{\rho}:=\bm{W}_{\rho}V\bm{W}_{\rho}\in\mathcal{B}(\mathcal{H})$.

  * 3)

$\pm V\geq 0$ in case (B) if and only if $\pm U\geq 0$. Since
$I_{\mathfrak{G}}$ is not compact, then $K$ (or $K^{\ast}$) regularizes the
component $\mathfrak{G}$ of $\mathcal{H}$, which is crucial to define the
resonances.

  * 4)

Under consideration (see Remark 3.5), Assumption 2.1 (B) can be extended by
$V$ of the form $V=(\Gamma_{1}\otimes K^{\ast})U(\Gamma_{2}\otimes K)$, where
$U$ is as above and $\Gamma_{1}$, $\Gamma_{2}$ acting in
$\ell^{2}(\mathbb{Z})$.

######  Proposition 2.2.

Let $\lambda_{q}\in\mathcal{T}_{\bullet}$. Under Assumption 2.1 ($\bullet$),
there exists $\varepsilon_{0}>0$ such that for all $|\omega|$ sufficiently
small, the operator-valued function

$D^{*}_{\varepsilon_{0}}(0)\cap\mathbb{C}_{1}\ni
k\mapsto\bm{W}_{-\rho}(H_{\omega,\bullet}-z_{q}(k))^{-1}\bm{W}_{-\rho}$

admits an analytic extension to $D_{\varepsilon_{0}}(0)$, with values in
$\mathcal{B}(\mathcal{H})$. We denote by
$\mathcal{R}^{(q)}_{\omega,\bullet}(k)$ this extension.

The proof of Proposition 2.2 is postponed to Section 5. Taking into account
the above result one defines the resonances of the operator
$H_{\omega,\bullet}$ near a threshold $\lambda_{q}\in\mathcal{T}_{\bullet}$ as
follows:

######  Definition 2.3.

The resonances of the operator $H_{\omega,\bullet}$ near a threshold
$\lambda_{q}\in\mathcal{T}_{\bullet}$ are defined as the points $z_{q}(k)$
such that $k\in D_{\varepsilon_{0}}(0)$ is a pole of the meromorphic extension
$\mathcal{R}^{(q)}_{\omega,\bullet}$ given by Proposition 2.2. The
multiplicity of a resonance $z_{q}(k_{0})$ is defined by

(2.4) ${\rm mult}(z_{q}(k_{0})):={\rm
rank}\frac{1}{2i\pi}\oint_{\gamma}\mathcal{R}^{(q)}_{\omega,\bullet}(k)dk,$

where $\gamma$ is a positively oriented circle centered on $k_{0}$, that not
contain any other pole of $\mathcal{R}^{(q)}_{\omega,\bullet}$. The set of
resonances of $H_{\omega,\bullet}$ will be denoted ${\rm
Res}(H_{\omega,\bullet})$.

Notice that by (2.1), the resonances near a threshold
$\lambda_{q}\in\mathcal{T}_{\bullet}$ are defined in a Riemann surface
$\mathcal{S}_{q}$, which is locally two sheeted.

##  3\. Main results

In this section one formulates our main results on the existence and the
asymptotic properties of the resonances of the operators $H_{\omega,A}$ and
$H_{\omega,B}$ near the spectral thresholds.

Let $a_{-1}$ and $b_{-1}$ be the operators in $\ell^{2}(\mathbb{Z})$ defined
by

(3.1) $\displaystyle(a_{-1}u)(n)$
$\displaystyle:=\sum_{m\in\mathbb{Z}}\frac{i}{2}W_{-\rho}(n)W_{-\rho}(m)u(m),$
(3.2) $\displaystyle(b_{-1}u)(n)$
$\displaystyle:=\sum_{m\in\mathbb{Z}}\frac{(-1)^{n+m+1}}{2}W_{-\rho}(n)W_{-\rho}(m)u(m).$

In case (A) or (B), for $q\neq p\in\\{0,1,...,d\\}$, define the projections in
$\mathcal{H}$

(3.3)
$\Pi_{q}:=\frac{2}{i}a_{-1}\otimes\pi_{q},\quad\Pi_{q,p}:=\frac{2}{i}a_{-1}\otimes\pi_{q}-{2}b_{-1}\otimes\pi_{p}.$

Notice that ${\rm rank}\,\Pi_{q}=\nu_{q}$ and ${\rm
rank}\,\Pi_{q,p}=\nu_{q}+\nu_{p}$. Introduce $E_{q}$ and $E_{q,p}$ defined in
$\mathcal{H}$ by

(3.4)
${E_{q}}:=\Pi_{q}V_{\rho}\Pi_{q}\quad\text{and}\quad{E_{q,p}}:=\Pi_{q,p}V_{\rho}\left(a_{-1}\otimes\pi_{q}+b_{-1}\otimes\pi_{p}\right).$

###  3.1.  Distribution of the resonances: non-accumulation case

Our first results consist on the existence, the number and the asymptotic
dependence on $\omega$ of the resonances of the operator $H_{\omega,A}$ near
the thresholds $\mathcal{T}_{A}$, and the ones of $H_{\omega,B}$ near the
thresholds $\mathcal{T}_{B}\setminus\\{0,4\\}$.

For $q\neq 0$, let
$\\{\alpha_{1}^{(q)},\alpha_{2}^{(q)},\cdots,\alpha_{r}^{(q)},1\leq
r\leq\nu_{q}\\}$ be the set of distinct eigenvalues of ${E_{q}}|_{{\rm
Ran}\,\Pi_{q}}$, each $\alpha_{j}^{(q)}$ of multiplicity $m_{q,j}$.
Analogously let
$\\{\beta_{1}^{(q)},\beta_{2}^{(q)},\cdots,\beta_{r}^{(q)},1\leq
r\leq\nu_{q}+\nu_{p}\\}$ be the set of distinct eigenvalues of
${E_{q,p}}|_{{\rm Ran}\,\Pi_{q,p}}$, each $\beta_{j}^{(q)}$ of multiplicity
$m_{q,p,j}$. Of course $\sum_{j=1}^{r}m_{q,j}=\nu_{q}$ and
$\sum_{j=1}^{r}m_{q,p,j}=\nu_{q}+\nu_{p}$.

######  Theorem 3.1 (Non-degenerate case).

Assume that $\mathfrak{G}$ is finite-dimensional and let Assumption 2.1 (A)
holds. Let $\lambda_{q}\in\mathcal{T}_{A}$ be a non-degenerate threshold.
Then, there exist $\varepsilon_{0},\delta_{0}>0$ such that for all
$|\omega|<\delta_{0}$ we have the following:

  * (i)

For any $z_{q}(k)\in{\rm Res}(H_{\omega,A})$ with $k=k_{q}(\omega)\in
D_{\varepsilon_{0}|\omega|}(0)$, there exists a unique
$\alpha_{j}^{(q)}\in\sigma({E_{q}}|_{{\rm Ran}\,\Pi_{q}})$ such that

(3.5)
${k_{q}(\omega)=-\frac{i}{2}\alpha_{j}^{(q)}\omega+\mathcal{O}\big{(}|\omega|^{1+\frac{1}{m_{q,j}}}\big{)}}.$

Conversely, for any $\alpha_{j}^{(q)}\in\sigma({E_{q}}|_{{\rm Ran}\,\Pi_{q}})$
there exists at least one and at most $m_{q,j}$ resonances
$z_{q}(k)=\lambda_{q}+k(\omega)^{2}$ of $H_{\omega,A}$ with $k(\omega)$
satisfying (3.5).

  * (ii)

If $M$ and $V$ are selfadjoint, then for any
$\alpha_{j}^{(q)}\in\sigma({E_{q}}|_{{\rm Ran}\,\Pi_{q}})$ there exist
$m_{q,j}$ resonances $z_{q}(k)$ (counting multiplicities) with
$k=k_{q}(\omega)$ satisfying (3.5). In particular

(3.6) $\sum_{z_{q}(k)\in{\rm Res}(H_{\omega,A}),\,k\in
D_{\varepsilon_{0}|\omega|}(0)}{\rm mult}(z_{q}(k))=\nu_{q}.$

######  Theorem 3.2 (Degenerate case).

Let $\mathfrak{G}$ be finite-dimensional and Assumption 2.1 (A) holds. Suppose
that $\lambda_{q}$ is degenerate and let $p\neq q\in\\{1,...,d\\}$ such that
$\lambda_{q}=\lambda_{p}+4$. Then, assertion (i) of Theorem 3.1 holds true
with $\frac{i}{2}\alpha_{j}^{(q)}$ replaced by $\beta_{j}^{(q)}$ and $m_{q,j}$
replaced by $m_{q,p,j}$.

The above results state that for any fixed threshold
$\lambda_{q}\in\mathcal{T}_{A}$ and $|\omega|$ small enough, the resonances
$z_{q}(k)=\lambda_{q}+k(\omega)^{2}$ near $\lambda_{q}$ are distributed (in
variable $k$) in clusters around the points
$-\frac{i}{2}\alpha_{j}^{(q)}\omega$ or $-\beta_{j}^{(q)}\omega$,
$j\in\\{1,...,r\\}$. If the threshold $\lambda_{q}$ is non-degenerate and $V$
and $M$ are selfadjoint, one has exactly $m_{q,j}$ resonance(s) in each
cluster and the total number of resonances near $\lambda_{q}$ is equal to
$\nu_{q}$ as shown by Figure 2.

######  Remark 3.3.

The same results occur in case (B) near the thresholds
$\lambda_{q}\in\mathcal{T}_{B}\setminus\\{0,4\\}$.

Figure 2.  Resonances of $H_{\omega,A}$ in variable $k$ near a non-degenerate
threshold $\lambda_{q}$ in the case $M$ and $V$ selfadjoint.

###  3.2.  Distribution of the resonances: accumulation case

For a compact selfadjoint operator $T$ and $I$ a real interval, let us
introduce
$n_{I}(T):=\rm{Tr}{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{I}(T)$,
the function that counts the number of eigenvalues of $T$ in $I$, including
multiplicities.

######  Theorem 3.4.

Let Assumption 2.1 (B) holds with $V\geq 0$ and $\omega$ small. Assume
moreover that $M$ is selfadjoint. Then, there exists $0<\varepsilon_{0}\ll 1$
such that the resonances $z_{0}(k)=k^{2}$ of $H_{\omega,B}$ with $k\in
D_{\varepsilon_{0}}^{\ast}(0)$ satisfy:

  * (i)

${\rm Im}(k/\omega)\leq 0$, $|{\rm Re}(k/\omega)|=o(|k/\omega|)$.

  * (ii)

Suppose that $E_{0}$ is of infinite rank. Then, the number of resonances of
$H_{\omega,B}$ near $0$ is infinite. More precisely, there exists a sequence
of positive numbers $(\varepsilon_{j})_{j}$ tending to zero such that,
counting multiplicities, we have as $j\to+\infty$

$\\#\big{\\{}z_{0}(k)\in{\rm
Res}(H_{\omega,B}):|\omega|\varepsilon_{j}<|k|<\varepsilon_{0}|\omega|\big{\\}}=n_{[2\varepsilon_{j},2]}(E_{0})(1+o(1)).$

######  Remark 3.5.

Note that Theorem 3.4 remains valid if one considers more general
perturbations of the form $V=(\Gamma\otimes K^{\ast})U(\Gamma^{\ast}\otimes
K)$ with $U=(U(n,m))_{(n,m)\in\mathbb{Z}^{2}}$, where $\Gamma$ and
$\Gamma^{\ast}$ acting in $\ell^{2}(\mathbb{Z})$ commute with the operator
$W_{-\rho}$.

Consider the domain
$\mathcal{C}_{\theta}(a,b):=\big{\\{}x+iy\in\mathbb{C}:a\leq x\leq
b,|y|\leq\theta|x|\big{\\}}$, for $a,b,\theta>0$. As corollaries of Theorem
3.4, one has the following results specifying in particular the distribution
of the discrete spectrum and the embedded eigenvalues of $H_{\omega,B}$ close
to the spectral threshold $0$.

######  Corollary 3.6.

Under the assumptions of Theorem 3.4, suppose that ${\rm rank}(E_{0})=+\infty$
and $Arg(\omega)\in\pm[0,\frac{\pi}{2})\cup\pm(\frac{\pi}{2},\pi)$. Then:

  * a)

$H_{\omega,B}$ has an infinite number of resonances close to $0$.

  * b)

If $Arg(\omega)\in\pm[0,\frac{\pi}{2})$: they are located in the second sheet
of $\mathcal{S}_{0}$ and accumulate around a semi-axis:
$k\in-i\omega\mathcal{C}_{\theta}(\varepsilon,\varepsilon_{0})$, $0<\theta\ll
1$, i.e. $Im\,k<0$. In particular, $V$ does not produce eigenvalues near $0$.

  * c)

If $Arg(\omega)\in\pm(\frac{\pi}{2},\pi)$: they are located in the first sheet
of $\mathcal{S}_{0}$:
$k\in-i\omega\mathcal{C}_{\theta}(\varepsilon,\varepsilon_{0})$, $0<\theta\ll
1$, i.e. $Im\,k>0$. In particular, $V$ does not produce embedded eigenvalues
near $0$. Furthermore, the only resonances are the discrete eigenvalues. They
are non real and accumulate near $0$ around a semi-axis.

######  Corollary 3.7.

Let the assumptions of Corollary 3.6 hold with $V\leq 0$. Then:

  * a)

$H_{\omega,B}$ has an infinite number of resonances close to $0$.

  * b)

If $Arg(\omega)\in\pm[0,\frac{\pi}{2})$: they are located in the first sheet
of $\mathcal{S}_{0}$: $k\in
i\omega\mathcal{C}_{\theta}(\varepsilon,\varepsilon_{0})$, $0<\theta\ll 1$,
i.e. $Im\,k>0$. In particular, $V$ does not produce embedded eigenvalues near
$0$. Furthermore, the only resonances are the discrete eigenvalues with:

$\bullet$ If $Arg(\omega)=0$, they are real and accumulate near $0$ from the
left.

$\bullet$ If $Arg(\omega)\in\pm(0,\frac{\pi}{2})$ they are non real and
accumulate near $0$ around a semi-axis.

  * c)

If $Arg(\omega)\in\pm(\frac{\pi}{2},\pi)$: they are located in the second
sheet of $\mathcal{S}_{0}$ and accumulate around a semi-axis: $k\in
i\omega\mathcal{C}_{\theta}(\varepsilon,\varepsilon_{0})$, $0<\theta\ll 1$,
i.e. $Im\,k<0$. In particular, $V$ does not produce eigenvalues near $0$.

Corollary 3.6 is summarized in Table 1 and a part of point c) with
$|\omega|=1$ is illustrated in Figure 3 in the physical $z$-plane.

$0$$\varepsilon^{2}$$\varepsilon_{0}^{2}$$e^{i\eta}\mathbb{R}_{+}$$\pi-
Arg(\omega)$${\rm Re}\,z$${\rm
Im}\,z$$y=\tan\big{(}2\eta-\pi\big{)}x$$\theta$$\theta$Absence of
eigenvalues$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\times$$\eta:=Arg(\omega)\in(\frac{\pi}{2},\pi),\>V\geq
0$ Figure 3. Here, the eigenvalues near $0$ are concentrated near the semi-
axis $y=\tan\big{(}2\eta-\pi\big{)}x$. Table 1.  Distribution of the
resonances and the complex eigenvalues near $0$.

######  Remark 3.8.

Define the unitary operator acting in $\ell^{2}(\mathbb{Z})$ by
$(J\phi)(n)=(-1)^{n}\phi(n)$. Then, as mentioned in the Introduction,
analogous of Theorem 3.4 and Corollaries 3.6 and 3.7 near the threshold $4$
can be naturally established by exploiting the identity $J\Delta
J^{-1}=-\Delta+4$, which allows to reduce the analysis near $0$. This part is
omitted to shorten the article.

##  4\. Illustrative examples

In this part, we give some models for which our main results can be applied.

###  4.1. Discrete Laplace operator on the strip (with Dirichlet boundary
condition)

Let $\mathfrak{G}={\mathbb{C}}^{N}$ and $(e_{k})_{k\in\\{1,\ldots,N\\}}$ its
canonical orthonormal basis. The operator $M$ is defined on $\mathfrak{G}$ by
$Me_{1}=2e_{1}-e_{2}$, $Me_{N}=-e_{N-1}+2e_{N}$ and

$Me_{k}=-e_{k-1}+2e_{k}-e_{k+1}\quad\mbox{for }k\in\\{2,\ldots,N-1\\}.$

Then, the operator $H_{0}$ may be considered as the Hamiltonian of the system
describing the behavior of a free particle moving in the discrete strip
$\mathbb{Z}\times\\{1,...,N\\}$.

The eigenvalues and a corresponding basis of (normalized) eigenvectors of $M$
are respectively given by

$\lambda_{j}=4\sin^{2}\left(\frac{\pi j}{2(N+1)}\right),\quad
v_{j}=\sum_{k=1}^{N}\sqrt{\frac{2}{N+1}}\sin\left(\frac{jk\pi}{N+1}\right)e_{k},\;\;j\in\\{1,\ldots,N\\}.$

###  4.2. Discrete Laplace operator on a semi-strip

Let $(e_{j})_{j\in\mathbb{Z}}$ be the canonical basis of
$\mathfrak{G}=\ell^{2}(\mathbb{Z})$ and consider the rank $N$ operator $M$
defined on $\ell^{2}(\mathbb{Z})$ by $Me_{1}=2e_{1}-e_{2}$,
$Me_{N}=-e_{N-1}+2e_{N}$ and

$Me_{k}=\begin{cases}-e_{k-1}+2e_{k}-e_{k+1}&\mbox{for
}k\in\\{2,\ldots,N-1\\},\\\ 0&\mbox{for
}k\in\mathbb{Z}\setminus\\{1,\ldots,N\\}.\end{cases}$

Then $H_{0}$ is the Hamiltonian describing a particle which can move in a
strip, or in straight lines in $\mathbb{Z}\times\mathbb{Z}$. In this case, the
particle cannot jump from the strip to the straight lines and vice versa, but
after introducing the perturbation, we can allow the interaction of these two
parts (Figures 5 and 5).

Figure 4. Case B: unperturbed situation, N=2

Figure 5. Case B: schematic perturbation, N=2

###  4.3. Coupling with a non-Hermitian model

The non-Hermitian Anderson model was first proposed for the analysis of vortex
pinning in type-II superconductors [16] and has also been applied to the study
of population dynamics [25]. Here, we consider a non-random version of this
model. Let $\mathfrak{G}=\ell^{2}({\mathbb{Z}}_{m})$ for some $m\geq 2$, where
${\mathbb{Z}}_{m}=\mathbb{Z}/m\mathbb{Z}$, and
$(e_{k})_{k\in{\mathbb{Z}}_{m}}$ its canonical orthonormal basis. Consider the
operator $M$ defined by

$\displaystyle Me_{k}$ $\displaystyle=e^{g}e_{k-1}+e^{-g}e_{k+1},\quad
k\in{\mathbb{Z}}_{m},g\in{\mathbb{R}}.$

The operator $M$ is diagonalizable, with eigenvalues and a corresponding basis
of eigenvectors given respectively by

$\displaystyle\lambda_{j}$
$\displaystyle=e^{g}e^{ij\theta_{m}}+e^{-g}e^{ij\theta_{m}}=2\cosh g\cos
j\theta_{m}+2i\sinh g\sin j\theta_{m}\quad,\quad\theta_{m}:=\frac{2\pi}{m},$
$\displaystyle v_{j}$
$\displaystyle=\sum_{k\in{\mathbb{Z}}_{m}}e^{ijk\theta_{m}}e_{k},\quad
j\in{\mathbb{Z}}_{m}.$

Similar to the previous example, if $g=0$, the operator $H_{0}$ is nothing but
the discrete Laplace operator on the tube ${\mathbb{Z}}\times{\mathbb{Z}}_{m}$
(Figure 6).

Figure 6.  Tube $\mathbb{Z}\times\mathbb{Z}_{5}$

###  4.4. Coupling with a ${\mathcal{P}T}$-symmetric Hamiltonian

The pertinence of $\mathcal{PT}$-symmetric Hamiltonians in physics is
discussed in e.g. [3, 23]. For illustrative purpose, we consider in the sequel
a minimal example of a non-Hermitian, $\mathcal{PT}$-symmetric system (see
[23] for details).

Let $\mathfrak{G}={\mathbb{C}}^{2}$ and $(e_{k})_{k\in\\{1,2\\}}$ its
canonical orthonormal basis. Identifying the operators with their matrix
representations in this basis, let $M$ be defined by

$\displaystyle
M=\kappa\sigma_{1}+i\gamma\sigma_{3}\quad\text{with}\;\;\quad\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),$

where $\kappa$ and $\gamma$ are non negative.

If $\kappa\neq\gamma$, direct calculations show that the eigenvalues and a
corresponding basis of eigenvectors of $M$ are respectively given by:

$\displaystyle\lambda_{j}$
$\displaystyle=(-1)^{j+1}\sqrt{\kappa^{2}-\gamma^{2}},$ $\displaystyle v_{j}$
$\displaystyle=(i\gamma+(-1)^{j+1}\sqrt{\kappa^{2}-\gamma^{2}})e_{1}+\kappa
e_{2},$

where $j\in\\{1,2\\}$. In this case, $M$ is diagonalizable but the
eigenprojectors are not orthogonal unless $\gamma=0$.

The linear operator $M$ commutes with the antilinear operator $\mathcal{PT}$,
where ${\mathcal{P}}$ stands for the linear (and unitary) operator
$\sigma_{1}$ while ${\mathcal{T}}$ stands for the (antilinear) complex
conjugation operator. If $\gamma<\kappa$, $\sigma(M)\subset{\mathbb{R}}$ while
if $\gamma>\kappa$, $\sigma(M)\subset i{\mathbb{R}}$.

If $\kappa=\gamma>0$, the $\mathcal{PT}$ symmetry is spontaneously broken in
the sense that $M$ is not diagonalizable anymore.

##  5\. Preliminary results and proof of Proposition 2.2

###  5.1. Study of the resolvent of the free Hamiltonian near the spectral
thresholds

The first step in our analysis is the study of the behavior of the resolvent
of the free Hamiltonian $H_{0,\bullet}$ near the spectral thresholds. In order
to unify the analysis for both cases (A) and (B), we introduce the operator
$\widehat{\pi}_{0,\bullet}$ on $\mathfrak{G}$ defined by

$\widehat{\pi}_{0,\bullet}:=\left\\{\begin{array}[]{lll}0_{\mathfrak{G}}\;\;\text{if}\;\bullet=A\\\
\pi_{0}\;\;\text{if}\;\bullet=B,\end{array}\right.$

where we recall that $\pi_{0}$ denotes the projection onto the infinite-
dimensional subspace ${\rm Ker}(M)$ in case (B). Using the fact that $M$ is
diagonalizable, for any $z\in\mathbb{C}\setminus\sigma(H_{0,\bullet})$, one
has

(5.1)
$(H_{0,\bullet}-z)^{-1}=\sum_{j=1}^{d}(\Delta+\lambda_{j}-z)^{-1}\otimes\pi_{j}+(\Delta-z)^{-1}\otimes\widehat{\pi}_{0,\bullet}.$

Let us recall the following basic properties of the one-dimensional discrete
Laplacian on $\ell^{2}(\mathbb{Z})$. Let $\mathcal{F}:\ell^{2}(\mathbb{Z})\to
L^{2}(\mathbb{T})$ be the unitary discrete Fourier transform defined by

$(\mathcal{F}u)(\theta)=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}e^{-in\theta}u(n),\quad
u\in\ell^{2}(\mathbb{Z}),\theta\in\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}.$

The operator $\Delta$ is unitarily equivalent to the multiplication operator
on $L^{2}(\mathbb{T})$ by the function $f:\theta\mapsto 2-2\cos(\theta)$. More
precisely, one has

(5.2) $(\mathcal{F}(\Delta u))(\theta)=f(\theta)(\mathcal{F}u)(\theta),\quad
u\in\ell^{2}(\mathbb{Z}),\theta\in\mathbb{T}.$

Hence, the operator $\Delta$ is selfadjoint in $\ell^{2}(\mathbb{Z})$ and its
spectrum is absolutely continuous and coincides with the range of the function
$f$, that is $\sigma(\Delta)=\sigma_{{\rm ac}}(\Delta)=[0,4]$.

For any $z\in\mathbb{C}\setminus[0,4]$, the kernel of $(\Delta-z)^{-1}$ is
given by (see for instance [22])

(5.3)
$R_{0}(z;n,m)=\frac{e^{-i\theta(z)|n-m|}}{2i\sin(\theta(z))},\quad(n,m)\in\mathbb{Z}^{2},$

where $\theta(z)$ is the unique solution to the equation $2-2\cos(\theta)=z$
lying in the region $\\{\theta\in\mathbb{C};-\pi\leq{\rm
Re}\,\theta\leq\pi,{\rm Im}\,\theta<0\\}$.

Using this explicit representation of the kernel, we can prove the following
result.

######  Lemma 5.1.

Let $z_{0}\in[0,4)$ and $\rho>0$. There exists $0<\varepsilon_{0}\ll 1$ such
that the operator-valued function

$D_{\varepsilon_{0}}^{*}(0)\cap\mathbb{C}_{1}\ni k\mapsto
W_{-\rho}(\Delta-(z_{0}+k^{2}))^{-1}W_{-\rho}$

admits an analytic extension to $D_{\varepsilon_{0}}^{*}(0)$ if $z_{0}=0$ and
to $D_{\varepsilon_{0}}(0)$ if $z_{0}\in(0,4)$, with values in the Hilbert-
Schmidt class operators $\mathfrak{S}_{2}(\ell^{2}(\mathbb{Z}))$. This
extension will be denoted $R_{0}(z_{0}+k^{2})$.

In the next result, we show that the weighted resolvent of the free
Hamiltonian $H_{0,\bullet}$ extends meromorphically near any
$\lambda_{q}\in\mathcal{T}_{\bullet}$ and we precise the nature of its
singularity at $\lambda_{q}$. Set

$\mathcal{H}_{\bullet}:=\left\\{\begin{array}[]{lll}\mathfrak{S}_{\infty}(\mathcal{H})\;\;\text{if}\;\bullet=A\\\
\mathcal{B}(\mathcal{H})\;\;\text{if}\;\bullet=B.\end{array}\right.$

######  Lemma 5.2.

Let $\lambda_{q}\in\mathcal{T}_{\bullet}$. There exists $0<\varepsilon_{0}\ll
1$ such that the operator-valued function

(5.4) $D^{*}_{\varepsilon_{0}}(0)\cap\mathbb{C}_{1}\ni
k\mapsto\bm{W}_{-\rho}(H_{0,\bullet}-z_{q}(k))^{-1}\bm{W}_{-\rho}$

admits an analytic extension to $D^{*}_{\varepsilon_{0}}(0)$, with values in
$\mathcal{H}_{\bullet}$, denoted $\mathcal{R}_{0,\bullet}^{(q)}(k)$. Moreover:

  * (i)

If $\lambda_{q}$ is non-degenerate, then

(5.5) $\mathcal{R}_{0,\bullet}^{(q)}(k)-\frac{a_{-1}\otimes\pi_{q}}{k}\in{\rm
Hol}\big{(}D_{\varepsilon_{0}}(0);\mathcal{H}_{\bullet}\big{)}.$

  * (ii)

If $\lambda_{q}$ is degenerate such that $\lambda_{q}=\lambda_{p}+4$ for some
$p\in\\{1,...,d\\}$, then

(5.6)
$\mathcal{R}_{0,\bullet}^{(q)}(k)-\frac{a_{-1}\otimes\pi_{q}+b_{-1}\otimes\pi_{p}}{k}\in{\rm
Hol}\big{(}D_{\varepsilon_{0}}(0);\mathcal{H}_{\bullet}\big{)}.$

###### Proof.

Let us start with the proof of the first part of the result on the analytic
extension. We first consider the case (A). Setting
$z_{j}^{(q)}:=\lambda_{q}-\lambda_{j}$ it follows from (5.1) that

(5.7)
$\bm{W}_{-\rho}(H_{0,A}-z_{q}(k))^{-1}\bm{W}_{-\rho}=\sum_{j=1}^{d}W_{-\rho}\big{(}\Delta-(z_{j}^{(q)}+k^{2})\big{)}^{-1}W_{-\rho}\otimes\pi_{j}.$

The above sum splits into the following two terms

(5.8) $\displaystyle\bm{W}_{-\rho}(H_{0,A}-z_{q}(k))^{-1}\bm{W}_{-\rho}=$
$\displaystyle\sum_{z_{j}^{(q)}\in[0,4]}W_{-\rho}\big{(}\Delta-(z_{j}^{(q)}+k^{2})\big{)}^{-1}W_{-\rho}\otimes\pi_{j}$
$\displaystyle+$
$\displaystyle\sum_{z_{j}^{(q)}\notin[0,4]}W_{-\rho}\big{(}\Delta-(z_{j}^{(q)}+k^{2})\big{)}^{-1}W_{-\rho}\otimes\pi_{j}.$

The second term in the RHS is clearly analytic with respect to $k$ in a small
neighborhood of $0$. On the other hand, by Lemma 5.1, the first term in the
RHS extends to an analytic function of $k$ in $D_{\varepsilon_{0}}^{*}(0)$ for
$\varepsilon_{0}>0$ small enough. The extension is clearly in
$\mathfrak{S}_{\infty}(\mathcal{H})$.

Consider now the case (B). The only difference with respect to the above case
comes from $q=0$. From (5.1) again one has

$\displaystyle\bm{W}_{-\rho}(H_{0,B}-z_{0}(k))^{-1}\bm{W}_{-\rho}$
$\displaystyle=\sum_{j=1}^{d}W_{-\rho}\big{(}\Delta-(-\lambda_{j}+k^{2})\big{)}^{-1}W_{-\rho}\otimes\pi_{j}+\bm{W}_{-\rho}(\Delta-k^{2})^{-1}\bm{W}_{-\rho}\otimes\pi_{0}.$

In this case, since the operator $\pi_{0}$ is not compact, the extension
belongs to $\mathcal{B}(\mathcal{H})$.

Turn now to the proof of the second part of the result. We only prove (i) in
case (A). The other assertion works similarly.

Recall from (5.7) that we have

$\displaystyle\bm{W}_{-\rho}(H_{0,A}-z_{q}(k))^{-1}$
$\displaystyle\bm{W}_{-\rho}=\sum_{j\neq
q}W_{-\rho}\big{(}\Delta-(z_{j}^{(q)}+k^{2})\big{)}^{-1}W_{-\rho}\otimes\pi_{j}$
$\displaystyle+W_{-\rho}(\Delta-k^{2})^{-1}W_{-\rho}\otimes\pi_{q}.$

Since $\lambda_{q}\neq\lambda_{j}+4$ for all $j\neq q$, it follows that
$z_{j}^{(q)}\notin\\{0,4\\}$ for any $j\neq q\in\\{1,...,d\\}$. Consequently,
by Lemma 5.1, the first term in the RHS of the above equation extends
analytically in a small neighborhood of $0$ with values in
$\mathfrak{S}_{\infty}(\mathcal{H})$. On the other hand, the kernel of the
operator $W_{-\rho}(\Delta-k^{2})^{-1}W_{-\rho}$ is given by

(5.9)
$\frac{e^{-\frac{\rho}{2}|n|}}{\|e^{-\frac{\rho}{2}|\cdot|}\|_{\ell^{2}(\mathbb{Z})}}R_{0}(k;n,m)\frac{e^{-\frac{\rho}{2}|m|}}{\|e^{-\frac{\rho}{2}|\cdot|}\|_{\ell^{2}(\mathbb{Z})}},$

where $R_{0}(k^{2};n,m)$ is defined by (5.3). One can write

(5.10)
$R_{0}(k^{2};n,m)=\frac{i}{k\sqrt{4-k^{2}}}+\frac{i(e^{i|n-m|2\arcsin\frac{k}{2}}-1)}{k\sqrt{4-k^{2}}}=\frac{i}{2k}+r(k;n,m),$

with

$r(k;n,m):=i\Big{(}\frac{1}{k\sqrt{4-k^{2}}}-\frac{1}{2k}\Big{)}+\frac{i(e^{i|n-m|2\arcsin\frac{k}{2}}-1)}{k\sqrt{4-k^{2}}}.$

One easily verifies that the function $r$ extends to a holomorphic function in
a small neighborhood of $0$. Therefore, putting together (5.9) and (5.10), one
obtains

(5.11)
$W_{-\rho}(\Delta-k^{2})^{-1}W_{-\rho}\otimes\pi_{q}=\frac{a_{-1}\otimes\pi_{q}}{k}+\mathcal{A}(k)\otimes\pi_{q},$

where $\mathcal{A}(k)$ acts on $\ell^{2}(\mathbb{Z})$ with kernel
$W_{-\rho}(n)r(k;n,m)W_{-\rho}(m)$. This ends the proof. ∎

######  Remark 5.3.

In case (B), replacing one of the weight operators $\bm{W}_{-\rho}$ in (5.4)
by $W_{-\rho}\otimes K$ with $K\in\mathfrak{S}_{\infty}(\mathfrak{G})$ ensures
that the analytic extension holds in $\mathfrak{S}_{\infty}(\mathcal{H})$. For
instance,

$D^{*}_{\varepsilon_{0}}(0)\cap\mathbb{C}_{1}\ni k\mapsto(W_{-\rho}\otimes
K)(H_{0,B}-z_{q}(k))^{-1}\bm{W}_{-\rho}$

admits an analytic extension to $D^{*}_{\varepsilon_{0}}(0)$, with values in
$\mathfrak{S}_{\infty}(\mathcal{H})$.

###  5.2. Proof of Proposition 2.2

The proof is a consequence of Lemma 5.2 and the analytic Fredholm extension
Theorem. From the resolvent identity

$(H_{\omega,\bullet}-z)^{-1}(I+\omega
V(H_{0,\bullet}-z)^{-1})=(H_{0,\bullet}-z)^{-1},$

it follows that

(5.12)
$\bm{W}_{-\rho}(H_{\omega,\bullet}-z_{q}(k))^{-1}\bm{W}_{-\rho}=\bm{W}_{-\rho}(H_{0,\bullet}-z_{q}(k))^{-1}\bm{W}_{-\rho}\big{(}I+\mathcal{P}_{\omega,\bullet}(z_{q}(k))\big{)}^{-1},$

where

(5.13)
$\mathcal{P}_{\omega,\bullet}(z):=\omega\bm{W}_{\rho}V(H_{0,\bullet}-z)^{-1}\bm{W}_{-\rho}.$

Lemma 5.2 implies that there exists $\varepsilon_{0}>0$ such that the
operator-valued function $k\mapsto\mathcal{P}_{\omega,\bullet}(z_{q}(k))$
defined by (5.13) extends to an analytic function in
$D^{*}_{\varepsilon_{0}}(0)$. In case (A), Lemma 5.2 again ensures that this
extension is with values in $\mathfrak{S}_{\infty}(\mathcal{H})$. On the other
hand, under Assumption 2.1 (B), assume for instance that $V\geq 0$. Then,
there exists a bounded operator $\mathcal{U}$ such that
$U=\bm{W}_{-\rho}\mathcal{U}\bm{W}_{-\rho}$ and it follows that

$\displaystyle\mathcal{P}_{\omega,B}(z_{q}(k))$
$\displaystyle=\omega\bm{W}_{\rho}V(H_{0,B}-z_{q}(k))^{-1}\bm{W}_{-\rho}$
$\displaystyle=\omega\bm{W}_{\rho}(1\otimes
K^{\ast})\bm{W}_{-\rho}\mathcal{U}\bm{W}_{-\rho}(1\otimes
K)(H_{0,B}-z_{q}(k))^{-1}\bm{W}_{-\rho}$ $\displaystyle=\omega(1\otimes
K^{\ast})\mathcal{U}(W_{-\rho}\otimes
K)(H_{0,B}-z_{q}(k))^{-1}\bm{W}_{-\rho}.$

According to Remark 5.3, $(W_{-\rho}\otimes
K)(H_{0,B}-z_{q}(k))^{-1}\bm{W}_{-\rho}$ extends to an analytic extension in a
small neighborhood of $0$ with values in $\mathfrak{S}_{\infty}(\mathcal{H})$.
Since the operator $(1\otimes K^{\ast})\mathcal{U}$ is bounded, it follows
that the analytic extension of $\mathcal{P}_{\omega,B}(z_{q}(k))$ is also with
values in $\mathfrak{S}_{\infty}(\mathcal{H})$. Therefore the analytic
Fredholm theorem ensures that

$D^{*}_{\varepsilon_{0}}(0)\cap\mathbb{C}_{1}\ni
k\mapsto\big{(}I+\mathcal{P}_{\omega,\bullet}(z_{q}(k))\big{)}^{-1}$

admits a meromorphic extension to $D^{*}_{\varepsilon_{0}}(0)$. We use the
same notation for the extended operator. Hence, the operator-valued function
$k\mapsto\bm{W}_{-\rho}(H_{\omega,\bullet}-z_{q}(k))^{-1}\bm{W}_{-\rho}$
extends to a meromorphic function of $k\in D^{*}_{\varepsilon_{0}}(0)$. This
ends the proof of Propositions 2.2.

##  6\. Proofs Theorems 3.1

Let us start with the following preliminary results which precise the nature
of the singularity of the meromorphic extension of the weighted resolvent at
the thresholds. Recall that $a_{-1}$ is the operator in $\ell^{2}(\mathbb{Z})$
with kernel $\frac{i}{2}W_{-\rho}(n)W_{-\rho}(m)$ and we introduce the
operator $b_{-1}$ in $\ell^{2}(\mathbb{Z})$ with kernel
$-\frac{(-1)^{n+m}}{2}W_{-\rho}(n)W_{-\rho}(m)$.

######  Proposition 6.1.

Let $\lambda_{q}\in\mathcal{T}_{\bullet}$ be such that
$\lambda_{q}\neq\zeta_{p}$ for all $p\neq q$. There exists $\varepsilon_{0}>0$
small enough such that for $k\in D^{*}_{\varepsilon_{0}}(0)$ the following
statements hold:

  * (i)

If $\lambda_{q}=\lambda_{q}$ then

(6.1)
$\bm{W}_{-\rho}(H_{0,\bullet}-z_{q}(k))^{-1}\bm{W}_{-\rho}-\frac{a_{-1}\otimes\pi_{q}}{k}\in\text{Hol}\,\big{(}D_{\varepsilon_{0}}(0);\mathfrak{E}\big{)}.$

  * (ii)

If $\lambda_{q}=\lambda_{q}+4$ then

(6.2)
$\bm{W}_{-\rho}(H_{0,\bullet}-z_{q}(k))^{-1}\bm{W}_{-\rho}-\frac{b_{-1}\otimes\pi_{q}}{k}\in\text{Hol}\,\big{(}D_{\varepsilon_{0}}(0);\mathfrak{E}\big{)}.$

Here $\mathfrak{E}$ is equals to
$\mathcal{S}_{\infty}(\ell^{2}(\mathbb{Z})\otimes\mathfrak{G})$ in the case
$\bullet=A$ and equals to $\mathcal{B}(\ell^{2}(\mathbb{Z};\mathcal{G}))$ in
the case $\bullet=B$.

###### Proof.

Let us start with the case (A). Suppose that $\lambda_{q}=\lambda_{q}$ and
recall from (5.1) that we have

$\displaystyle\bm{W}_{-\rho}(H_{0,A}-z_{q}(k))^{-1}$
$\displaystyle\bm{W}_{-\rho}=\sum_{j\neq
q}W_{-\rho}\big{(}\Delta-(\lambda_{q}-\lambda_{j}+k^{2})\big{)}^{-1}W_{-\rho}\otimes\pi_{j}$
$\displaystyle+W_{-\rho}\big{(}\Delta-k^{2}\big{)}^{-1}W_{-\rho}\otimes\pi_{q}.$

Since $\lambda_{q}\neq\lambda_{j}+4$ for all $j\neq q$ it follows that
$\lambda_{q}-\lambda_{j}\notin\\{0,4\\}$ for any $j\neq q\in\\{1,...,d\\}$.
Consequently, by Lemma 5.1, the first term in the RHS of the above equation
extends analytically in a small neighborhood of $0$. On the other hand, taking
the Laurent expansion of $k\mapsto
W_{-\rho}\big{(}\Delta-k^{2}\big{)}^{-1}W_{-\rho}\otimes\pi_{q}$ near $k=0$
using the fact that the kernel of $\big{(}\Delta-k^{2}\big{)}^{-1}$ is given
by according to (5.3) one gets

$W_{-\rho}\big{(}\Delta-k^{2}\big{)}^{-1}W_{-\rho}\otimes\pi_{q}-\frac{1}{k}a_{-1}\otimes\pi_{q}\in\text{Hol}\,\big{(}D_{\varepsilon_{0}}(0);\mathcal{S}_{\infty}(\ell^{2}(\mathbb{Z})\otimes\mathfrak{G})\big{)},$

which proves the claim of (i). The proof of (ii) is similar except that in
this case the kernel of $(\Delta-(4+k^{2}))^{-1}$ is given by
$\frac{-(-1)^{n+m}}{2}W_{-\rho}(n)W_{-\rho}(m)$.

Consider now the case (B) and suppose that $\lambda_{q}=\lambda_{q}$. From
(5.1) we have

$\displaystyle\bm{W}_{-\rho}(H_{0,B}-z_{q}(k))^{-1}$
$\displaystyle\bm{W}_{-\rho}=\sum_{j=1}^{d}W_{-\rho}\big{(}\Delta-(\lambda_{q}-\lambda_{j}+k^{2})\big{)}^{-1}W_{-\rho}\otimes\pi_{j}$
$\displaystyle+W_{-\rho}\big{(}\Delta-(\lambda_{q}+k^{2})\big{)}^{-1}W_{-\rho}\otimes\pi_{0}.$

Assume first that $\lambda_{q}\neq 0$. It follows from Lemma 5.1 that the
second term in the RHS of the above equation extends analytically in a small
neighborhood of $0$. The first term can be treated as above and then we get
the claim. The case $\lambda_{q}=0$ follows again from Lemma 5.1 and the above
analysis. ∎

##  7\. Proofs of Theorems 3.1 and 3.2

###  7.1. Proof of part (i) of Theorem 3.1

Let $\lambda_{q}\in\mathcal{T}_{A}$ be a fixed threshold and assume that
$\lambda_{q}$ is non-degenerate in the sense of Definition 2.1.

According to Lemma 5.2, there exists $\varepsilon_{0}>0$ and an analytic
function $\mathcal{G}$ in $D_{\varepsilon_{0}}(0)$ with values in
$\mathfrak{S}_{\infty}(\mathcal{H})$ such that for all $k\in
D_{\varepsilon_{0}}^{*}(0)$ we have

(7.1)
$\mathcal{R}_{0,A}^{(q)}(k)=\frac{a_{-1}\otimes\pi_{q}}{k}+\mathcal{G}(k).$

It follows from equation (5.12) that for all $k\in
D_{\varepsilon_{0}}^{*}(0)$,

(7.2)
$\mathcal{R}_{\omega,A}^{(q)}(k)=\Big{(}\frac{a_{-1}\otimes\pi_{q}}{k}+\mathcal{G}(k)\Big{)}\left[I+\mathcal{P}_{\omega,A}(z_{q}(k))\right]^{-1},$

where $\mathcal{P}_{\omega,A}(z_{q}(k))$ is defined by (5.13). More precisely,
one has

(7.3)
$\left[I+\mathcal{P}_{\omega,A}(z_{q}(k))\right]^{-1}=\big{(}I+\omega{V}_{\rho}\mathcal{R}_{0,A}^{(q)}(k)\big{)}^{-1}.$

Since $\mathcal{G}$ is analytic near $0$, it follows that for $|\omega|$ small
enough, the operator-valued function $I+\omega{V}_{\rho}\mathcal{G}(k)$ is
invertible. Using (7.1), one writes

(7.4)
$\left[I+\mathcal{P}_{\omega,A}(z_{q}(k))\right]^{-1}=\Big{(}I+\frac{\omega}{k}\mathcal{L}_{\omega}(k)\Big{)}^{-1}\left(I+\omega{V}_{\rho}\mathcal{G}(k)\right)^{-1},$

where $\mathcal{L}_{\omega}(k)$ is the operator in $\mathcal{H}$ defined by

$\mathcal{L}_{\omega}(k):=\left(I+\omega{V}_{\rho}\mathcal{G}(k)\right)^{-1}{V}_{\rho}(a_{-1}\otimes\pi_{q}).$

Putting together (7.2) and (7.4), we obtain, for all $k\in
D_{\varepsilon_{0}}^{*}(0)$,

(7.5)
$\mathcal{R}_{\omega,A}^{(q)}(k)=\Big{(}\frac{a_{-1}\otimes\pi_{q}}{k}+\mathcal{G}(k)\Big{)}\Big{(}I+\frac{\omega}{k}\mathcal{L}_{\omega}(k)\Big{)}^{-1}\left(I+\omega{V}_{\rho}\mathcal{G}(k)\right)^{-1}.$

Then, the poles of $k\mapsto\mathcal{R}_{\omega,A}^{(q)}(k)$ near $0$ coincide
with those of the operator-valued function

(7.6) $k\mapsto
J_{\omega}(k):=\Big{(}\frac{a_{-1}\otimes\pi_{q}}{k}+\mathcal{G}(k)\Big{)}\Big{(}I+\frac{\omega}{k}\mathcal{L}_{\omega}(k)\Big{)}^{-1}.$

We shall make use of the following elementary result whose proof is omitted.

######  Lemma 7.1.

Let $\mathcal{K}$ be a Hilbert space and consider two linear operators
$A,\Pi:\mathcal{K}\to\mathcal{K}$ such that $\Pi^{2}=\Pi$ and $A\Pi=A$. Then,
$I+A$ is invertible if and only if $\Pi(I+A)\Pi:{\rm Ran}\,\Pi\to{\rm
Ran}\,\Pi$ is invertible, and in this case one has

$(I+A)^{-1}=(I-\widetilde{\Pi}A\Pi)B^{-1}+\widetilde{\Pi},$

where $\widetilde{\Pi}:=I-\Pi$ and $B^{-1}:=(\Pi(I+A)\Pi)^{-1}\oplus 0$ with
respect to the decomposition $\mathcal{K}={\rm Ran}\,\Pi\oplus{\rm
Ran}\,\widetilde{\Pi}$.

Let $\Pi_{q}$ be the projection on $\mathcal{H}$ defined by (3.3). Applying
the above result with $A=\frac{\omega}{k}\mathcal{L}_{\omega}(k)$ and
$\Pi=\Pi_{q}$, we get

$\Big{(}I+\frac{\omega}{k}\mathcal{L}_{\omega}(k)\Big{)}^{-1}=\Big{(}I-\frac{\omega}{k}\widetilde{\Pi}_{q}\mathcal{L}_{\omega}(k)\Pi_{q}\Big{)}\biggl{(}\left(\Pi_{q}\left(I+\frac{\omega}{k}\mathcal{L}_{\omega}(k)\right)\Pi_{q}\right)^{-1}\oplus
0\biggr{)}+\widetilde{\Pi}_{q}.$

Here, $\widetilde{\Pi}_{q}:=I-\Pi_{q}$. Therefore, a straightforward
computation yields

$\displaystyle J_{\omega}(k)$
$\displaystyle=\left(\frac{i}{2}\Pi_{q}-\omega\mathcal{G}(k)\Big{(}\frac{k}{\omega}-\widetilde{\Pi}_{q}\mathcal{L}_{\omega}(k)\Pi_{q}\Big{)}\right)\big{(}\left[\Pi_{q}\left(k+\omega\mathcal{L}_{\omega}(k)\right)\Pi_{q}\right]^{-1}\oplus
0\big{)}$ $\displaystyle+\mathcal{G}(k)\widetilde{\Pi}_{q}.$

Since
$\left[\Pi_{q}\left(k+\omega\mathcal{L}_{\omega}(k)\right)\Pi_{q}\right]^{-1}:{\rm
Ran}\,\Pi_{q}\to{\rm Ran}\,\Pi_{q}$, it follows that the operator
$\left[\Pi_{q}\left(k+\omega\mathcal{L}_{\omega}(k)\right)\Pi_{q}\right]^{-1}\oplus
0$ is stable by $\Pi_{q}:\mathcal{H}\to\mathcal{H}$. Consequently,

(7.7)
$\begin{split}J_{\omega}(k)&=\left(\frac{i}{2}-\omega\mathcal{G}(k)\Big{(}\frac{k}{\omega}-\widetilde{\Pi}_{q}\mathcal{L}_{\omega}(k)\Pi_{q}\Big{)}\right)\big{(}\left[\Pi_{q}\left(k+\omega\mathcal{L}_{\omega}(k)\right)\Pi_{q}\right]^{-1}\oplus
0\big{)}\\\ &+\mathcal{G}(k)\widetilde{\Pi}_{q}.\end{split}$

Using the analyticity of $\mathcal{G}$ and $\mathcal{L}_{\omega}$ near $0$,
one sees that
$\frac{i}{2}-\omega\mathcal{G}(k)\big{(}\frac{k}{\omega}-\widetilde{\Pi}_{q}\mathcal{L}_{\omega}(k)\Pi_{q}\big{)}$
is invertible for $|k|$ of order of $|\omega|\ll 1$ small enough. Therefore,
we conclude that the poles near $0$ of $J_{\omega}$ are the same to those of
the operator-valued function

$k\mapsto\left(\Pi_{q}\left(k+\omega\mathcal{L}_{\omega}(k)\right)\Pi_{q}\right)^{-1}:{\rm
Ran}\,\Pi_{q}\to{\rm Ran}\,\Pi_{q}.$

Let $M_{\omega}(k)$ be the matrix of the operator
$\Pi_{q}\left(k+\omega\mathcal{L}_{\omega}(k)\right)\Pi_{q}:{\rm
Ran}\,\Pi_{q}\to{\rm Ran}\,\Pi_{q}$. We have

$\displaystyle\Pi_{q}\left(k+\omega\mathcal{L}_{\omega}(k)\right)\Pi_{q}$
$\displaystyle=k\Pi_{q}+\omega\Pi_{q}\left(I+\omega{V}_{\rho}\mathcal{G}(k)\right)^{-1}{V}_{\rho}(a_{-1}\otimes\pi_{q})\Pi_{q}$
(7.8)
$\displaystyle=k\Pi_{q}+\frac{i}{2}\omega\Pi_{q}{V}_{\rho}\Pi_{q}+\omega^{2}\Pi_{q}S_{\omega}(k)\Pi_{q},$

where $k\mapsto S_{\omega}(k):=\frac{i}{2}\sum_{n\geq
1}(-1)^{n}\omega^{n-1}({V}_{\rho}\mathcal{G}(k))^{n}{V}_{\rho}$ is an
operator-valued function which is analytic near $k=0$ for $|\omega|>0$ small
enough and $\|S_{\omega}(k)\|=\mathcal{O}(1)$ uniformly w.r.t. $k$.

The usual expansion formula for the determinant allows to write

${\rm
det}(M_{\omega}(k))=\omega^{\nu_{q}}\biggl{(}\prod_{j=1}^{r}\Big{(}\frac{k}{\omega}+\bm{\alpha}_{j}^{(q)}\Big{)}^{m_{q,j}}+\omega
s_{\omega}(k)\biggr{)},$

with $\bm{\alpha}_{j}^{(q)}:=\frac{i}{2}\alpha_{j}^{(q)}$, where
$\\{{\alpha_{j}^{(q)}}\\}_{j=1}^{r}$ are the distinct eigenvalues of
$E_{q}:=\Pi_{q}{V}_{\rho}\Pi_{q}:{\rm Ran}\,\Pi_{q}\to{\rm Ran}\,\Pi_{q}$ and
$s_{\omega}$ is an analytic scalar-valued function satisfying

(7.9) $|s_{\omega}(k)|\leq C_{0},$

for some constant $C_{0}>0$ independent of $k$ and $\omega$. We are therefore
led to study the roots of the equation

(7.10)
$\prod_{j=1}^{r}\Big{(}\frac{k}{\omega}+\bm{\alpha}_{j}^{(q)}\Big{)}^{m_{q,j}}+\omega
s_{\omega}(k)=0.$

On the one hand, by a simple contradiction argument one shows that all the
roots of the above equation satisfy (3.5). On the other hand, let
$j_{0}\in\\{1,...,r\\}$ and let $C>0$ be a constant independent of $k$ and
$\omega$. We set

$\delta_{j_{0},C}:=C|\omega|^{1+\frac{1}{m_{q,j_{0}}}}.$

There exists a constant $C^{\prime}>0$ such that for any $k\in\partial
D_{\delta_{j_{0},C}}(-\bm{\alpha}_{j_{0}}\omega)$, one has

$\prod_{j=1,j\neq
j_{0}}^{r}\Big{|}\frac{k}{\omega}+\bm{\alpha}_{j}^{(q)}\Big{|}^{m_{q,j}}\geq
C^{\prime}.$

Consequently,

$\prod_{j=1}^{r}\Big{|}\frac{k}{\omega}+\bm{\alpha}_{j}^{(q)}\Big{|}^{m_{q,j}}\geq
CC^{\prime}|\omega|>|\omega s_{\omega}(k)|,\quad\forall k\in\partial
D_{\delta_{j_{0},C}}(-\bm{\alpha}_{j_{0}}\omega),$

where $C>0$ is chosen such that $CC^{\prime}>C_{0}$, with $C_{0}$ given by
(7.9).

Since both terms in (7.10) are analytic functions of $k$ near $k=0$, it
follows by Rouché Theorem that for $|\omega|$ small, ${\rm
det}(M_{\omega}(k))$ admits exactly $m_{q,j_{0}}$ zeros in
$D_{\delta_{j_{0},C}}(-\bm{\alpha}_{j_{0}}\omega)$, counting multiplicities.
This ends the proof of statement (i).

###  7.2. Proof of part (ii) of Theorem 3.1

Fix $q\in\\{1,...,d\\}$. Equation (3.5) implies that in variable $k$, the
resonances of $H_{\omega,A}$ are distributed in “clusters” around the points
$-\frac{i}{2}\omega\alpha_{j}^{(q)}$, $j\in\\{1,...,r\\}$. Fix
$j\in\\{1,...,r\\}$ and let $C>0$ and $1<\delta<1+\frac{1}{m_{q,j}}$ so that
the disk $D_{C|\omega|^{\delta}}(-\frac{i}{2}\omega\alpha_{j}^{(q)})$ contains
all the resonances of the j-th cluster and only them. Set
$\Gamma_{j}:=\partial
D_{C|\omega|^{\delta}}(-\frac{i}{2}\omega\alpha_{j}^{(q)})$. We will show that

(7.11) ${\rm
rank}\frac{1}{2i\pi}\oint_{\Gamma_{j}}\mathcal{R}^{(q)}_{\omega,A}(k)dk={\rm
rank}\frac{1}{2i\pi}\oint_{\Gamma_{j}}(k\Pi_{q}+\frac{i}{2}\omega\Pi_{q}{V}_{\rho}\Pi_{q})^{-1}\,dk.$

Since the RHS is equal to the multiplicity of
$-\frac{i}{2}\omega\alpha_{j}^{(q)}$, part ii) of Theorem 3.1 follows.

Equality (7.11) is a consequence of the following Lemma and standard arguments
(see for instance [27, Page 14] and Lemma 4.1 of [20, Chapter 1]).

######  Lemma 7.1.

As $|\omega|\to 0$, one has

  1. a)

$\Big{\|}\oint_{\Gamma_{j}}\mathcal{R}^{(q)}_{\omega,A}(k)-\left(k\Pi_{q}+\frac{i}{2}\omega\Pi_{q}{V}_{\rho}\Pi_{q}\right)^{-1}\,dk\Big{\|}=o(1)$.

  2. b)

$\Big{\|}\left(\frac{1}{2i\pi}\oint_{\Gamma_{j}}\mathcal{R}^{(q)}_{\omega,A}(k)dk\right)^{2}-\frac{1}{2i\pi}\oint_{\Gamma_{j}}\mathcal{R}^{(q)}_{\omega,A}(k)dk\Big{\|}=o(1).$

###### Proof.

We simplify the notations by putting

$\mathcal{A}_{0}(k):=k\Pi_{q}+\frac{i}{2}\omega\Pi_{q}{V}_{\rho}\Pi_{q}\quad\text{and}\quad\mathcal{A}(k):=\Pi_{q}\left(k+\omega\mathcal{L}_{\omega}(k)\right)\Pi_{q}$

as operators acting from ${\rm Ran}\,\Pi_{q}\to{\rm Ran}\,\Pi_{q}$. Using
(7.5), (7.6) and (7.7), one gets

(7.12)
$\oint_{\Gamma_{j}}\mathcal{R}^{(q)}_{\omega,A}(k)dk=\oint_{\Gamma_{j}}T_{\omega}(k)(\mathcal{A}(k)^{-1}\oplus
0)U_{\omega}(k)dk,$

where $T_{\omega}(k)=I+\omega\widetilde{T}_{\omega}(k)$ and
$U_{\omega}(k)=I+\omega\widetilde{U}_{\omega}(k)$ are holomorphic operator-
valued functions in $k$ and uniformly bounded inside $\Gamma_{j}$. One writes

(7.13) $\begin{split}T_{\omega}(k)(\mathcal{A}(k)^{-1}&\oplus
0)U_{\omega}(k)=\mathcal{A}(k)^{-1}\oplus
0+\omega\widetilde{T}_{\omega}(k)(\mathcal{A}(k)^{-1}\oplus 0)\\\
&+\omega(\mathcal{A}(k)^{-1}\oplus
0)\widetilde{U}_{\omega}(k)+\omega^{2}\widetilde{T}_{\omega}(k)(\mathcal{A}(k)^{-1}\oplus
0)\widetilde{U}_{\omega}(k).\end{split}$

On the other hand, from (7.1), one has for any $k\in\Gamma_{j}$,

(7.14)
$\mathcal{A}(k)=\mathcal{A}_{0}(k)\big{(}I+\omega^{2}\mathcal{A}_{0}(k)^{-1}\widetilde{S}_{\omega}(k)\big{)},$

with $\widetilde{S}_{\omega}(k):=\Pi_{q}S_{\omega}(k)\Pi_{q}$. The fact that
$M$ and $V$ are selfadjoint in $\mathcal{H}$ implies that
$\Pi_{q}{V_{\rho}}\Pi_{q}$ is selfadjoint. Thus, for any $k\in\Gamma_{j}$, we
have

(7.15)
$\|\mathcal{A}_{0}(k)^{-1}\|=\|(k\Pi_{q}+\frac{i}{2}\omega\Pi_{q}{V}_{\rho}\Pi_{q})^{-1}\|=\frac{1}{{\rm
dist}(k,\sigma(-\frac{i}{2}\omega\Pi_{q}{V}_{\rho}\Pi_{q}))}=\mathcal{O}(|\omega|^{-\delta}).$

Therefore,
$\|\big{(}I+\omega^{2}\mathcal{A}_{0}(k)^{-1}\widetilde{S}_{\omega}(k)\big{)}^{-1}\|=\mathcal{O}(1)$
for $|\omega|$ small enough uniformly w.r.t. $k\in\Gamma_{j}$. It follows from
(7.14) that we have

(7.16)
$\begin{split}\Big{\|}\oint_{\Gamma_{j}}&\,\omega\widetilde{T}_{\omega}(k)(\mathcal{A}(k)^{-1}\oplus
0)\,dk\Big{\|}\\\ &\leq{\rm
Cst.}\,|\omega||\omega|^{\delta}\int_{0}^{2\pi}\|\mathcal{A}_{0}(k_{\omega}(t))^{-1}\|\,dt=\mathcal{O}(|\omega|),\end{split}$

where
$k_{\omega}(t):=-\frac{i}{2}\omega\alpha_{j}^{(q)}+e^{it}C|\omega|^{\delta}$.
The integral over $\Gamma_{j}$ of each one of the two last terms in the RHS of
(7.13) can be estimated in the same way and we obtain

(7.17)
$\begin{split}&\Big{\|}\oint_{\Gamma_{j}}\omega(\mathcal{A}(k)^{-1}\oplus
0)\widetilde{U}_{\omega}(k)dk\Big{\|}=\mathcal{O}(|\omega|),\\\
&\Big{\|}\oint_{\Gamma_{j}}\omega^{2}\widetilde{T}_{\omega}(k)(\mathcal{A}(k)^{-1}\oplus
0)\widetilde{U}_{\omega}(k)dk\Big{\|}=\mathcal{O}(|\omega|).\end{split}$

Now, using (7.15) again, we get
$\|\big{(}I+\omega^{2}\mathcal{A}_{0}(k)^{-1}\widetilde{S}_{\omega}(k)\big{)}^{-1}-I\|=\mathcal{O}(|\omega|^{2-\delta})$,
which yields

$\displaystyle\Big{\|}\oint_{\Gamma_{j}}\mathcal{A}(k)^{-1}$
$\displaystyle-\mathcal{A}_{0}(k)^{-1}\,dk\Big{\|}$
$\displaystyle=\Big{\|}\oint_{\Gamma_{j}}\big{[}(I+\omega^{2}\mathcal{A}_{0}(k)^{-1}\widetilde{S}_{\omega}(k))^{-1}-I\big{]}\mathcal{A}_{0}(k)^{-1}\,dk\Big{\|}$
(7.18) $\displaystyle\leq{\rm
Cst.}\,|\omega|^{2-\delta}|\omega|^{\delta}\int_{0}^{2\pi}\|\mathcal{A}_{0}(k_{\omega}(t))^{-1}\|\,dt=\mathcal{O}(|\omega|^{2-\delta}).$

Putting together (7.12), (7.13), (7.16), (7.17) and (7.2) we get statement a).

Let us now prove statement $b)$. We set
${\widetilde{\Gamma}}_{j}(t)=\\{-\frac{i}{2}\omega\alpha_{j}^{(q)}+e^{-it}\varepsilon|\omega|^{\delta},t\in[0,2\pi]\\}$
with $0<\epsilon<C$ independent of $\omega$. For $k\in\Gamma_{j}$ and
$\widetilde{k}\in\widetilde{\Gamma}_{j}$, using (7.13) and proceeding as
above, one obtains

(7.19)
$\Big{\|}\oint_{\Gamma_{j}}\oint_{\widetilde{\Gamma}_{j}}\mathcal{R}^{(q)}_{\omega,A}(k)\mathcal{R}^{(q)}_{\omega,A}(\widetilde{k})-\mathcal{A}(k)^{-1}\mathcal{A}(\widetilde{k})^{-1}d\widetilde{k}dk\Big{\|}=\mathcal{O}(|\omega|).$

Next, from the resolvent identity one writes

(7.20)
$\mathcal{A}(k)^{-1}\mathcal{A}({\widetilde{k}})^{-1}=\frac{\mathcal{A}(k)^{-1}}{{\widetilde{k}}-k}-\frac{\mathcal{A}({\widetilde{k}})^{-1}}{{\widetilde{k}}-k}-\omega^{2}\mathcal{A}({k})^{-1}\frac{(\widetilde{S}_{\omega}({\widetilde{k}})-(\widetilde{S}_{\omega}({k}))}{{\widetilde{k}}-k}\mathcal{A}({\widetilde{k}})^{-1}.$

Further, since $\epsilon<C$, then for $k\in\Gamma_{j}$ the map
$\widetilde{k}\mapsto\frac{1}{\widetilde{k}-k}$ is holomorphic inside
$\widetilde{\Gamma}_{j}$ so that
$\oint_{{\widetilde{\Gamma}}_{j}}\frac{d\widetilde{k}}{\widetilde{k}-k}=0$.
Thus, one gets

(7.21)
$\oint_{\Gamma_{j}}\oint_{{\widetilde{\Gamma}}_{j}}\frac{\mathcal{A}(k)^{-1}}{{\widetilde{k}}-k}d{\widetilde{k}}dk-\oint_{{\Gamma}_{j}}\oint_{{\widetilde{\Gamma}}_{j}}\frac{\mathcal{A}({\widetilde{k}})^{-1}}{{\widetilde{k}}-k}d{\widetilde{k}}dk=2i\pi\oint_{\Gamma_{j}}{\mathcal{A}(k)^{-1}}dk.$

Using the fact that
$\|\big{(}I+\omega^{2}\mathcal{A}_{0}(k)^{-1}\widetilde{S}_{\omega}(k)\big{)}^{-1}\|=\mathcal{O}(1)$
as above together with (7.14), we obtain by setting
$\widetilde{k}_{\omega}(t^{\prime}):=-\frac{i}{2}\omega\alpha_{j}^{(q)}+e^{-it^{\prime}}\varepsilon|\omega|^{\delta}$
for $t^{\prime}\in[0,2\pi]$,

$\displaystyle|\omega|^{2}\Big{\|}\oint_{\Gamma_{j}}\oint_{{\widetilde{\Gamma}}_{j}}$
$\displaystyle\mathcal{A}({k})^{-1}\frac{(\widetilde{S}_{\omega}({\widetilde{k}})-(\widetilde{S}_{\omega}({k}))}{{\widetilde{k}}-k}\mathcal{A}({\widetilde{k}})^{-1}d{\widetilde{k}}d{k}\Big{\|}$
$\displaystyle\leq$ $\displaystyle{\rm
Cst.}\,|\omega|^{2}|\omega|^{2\delta}\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{\|\mathcal{A}_{0}({k_{\omega}(t)})^{-1}\|\|\mathcal{A}_{0}({\widetilde{k}_{\omega}(t^{\prime})})^{-1}\|}{{|\widetilde{k}_{\omega}(t^{\prime})}-k_{\omega}(t)|}dt^{\prime}dt$
$\displaystyle\leq$ $\displaystyle{\rm
Cst.}\,|\omega|^{2}|\omega|^{2\delta}\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{|\omega|^{-2\delta}}{|\omega|^{\delta}}dt^{\prime}dt$
(7.22) $\displaystyle=$ $\displaystyle\mathcal{O}(|\omega|^{2-\delta}).$

Putting together (7.19)-(7.2) and using (7.13) and (7.16), we get statement
$b)$. ∎

###  7.3. Proof of of Theorem 3.2

The proof can be performed in a similar manner as that of (i) of Theorem 3.1.
The only difference is to define the operator $\mathcal{L}_{\omega}(k)$ by

$\mathcal{L}_{\omega}(k)=\left(I+\omega{V}_{\rho}\mathcal{G}(k)\right)^{-1}{V}_{\rho}\left(a_{-1}\otimes\pi_{q}+b_{-1}\otimes\pi_{p}\right)$

to get the analogous equation of (7.5) in this case. Then, Lemma 7.1 can be
applied with $\Pi=\Pi_{q,p}$ and (7.7) is also obtained. The rest of the proof
is similar.

##  8\. Proof of Theorem 3.4

###  8.1. Resonances as characteristic values

We start this section by giving an alternative definition of the multiplicity
of a resonance. Our first task will be to show that this new definition
coincides with the one given in (2.4).

To begin with, let us recall some definitions and results on characteristic
values. For more details, one refers to [13] and the book [14, Section 4]. Let
$\mathscr{U}$ be a neighborhood of a fixed point $w\in\mathbb{C}$, and
$F:\mathscr{U}\setminus\\{w\\}\longrightarrow{\mathcal{B}}(\mathcal{K})$ be a
holomorphic operator-valued function. The function $F$ is said to be finite
meromorphic at $w$ if its Laurent expansion at $w$ has the form

$F(z)=\sum_{n=m}^{+\infty}(z-w)^{n}A_{n},\quad m>-\infty,$

where (if $m<0$) the operators $A_{m},\ldots,A_{-1}$ are of finite rank.

Assume that the set $\mathscr{U}$ is open and connected, $F$ is finite
meromorphic and Fredholm at each point of $\mathscr{U}$, and there exists
$w_{0}\in\mathscr{U}$ such that $F(w_{0})$ is invertible. Then, there exists a
closed and discrete subset $Z^{\prime}$ of $\mathscr{U}$ such that $F(z)$ is
invertible for each $z\in\mathcal{D}\backslash Z^{\prime}$ and

$F^{-1}:\mathscr{U}\backslash Z^{\prime}\longrightarrow{\rm GL}(\mathcal{K})$

is finite meromorphic and Fredholm at each point of $\mathscr{U}$ [14,
Proposition 4.1.4].

######  Definition 8.1.

The points of $Z^{\prime}$ where the function $F$ or $F^{-1}$ is not
holomorphic are called the characteristic values of $F$. The index of $F$ with
respect to the contour $\partial\Omega$ is defined by

(8.1) $Ind_{\partial\Omega}\hskip
1.42262ptF:=\frac{1}{2i\pi}\textup{Tr}\oint_{\partial\Omega}F^{\prime}(z)F(z)^{-1}dz,$

where $\partial\Omega$ is the boundary of a connected domain
$\Omega\subseteq\mathcal{D}$ not intersecting $Z^{\prime}$. This number is
actually an integer (see section [14, Section 4]).

Now, let $\lambda_{q}\in\mathcal{T}_{\bullet}$ be fixed so that $z_{q}(k)$ is
defined by (2.1). It follows from identity (5.12) and Lemma 5.2 that
$z_{0}=z_{q}(k_{0})\in{\rm
Res}(H_{\omega,\bullet})\setminus\mathcal{T}_{\bullet}$ if and only if
$k_{0}\in D^{*}_{\varepsilon_{0}}(0)$ is a characteristic value of the
operator-valued function $I+\mathcal{P}_{\omega,\bullet}(z_{q}(\cdot))$, where
$\mathcal{P}_{\omega,\bullet}(\cdot)$ is defined by (5.13).

######  Definition 8.2.

For $z_{0}\in{\rm Res}(H_{\omega,\bullet})$, we define

(8.2)
$\textup{mult}_{T}(z_{0}):=Ind_{\gamma_{z}}\,\big{(}I+\mathcal{P}_{\omega,\bullet}(z_{q}(\cdot))\big{)}.$

Here, $\gamma_{z}$ is a positively oriented circle chosen sufficiently small
so that $k_{0}$ is the only characteristic value enclosed by $\gamma_{z}$.

The following result states that both definitions (8.2) and (2.4) of the
multiplicity of a resonance coincide.

######  Lemma 8.3.

Let Assumption 2.1 $(\bullet)$ holds and let $z_{0}\in{\rm
Res}(H_{\omega,\bullet})$. Then, one has

$\textup{mult}_{T}(z_{0})=\textup{mult}(z_{0}).$

###### Proof.

This result is a consequence of Proposition 9.1 applied to
$P=H_{\omega,\bullet}$, $Q=H_{0,\bullet}$ and $S=\omega V$. Set
$B:=\mathcal{H}=\ell^{2}(\mathbb{Z})\otimes\mathfrak{G}$,
$B_{0}:=\ell^{2}_{\rho}(\mathbb{Z})\otimes\mathfrak{G}$ and
$B_{1}:=\ell^{2}_{-\rho}(\mathbb{Z})\otimes\mathfrak{G}$, where
$\ell^{2}_{\pm\rho}(\mathbb{Z}):=\\{u:\mathbb{Z}\to\mathbb{C}:\sum_{n\in\mathbb{Z}}|u(n)|^{2}e^{\pm\rho|n|}<\infty\\}$.
It is standard to see that $B_{0}$ is dense in $B$, $B$ is dense in $B_{1}$
and that the corresponding inclusions are continuous. From Assumption 2.1
($\bullet$), the operator $V$ is bounded. Then, Conditions (I) and (II) are
satisfied since $H_{0,\bullet}+\omega V$ is bounded. Thus, all the
requirements of Proposition 9.1 are met and the result follows. ∎

Notations are those introduced in the proof of Proposition 2.2. Under
Assumption 2.1 (B) with $V\geq 0$, define the operator-valued function

(8.3) $k\mapsto
X_{{\omega}}(z_{0}(k)):=\omega\mathcal{U}^{\frac{1}{2}}(W_{-\rho}\otimes
K)(H_{0,B}-k^{2})^{-1}[\mathcal{U}^{\frac{1}{2}}(W_{-\rho}\otimes K)]^{\ast},$

which is analytic in $D_{\varepsilon_{0}}^{\ast}(0)$ with values in compact
operators, as follows immediately from Remark 5.3. In particular, if one sets
$L:=\mathcal{U}^{\frac{1}{2}}(W_{-\rho}\otimes K)$, then $V=L^{\ast}L$ and
identity

$\big{(}I+\omega L(H_{0,B}-z_{0}(k))^{-1}L^{\ast}\big{)}\big{(}I-\omega
L(H_{\omega,B}-z_{0}(k))^{-1}L^{\ast}\big{)}=I,$

implies that $z_{0}(k)=k^{2}\in\mathcal{S}_{0}$ is a resonance of
$H_{\omega,B}$ if and only if $k$ is a characteristic value of
$I+X_{{\omega}}(z_{0}(\cdot))$. Moreover, the following holds:

######  Lemma 8.4.

The multiplicity of $z_{0}=k_{0}^{2}$ given by (8.2) coincides with the
multiplicity of $k_{0}$ as a characteristic value of
$I+X_{{\omega}}(z_{0}(\cdot))$. That is

$\textup{mult}_{T}(z_{0})=Ind_{\gamma_{z}}\,\big{(}I+X_{{\omega}}(z_{0}(\cdot))\big{)}.$

###### Proof.

Under Assumption 2.1 (B), as in the proof of Lemma 5.2 and Remark 5.3, one can
show that for $k\in D_{\varepsilon_{0}}^{\ast}(0)$ the operators
$\mathcal{P}_{\omega,B}(z_{0}(k))$ and $X_{{\omega}}(z_{0}(k))$ are in the
Schatten class ${\mathfrak{S}}_{p}(\mathcal{H})$. Therefore, one can define
the $p$-regularized determinant of $I+F_{j}(k)$, $j=1$, $2$ where
$F_{1}(k)=\mathcal{P}_{\omega,B}(z_{0}(k))$ and
$F_{2}(k)=X_{{\omega}}(z_{0}(k))$, by

$f_{j}(k)={\rm det}_{p}(I+F_{j}(k)):={\rm
det}\Big{[}(I+F_{j}(k))e^{\sum_{s=0}^{p-1}\frac{(-F_{j}(k))^{s}}{s}}\Big{]}.$

Moreover, $D_{\varepsilon_{0}}^{\ast}(0)\ni k\mapsto f_{j}(k)$ is a
holomorphic function and $k_{0}$ is a zero of $f_{j}$. The Residue theorem
implies that the multiplicity of $k_{0}$ as zero of $f_{j}$ is equal to

$\displaystyle\frac{1}{2i\pi}\int_{\gamma_{z}}\frac{f_{j}^{\prime}(k)}{f_{j}(k)}dk=\frac{1}{2i\pi}\int_{\gamma_{z}}\partial_{k}\ln
f_{j}(k)dk$

$\displaystyle=\frac{1}{2i\pi}\int_{\gamma_{z}}{\rm
Tr}\Big{(}(I+F_{j}(k))^{\prime}(I+F_{j}(k))^{-1}-\sum_{s=0}^{m-1}(I+F_{j}(k))^{\prime}(-F_{j}(k))^{s-1}\Big{)}dk=Ind_{\gamma_{z}}\,\big{(}I+F_{j}(k)\big{)}.$

Now, it suffices to note that the functions $f_{1}$ and $f_{2}$ satisfy for
$k\in D_{\varepsilon_{0}}^{\ast}(0)\cap\mathbb{C}_{1}$

$f_{1}(k)={\rm det}_{p}\big{(}I+\omega
V(H_{0,B}-z_{0}(k))^{-1}\big{)}=f_{2}(k).$

Thus $f_{1}$ and $f_{2}$ coincide in $D_{\varepsilon_{0}}^{\ast}(0)$ and the
result follows. ∎

Introduce the operator

(8.4)
$\mathcal{Q}_{0}:=-iL\bm{W}_{\rho}(a_{-1}\otimes\pi_{0})\bm{W}_{\rho}L^{\ast},$

and let $P_{0}$ be the orthogonal projection onto Ker$({\mathcal{Q}}_{0})$.
Then, according to Lemma 5.2

(8.5)
$\mathcal{T}_{{\omega}}(z_{0}(k))=\frac{i\omega}{k}\mathcal{Q}_{0}+\omega
T_{V}(k)$

for $T_{V}(k)=L\bm{W}_{\rho}\mathcal{G}(k)\bm{W}_{\rho}L^{\ast}$
($\mathcal{G}(k)$ being defined in (7.1)).

###  8.2. Proof of Theorem 3.4

From the above discussion, the poles different from zero of
$\bm{W}_{-\rho}(H_{\omega,B}-z_{0}(k))^{-1}\bm{W}_{-\rho}$ coincide with the
characteristic values $k$ of $I+X_{{\omega}}(z_{0}(\cdot))$.

Set

$\mathcal{A}_{{\omega}}(\zeta):=\mathcal{Q}_{0}-\omega\zeta
T_{V}(-i\omega\zeta).$

which is analytic near $0$. Then, (8.5) yields
$I+X_{{\omega}}(z_{0}(k))=I-\frac{\mathcal{A}_{{\omega}}(\zeta)}{\zeta},$ with
$\zeta=i\omega^{-1}k$.

Now notice the following:

  * •

$\mathcal{A}_{{\omega}}(0)=\mathcal{Q}_{0}$ is selfadjoint.

  * •

Assumption 2.1 (B) implies that $\mathcal{A}_{\omega}(\zeta)$ is compact-
valued.

  * •

$I-\mathcal{A}_{\omega}^{\prime}(0)P_{0}=I-\mathcal{\omega}T_{V}(0)P_{0},$ so
for $\omega\ll 1$ the operator $I-\mathcal{A}_{\omega}^{\prime}(0)P_{0}$ is
invertible.

Thus the conditions of [5, Corollary 3.4. (i) and (ii)] are met. This result
states that in our situation, for $\varepsilon_{0}>0$ small enough and for any
$0<\varepsilon<\varepsilon_{0}$, the characteristic values
$\zeta=i\omega^{-1}k$ such that
$\varepsilon|\omega|<|k|<\varepsilon_{0}|\omega|$, satisfy

(8.6) ${\rm Re}(ik/\omega)\geq 0,\qquad|{\rm Im}(ik/\omega)|=o(|k/\omega|),$

which in turn imply (i) of Theorem 3.4.

Now, let us denote the set of characteristic values of $I+X_{{\omega}}$ by
${\rm Char}(\bullet)$. Equation (8.6) shows that for
$\varepsilon|\omega|<|k|<\varepsilon_{0}|\omega|$, these characteristic values
$\zeta=i\omega^{-1}k$ are concentrated in a sector
$\mathcal{C}_{\theta}(\varepsilon,\varepsilon_{0})$, $\theta>0$. In
particular, using Lemmas 8.3 and 8.4, one has that (counting multiplicities)

(8.7) $\begin{split}\\#\big{\\{}z_{0}(k)\in&{\rm
Res}(H_{\omega,B}):\varepsilon|\omega|<|k|<\varepsilon_{0}|\omega|\big{\\}}\\\
&=\\#\big{\\{}z=i\omega^{-1}k\in{\rm
Char}(\bullet)\cap\mathcal{C}_{\theta}(\varepsilon,\varepsilon_{0})\big{\\}}+\mathcal{O}(1),\end{split}$

for $\varepsilon\searrow 0$.

Further, by Assumption 2.1 (B),
$\mathcal{A}_{{\omega}}(0)=\mathcal{Q}_{0}\in{\mathfrak{S}}_{p}(\mathcal{H})$.
If it also has infinite rank, then [5, Corollary 3.9] implies that there
exists a sequence $(\varepsilon_{j})_{j}$ of positive numbers tending to zero
such that

(8.8) $\\#\big{\\{}z=i\omega^{-1}k\in{\rm
Char}(\bullet)\cap\mathcal{C}_{\theta}(\varepsilon_{j},\varepsilon_{0})\big{\\}}=\textup{Tr}\chi_{[\varepsilon_{j},1]}(\mathcal{Q}_{0})(1+o(1)),\quad
j\to\infty.$

To conclude the proof, it suffices to notice that one has

$\mathcal{Q}_{0}=\frac{1}{2}(L{\bf W}_{\rho}\Pi_{0})(L{\bf
W}_{\rho}\Pi_{0})^{*}\quad\text{and}\quad E_{0}=(L{\bf
W}_{\rho}\Pi_{0})^{*}(L{\bf W}_{\rho}\Pi_{0}),$

so that both operators has the same rank and
$n_{[\varepsilon_{j},1]}(\mathcal{Q}_{0})=n_{[\varepsilon_{j},1]}(\frac{1}{2}E_{0})=n_{[2\varepsilon_{j},2]}(E_{0})$.

##  9\. Appendix

In this appendix we prove an abstract result concerning the multiplicity of
resonances. We use the abstract setting for the theory of resonances as it
appears in [1]. The presentation in this section is given in a more general
framework than the required for our study, so it can be applied in other
settings as well [29, 7].

Let $B$ be a Banach space and $B_{0},B_{1}$ two reflexive Banach spaces such
that $B_{0}\subset B\subset B_{1}$, $B_{0}$ is dense in $B$ and $B$ is dense
in $B_{1}$. Further, the natural injections
$\mathcal{I}_{0}:B_{0}\hookrightarrow B$ and $\mathcal{I}:B\hookrightarrow
B_{1}$ are continuous.

Let $Q:Dom(Q)\subset B\to B$ be a closed linear operator and $S:B\to B_{0}$
linear, such that $S:B\to B$ is bounded and extends as a bounded operator from
$B_{1}$ to $B_{0}$. Define

$P:=Q+S\quad\text{with}\quad Dom(P)=Dom(Q).$

Let $D\subset\mathbb{C}$ be an open subset of the resolvent set of $Q$ and the
resolvent set of $P$, which we suppose non empty. We assume that $D\ni
z\to\mathcal{I}(P-z)^{-1}\mathcal{I}_{0}\in\mathcal{B}(B_{0},B_{1})$ has a
finite meromorphic extension to $D^{+}\supset D$. In the same way we assume
that $D\ni
z\to\mathcal{I}(Q-z)^{-1}\mathcal{I}_{0}\in\mathcal{B}(B_{0},B_{1})$ has an
analytic extension to $D^{+}$. Denote these extensions by $R(z),R_{0}(z)$
respectively.

Therefore, if $z_{0}\in D^{+}$ is a pole of $R(z)$, we have the expansions

(9.1) $\begin{split}R_{0}(z)&=\sum_{j=0}^{\infty}M_{j}(z-z_{0})^{j},\\\
R(z)&=A_{-L}(z-z_{0})^{-L}+\cdots+A_{-1}(z-z_{0})^{-1}+{\rm
Hol}(z),\end{split}$

with ${\rm rank}(A_{-j})<\infty$ for $0<j\leq L$.

We assume the following conditions:

Condition I: The operator $\mathcal{P}$ such that
$Dom({\mathcal{P}}):=\mathcal{I}Dom(P)$ and $\mathcal{P}u:=\mathcal{I}{P}u$,
is closable in $B_{1}$.

Condition II: The set $\mathcal{F}_{0}:=\\{u\in B_{0}\cap Dom(P):Pu\in
B_{0}\\}$ is dense in $B_{0}$.

Let $P_{1}$ be the closure of $\mathcal{P}$. From [1, Proposition 5.2] the
image of $R(z)$ is contained in $Dom(P_{1})$, and using (9.1) it follows from
[1, Theorem 5.5] that

(9.2) $A_{-j-1}=(P_{1}-z_{0})^{j}A_{-1},$ (9.3) ${\rm
Ran}(A_{-j-1})\subset{\rm Ran}(A_{-j}),\quad j\geq 1.$

######  Proposition 9.1.

Let $z_{0}\in D^{+}$ be a pole of $R(z)$ and let $\gamma_{z_{0}}$ be a
positively oriented curve containing $z_{0}$ and no other pole of $R(z)$.
Then,

${\rm rank}(A_{-1})=Ind_{\gamma_{z_{0}}}(I+SR_{0}(z)).$

###### Proof.

The proof follows some ideas of [4, Proposition 3]. Using the resolvent
identity $R(z)-R_{0}(z)=R(z)SR_{0}(z)=-R_{0}(z)SR(z)$, valid for $z\in D$,
algebraic computations show that

(9.4) $A_{-1}=-\sum_{j\geq 0}\sum_{k<0}A_{k}SM_{j-k}SA_{-j-1},$

$\text{where}\,\,A_{-j}=0\,\,\text{if}\,\,j>L$ and $M_{j-k}=0$ if $j-k<0$.
This sum is then finite (see the proof of [4, Lemma 2] for more details).

Using (9.2) and (9.3) one can define the operator $\Pi_{-1}:{\rm
Ran}(A_{-1})\to{\rm Ran}(A_{-1})$

(9.5) $\Pi_{-1}:=-\sum_{j\geq 0}\sum_{k<0}A_{k}SM_{j-k}S(P_{1}-z_{0})^{j}.$

It follows from (9.5) and (9.2) that one has

(9.6) $A_{-1}=\Pi_{-1}A_{-1}.$

Moreover, $\Pi_{-1}^{2}=\Pi_{-1}$. Indeed, let $f\in{\rm Ran}(A_{-1})$ and by
(9.3), take $g_{k,l}$ such that
$A_{-1}g_{k,l}=A_{k}SM_{j-k}S(P_{1}-z_{0})^{j}f$. Then one has

$\Pi_{-1}f=-\sum_{j\geq
0}\sum_{k<0}A_{k}SM_{j-k}S(P_{1}-z_{0})^{j}f=-A_{-1}\sum_{j\geq
0}\sum_{k<0}g_{k,l},$

and (9.6) implies $\Pi_{-1}^{2}f=-\Pi_{-1}A_{-1}\sum_{j\geq
0}\sum_{k<0}g_{k,l}=-A_{-1}\sum_{j\geq 0}\sum_{k<0}g_{k,l}=\Pi_{-1}f.$

Using (9.3) and (9.6) one can show that ${\rm Ran}(A_{-1})={\rm
Ran}(\Pi_{-1})$. Hence ${\rm rank}(A_{-1})={\rm Tr}(\Pi_{-1}).$ Now, using
(9.2) and the cyclicity of the trace, we have

$\displaystyle{\rm Tr}(\Pi_{-1})=$ $\displaystyle-{\rm Tr}\sum_{j\geq
0}\sum_{-L\leq k<0}SM_{j-k}S(P_{1}-z_{0})^{j}A_{k}$ $\displaystyle=$
$\displaystyle-{\rm Tr}\sum_{j\geq 0}\sum_{j-L\leq k<0}SM_{j-k}SA_{k-j}.$

On the other hand, we can see that $(I+SR_{0}(z))(I-SR(z))=I$. Then, we have

$\displaystyle Ind_{\gamma_{z_{0}}}(I+SR_{0}(z))=$ $\displaystyle-{\rm
Tr}\frac{1}{2\pi i}\oint_{\gamma_{z_{0}}}SR(z)S\partial_{z}R_{0}(z)dz$
$\displaystyle=$ $\displaystyle-{\rm Tr}\sum_{L\geq l\geq 1}lSA_{-l}SM_{l}$
$\displaystyle=$ $\displaystyle-{\rm Tr}\sum_{L\geq l\geq 1}lSM_{l}SA_{-l}.$

Noticing that $\sum_{L\geq l\geq 1}lSM_{l}SA_{-l}=\sum_{j\geq 0}\sum_{j-L\leq
k<0}SM_{j-k}SA_{k-j}$, this ends the proof. ∎

Acknowledgements

The first author acknowledges the financial support of the project 042133MR-
POSTDOC of the Universidad de Santiago de Chile.

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