Resonances on regular tree graphs
Olivier Bourget Diomba Sambou Amal Taarabt




We investigate the distribution of the resonances near spectral thresholds
of Laplace operators on regular tree graphs with k-fold branching, k ≥ 1,
perturbed by nonself-adjoint exponentially decaying potentials.
We establish results on the absence of resonances which in particular involve absence
of discrete spectrum near some sectors of the essential spectrum of the operators. 


Robust two-qubit gates in a linear ion crystal using a frequency-modulated driving force
    Kenneth R. Brown
    December 30, 2023
========================================================================================














§ INTRODUCTION






A great interest has been focused in the last decades on spectral analysis of Laplace 
operators on regular trees. This includes local perturbations <cit.>, random settings <cit.> (see also the references therein), and quantum ergodicity regimes 
<cit.>. For complementary results, we refer the reader to the papers <cit.>, 
and for the relationships between the Laplace operator on trees and quantum graphs, see <cit.>. 
However, it seems that resonances have not been systematically studied in the context of 
(regular) trees.



In this paper, we use resonance methods to obtain better understanding of local spectral 
properties for perturbed Schrödinger operators on regular tree graphs with k-fold branching, 
k ≥ 1, as we describe below (cf. Section <ref>). Our techniques are similar to 
those used in <cit.> (and references therein), where self-adjoint perturbations
are considered. Actually, these methods can be extended to nonself-adjoint models, see for 
instance <cit.>. Here, we are focused on some nonself-adjoint perturbations of the Laplace 
operator on regular tree graphs. In particular, we shall derive as a by-product, a description 
of the eigenvalues distribution near the spectral thresholds of the operator. 



Since a nonself-adjoint framework is involved in this article, it is convenient to clarify the 
different notions of spectra we use. Let T be a closed linear operator acting on a separable 
Hilbert space ℋ, and z be an isolated point of σ(T) the spectrum of T. 
If γ is a small contour positively oriented containing z as the only point of σ(T), 
we recall that the Riesz projection P_z associated to z is defined by 

    P_z := 1/2iπ∮_γ (T - ζ)^-1 dζ.

The algebraic multiplicity of z is then defined by 

    (z) := rank (P_z),

and when it is finite, the point z is called a discrete eigenvalue of the operator T. 
Note that we have the inequality m  (z) ≥dim ( Ker 
(T - z) ), which is the geometric multiplicity of z. The equality holds  
if T = T^∗. So, we define the discrete spectrum of T as

    σ_ disc(T) := { z ∈σ(T) : z }.

We recall that if a closed linear operator has a closed range and both its kernel and 
cokernel are finite-dimensional, then it is called a Fredholm operator. Hence, we define 
the essential spectrum of T as

    σ_ ess(T) := { z ∈ : }.
 
Note that σ_ ess(T) is a closed subset of σ(T). 




The paper is organized as follows. In Section <ref>, we present our model.
In section <ref>, we state our main results Theorem <ref> and Corollaries
<ref>, <ref>. Section <ref> is devoted to preliminary results we need due 
to Allard and Froese. In Section <ref>, we establish a formula giving a kernel 
representation of the resolvent associated to the operator we consider and which is 
crucial for our analysis. In Section <ref>, we define and characterize 
the resonances near the spectral thresholds, while in Section <ref> we give the 
proof of our main results. Section <ref> gathers useful tools on the characteristic 
values concept of finite meromorphic operator-valued functions.








§ PRESENTATION OF THE MODEL



We consider an infinite graph 
𝔾 = (𝒱,ℰ) with vertices 𝒱 and edges ℰ, 
and we let ℓ^2(𝒱) be the Hilbert space

    ℓ^2(𝒱) := {ϕ : 𝒱→ : 
    ‖ϕ‖^2 := ∑_v ∈𝒱|ϕ(v) 
    |^2 < ∞},


with the inner product 

    ⟨ϕ,ψ⟩ := ∑_v ∈𝒱ϕ(v) ψ(v).

On ℓ^2(𝒱), we consider the symmetric Schrödinger operator 
defined by

    (ϕ)(v) := - ∑_w   :   w ∼ v( ϕ(w) - ϕ(v) ),

where w ∼ v means that the vertices w and v are connected by an edge. If 
we define on ℓ^2(𝒱) the symmetric operator L by

    (L ϕ)(v) := ∑_w   :   w ∼ vϕ(w),

then it is not difficult to see that the operator  can be written as

    = -L + d,

where d is the multiplication operator by the function (also) noted 
d : 𝒱→, with d(v) denoting the number of edges connected with
the vertex v. Note that when d is bounded, then so is the symmetric operators  
and L, hence self-adjoint. In a regular rooted tree graph with k-fold branching, k ≥ 1, 
(see Figure 2.1 for a binary tree graph), we have d = k + 1 - d_0 with

    d_0(v) =  
    1    if  v  coincides with the root of the tree, 
     
    0    otherwise.

This is the same model described in <cit.> and we refer to 
this paper for more details. 
In (<ref>), d can be viewed as a perturbation of the operator L. It is well know 
(see Lemma <ref>) that the spectrum of the operator L is absolutely continuous, 
coincides with the essential spectrum and is equal to

    σ (L) = σ_ (L) = σ_ (L) = 
    [ -2√(k),2√(k)].







On ℓ^2(𝒱), we define the perturbed operator

    _M :=  + M,

where M is identified with the multiplication operator by the bounded potential function 
(also) noted M. In a regular rooted tree graph with k-fold branching, according to 
above, the operator _M can be written as

    _M = -L + k + 1 - d_0 + M.

In (<ref>), the degree term d_0 can be included in the potential perturbation so 
that _M can be viewed as a perturbation of the operator -L + k + 1. Hence, from 
now on, the operator _M will be written as

    _M = -L + k + 1 + M   with  M := - d_0 + M.

In the sequel, we set 

    t_±(k) := ± 2√(k) + k + 1,

and we shall simply write t_± when no confusion can arise. Then, from (<ref>), 
it follows that the spectrum of the operator -L + k + 1 satisfies

    σ (-L + k + 1) = σ_ (-L + k + 1) = σ_ (-L + k + 1) = 
    [ t_-(k),t_+(k) ],

where the t_±(k) play the role of thresholds of this spectrum.



Now, let us choose some vertex v_0 = 0 ∈𝒱 as the origin of the graph 
𝔾 = (𝒱,ℰ). For v ∈𝒱, we define 
| v | as the length of the shortest path connecting 0 to v. Hence, 
| v | defines in the graph the distance from 0 to v. For r > 0, 
let S_r be the sphere of radius r in the graph defined by

    S_r := { v ∈𝒱 : | v | = r }.








In this case, we have

    𝒱 = _0^∞ S_r,

where  means a disjoint union, so that we have

    ℓ^2(𝒱) = ⊕_0^∞ℓ^2(S_r).

In this paper, we are interested in the case of regular rooted tree graphs (<ref>) 
with k-fold branching, k ≥ 1. Moreover, the potential M will be assumed to satisfy 
the following assumption:



Assumption (A): For v ∈𝒱, we have

    | M(v)|≤ Const.  e^-δ| v |,    with  δ > 0    if   k = 1, 
    δ≥ 6 ln (k)     otherwise.



We point out that in Assumption (A) above, there is no restriction on the 
perturbation potential M concerning its self-adjointness or not.
The case k = 1 includes in particular the case of the Laplacian on 
ℓ^2(,) without any boundary condition at 0.


As mentioned above, in this article we investigate the resonances (or eigenvalues) 
distribution for the operator _M near the spectral thresholds t_±(k) given by 
(<ref>). As this will be observed, the work of Allard and Froese <cit.> will 
play an important role in our analysis (cf. Section <ref> for more details). 
More precisely, in order to establish a suitable representation of the resolvent 
associated to the operator -L + k + 1, k ≥ 1 (cf. Theorem <ref>). 
Under Assumption (A), the perturbation potential M satisfies the 
decay assumption of M. So, if we let Λ_n to denote the orthogonal projection 
onto ⊕_r=0^n ℓ^2(S_r), then with the aid of the Schur lemma, it can
be shown that 

    ‖M - MΛ_n ‖n →∞⟶ 0.

Since ℓ^2(S_r) is a k^r-dimensional space, then (<ref>) implies that 
the operator 
M is the limit in norm of a sequence of finite rank operators. 
Therefore, M is a compact operator and in particular it is relatively 
compact with respect to the operator -L + k + 1. Thus, since the operator 
-L + k + 1 is self-adjoint, then by <cit.> we have a 
disjoint union

    σ(_M) = σ_ ess(_M) σ_ disc(_M).
 
Moreover, Weyl's criterion on the invariance of the essential spectrum implies that

    σ_ (_M) = σ_ 
    (-L + k + 1) = [ t_-(k),t_+(k) ].

However, the (complex) discrete spectrum σ_ disc(_M)
generated by the potential M can only accumulate at the points of 
σ_ (_M).


When M = M^∗, 
σ_ (_M) is just the set of real eigenvalues 
of _M respectively from the right and the left of t_±(k). 


Exploiting the exponential decay of the potential M, we extend 
(cf. Section <ref>) meromorphically in Banach weighted spaces the resolvent of the 
operator _M near z = t_±(k), in some two sheets Riemann surfaces 
ℳ_t_± respectively. The first main difficulty to overcome 
is to establish a good representation of the kernel of the resolvent associated to the
operator -L + k + 1, k ≥ 1 (cf. Section <ref>). We thus define the resonances of 
the operator _M near z = t_±(k) as the poles of the above 
meromorphic extensions. Notice that this set of resonances contains the eigenvalues 
of the operator _M localized near the spectral thresholds t_±(k). 
Otherwise, in the two sheets Riemann surfaces ℳ_t_±,
the resonances will be parametrized by z_t_±(λ) for λ 
sufficiently small for technical reasons. Furthermore, the point λ = 0 corresponds to the threshold 
z = t_±(k) (cf. Section <ref> for more details). Actually, the resonances 
verifying

    z_t_±(λ) ∈ℳ_t_±,   (λ) < 0,

live in the non physical plane while those verifying 

    z_t_±(λ) ∈ℳ_t_±,   (λ) ≥ 0,
 
coincide with the discrete and the embedded eigenvalues of the operator _M 
near t_± and are localized in the physical plane. 
We state Theorem <ref> where we establish an absence of resonances of the operator 
_M near the spectral thresholds t_±(k). In particular, this implies 
results on the absence of discrete spectrum and embedded eigenvalues near t_±(k) 
(cf. Corollaries <ref> and <ref>). To prove these results, we first reduce the analysis 
of resonances near the thresholds t_±(k) to that of the noninvertibility of some
nonself-adjoint compact operators near λ = 0 (cf. Propositions <ref> and 
<ref>). This can be seen as a Birman-Schwinger principle in a nonself-adjoint context.
Afterwards, the reduction made on the problem is reformulated in terms of characteristic
values problems (cf. Propositions <ref> and <ref>). This allows us to apply 
powerful results (cf. Section <ref>) on the theory of characteristic 
values of finite meromorphic operator-valued functions to conclude. 













§ STATEMENT OF THE MAIN RESULTS




Let us first fix some notations. If λ∈, as usual 
|λ| <<1 means that λ is chosen small enough. The set 
of resonances of the operator _M near the spectral thresholds 
t_±(k) given by (<ref>) will be respectively denoted by

    Res_t_±  (_M).

We also recall that near z = t_±(k), the 
resonances are defined in some Riemann surfaces ℳ_t_± and coincide 
with z_t_±(λ), 0 < |λ| <<1. More precisely, 
they are parametrized respectively by

    z_t_±(λ) := t_±(k) ∓λ^2√(k)∈ℳ_t_±.

Furthermore, the embedded eigenvalues and the discrete spectrum of the operator 
_M near t_±(k) are the resonances 
z_t_±(λ) ∈ℳ_t_± with (λ) ≥ 0. 



Now, for 0 < r <<1, let us introduce the punctured neighborhood of λ = 0

    Ω_r^∗ := {λ∈ : 0 < |λ| < r }.

We then can state our first main result that gives an absence of resonances of 
the operator _M near the thresholds t_±(k), in small domains 
of the form t_±(k) ∓√(k) Ω_r^∗^2.

[Absence of resonances]
Assume that the potential M satisfies Assumption (A). Then, for any r > 0 small
enough and any punctured neighborhood Ω_r^∗, we have

    #{ z_t_±(λ) ∈ Res_t_±(_M) : λ∈Ω_r^∗} = 0,

the resonances being counted with their multiplicity given by 
(<ref>) and (<ref>).


Notice that Theorem <ref> just says that the operator _M has no 
resonances in a punctured neighborhood of t_±(k) in the two-sheets Riemann surfaces ℳ_t_± where they are defined.






Since near t_±(k) the discrete spectrum of the operator _M 
corresponds to resonance points z_t_±(λ) ∈ℳ_t_± 
with (λ) ≥ 0, then a first consequence of Theorem <ref> is the 
following result giving a non cluster phenomena of real or non real eigenvalues 
near t_±(k).


Assume that the potential M satisfies Assumption (A). Then, there is no sequence 
(ν_j)_j of non real or real eigenvalues of the operator _M 
accumulating at t_±(k).


Now, thanks to the parametrizations (<ref>), the embedded eigenvalues of the 
operator _M near t_±(k) respectively from the left and the 
right are the resonances z_t_±(λ) = t_±(k) ∓λ^2√(k) 
with λ∈_+ sufficiently small. Therefore, as a second consequence of 
Theorem <ref> together with <cit.>, we have the following:


Assume that the potential M satisfies Assumption (A). Then, for any r > 0 small
enough, the operator _M has no embedded eigenvalues in 

    ( t_-(k),t_-(k) + r^2 ) ∪( t_+(k) - r^2,t_+(k) ).
 
In particular, for M = M^∗, the set of embedded eigenvalues of the operator 
_M in 

( t_-(k),t_+(k) ) is finite. 














§ ON A DIAGONALIZATION OF THE OPERATOR L FOR A REGULAR TREE GRAPH




In this section, we summarize some results and tools we need and which are developed in <cit.>. We shall essentially follow <cit.>
and we refer to the cited papers for more details.



Define the operator Π on ℓ^2(𝒱) by

    (Πϕ)(v) := ∑_w   :   w → vϕ(w),

where for two vertices v and w, v → w means that they are connected
by an edge with | w | = | v | + 1. Using the inner product defined
on the Hilbert space ℓ^2(𝒱) by (<ref>), it can be easily checked 
that the adjoint operator Π^∗ is given by

    (Π^∗ϕ)(v) := ∑_w   :   v → wϕ(w).

If we let L_S be the spherical Laplacian defined on ℓ^2(𝒱) by

    (L_S ϕ)(v) := ∑_w   :   w ∼ v 
    | w | = | v |ϕ(w),

then the operator L given by (<ref>)
can be written as

    L = Π + Π^∗ + L_S.

In a regular rooted tree graph with k-fold branching, since there are 
no edges connecting vertices within each sphere, then L_S = 0 so that

    L = Π + Π^∗.

To diagonalize the operator L given by (<ref>), invariant subspaces M_n, 
n ≥ 0, for Π are firstly constructed in <cit.>. More precisely, we have the 
following lemma:

<cit.>
The Hilbert space ℓ^2(𝒱) can be decomposed as an orthogonal direct sum

    ℓ^2(𝒱) = ⊕_n = 0^∞ M_n = 
    ⊕_n = 0^∞⊕_r = n^∞ Q_n,r,

where the subspaces M_n are  L-invariant.


By construction in this Lemma <ref>, we have Q_0,0 := ℓ^2(S_0) and  
ℓ^2(S_r) = ⊕_ℓ = 0^r Q_ℓ,r. A schematic interpretation
yields a triangular diagram as in Figure 4.1 below.







According to Lemma <ref>, for any n ≥ 0, the subspace M_n is invariant for the 
operator L. Thus, L can be decomposed as

    L = ⊕_n = 0^∞ L_n,

the operators L_n, n ≥ 0, being the restriction of L to M_n. So, in order to 
diagonalize the operator L, it suffices to do it for each operator L_n for n ≥ 0. 
Consider a vector ϕ∈ M_n, i.e.

    ϕ = ⊕_j = 0^∞ϕ_n,n+j,

with ϕ_n,n+j∈ Q_n,n+j for any j ≥ 0. The idea is to construct an
isomorphism between the subspace M_n and the space of Q_n,n-valued sequences, 
namely the space ℓ^2(^+,Q_n,n). By construction of the Q_n,r, n ≥ 0,
r ≥ 0 (see for instance <cit.>), for any j ≥ 0, the operator 
( 1/√(k)Π)^j defines an isometry between Q_n,n+j and
Q_n,n, and (<ref>) can be written as

    ϕ = ⊕_j = 0^∞( 1/√(k)Π)^j χ_n + j,

where χ := (χ_n + j)_j ≥ 0 defines a sequence of vectors lying in 
Q_n,n. Therefore, under the above considerations, the operator

    W : M_n ⟶ℓ^2(^+,Q_n,n),    W ϕ = χ,

defines an isomorphism between the spaces M_n and ℓ^2(^+,Q_n,n). Indeed, 
we have

    ⟨ϕ,ϕ⟩ = ∑_j = 0^∞⟨ϕ_n + j,ϕ_n + j⟩ = ∑_j = 0^∞⟨χ_n + j,χ_n + j⟩ = ⟨χ,χ⟩_ℓ^2(^+,Q_n,n).

Now, let :=  /2π be the torus and define the unitary operator 

    U : M_n ≅ℓ^2(^+,Q_n,n) ⟶ L^2_ odd (𝕋,Q_n,n)

acting as

    U ( (χ_n,χ_n + 1, … ) ) := 1/√(π)∑_j = 0^∞χ_n + jsin( (j + 1) θ).

Notice that the inner product in L^2_ odd (𝕋,Q_n,n) is defined by

    ⟨ f,g ⟩_L^2_ odd := ∫_𝕋⟨ f(θ),g(θ) 
    ⟩  dθ.

Hence, a direct computation shows that 

    ⟨ U χ,U χ⟩_L^2_ odd = 
    ⟨χ,χ⟩_ℓ^2(^+,Q_n,n) = ⟨ϕ,ϕ⟩,

where the last equality corresponds to (<ref>). Moreover, we have the following lemma:

<cit.>
For any n ≥ 0, we have

    U L_n U^∗ = 2√(k)cos (θ).



In particular, Lemma <ref> shows that for any n ≥ 0, the spectrum of the 
operator L_n is equal to [ -2√(k),2√(k)] and is absolutely 
continuous. Hence, this implies (<ref>).










§ REPRESENTATION OF THE WEIGHTED RESOLVENT A (-L + K + 1 - Z)^-1 B^∗




In this section, we give a suitable representation of the weighted resolvent 
A (-L + k + 1 - z)^-1 B^∗ which turns to be useful in our analysis, where A 
and B are bounded operators on ℓ^2(𝒱).







For n ≥ 0, let P_n be the orthogonal projection of ℓ^2(𝒱) 
onto M_n, the subspace defined in Lemma <ref>. Since M_n can be decomposed as

    M_n = ⊕_j = 0^∞ Q_n,n + j,

then if we let (E_m^n,n+j)_0 ≤ m ≤ N_j denote an orthonormal basis of the
finite-dimensional space Q_n,n + j for any j fixed, we have

    P_n = ∑_j  ≥  0∑_0 ≤ m ≤ N_j⟨·,E_m^n,n+j⟩ 
    E_m^n,n+j.

Notice that for any j ≥ 0 fixed, we have 

    1 + N_j =   Q_n,n+j < k^n + j =  ℓ^2(S_n + j),

since Q_n,n+j⊂ℓ^2(S_n + j). Furthermore, according to Section 
<ref>, for any 0 ≤ m ≤ N_j fixed, there exists a unique vector 
χ_m^n,n+j∈ Q_n,n such that

    E_m^n,n+j = ( 1/√(k)Π)^j χ_m^n,n + j.

Our goal in this section is to prove the following result:


Let A and B be two bounded operators on ℓ^2(𝒱). Then, for any z 
in the resolvent set of the operator -L + k + 1 and any φ∈ℓ^2(𝒱), 
we have

    A (-L + k + 1 - z)^-1 B^∗φ (v) = ∑_v' ∈𝒱 K(v,v') φ (v'),

where the kernel K(v,v') is given by

    K(v,v') :=    1/2 √(k)√(2/π)∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ( B E_m^n,n+j) (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩
       ×( - i e^i(j + ℓ + 2)Φ/√(u)√(4 - u) + i e^i| j - ℓ|Φ/√(u)√(4 - u))
    ( A E_q^n,n + ℓ) (v) ,

throughout the double change of variables

    z + 2√(k) - (k + 1)/√(k) = u = 4 sin^2 ( Φ/2), 
       (Φ) > 0.




Let φ∈ℓ^2(𝒱) and z ∈ρ (-L + k + 1) the resolvent set of 
the operator -L + k + 1. Thanks to (<ref>), we have

    (-L + k + 1 - z)^-1    = ∑_n  ≥  0 P_n (-L_n + k + 1 - z)^-1 P_n 
        = ∑_n  ≥  0 P_n U^∗ U (-L_n + k + 1 - z)^-1 U^∗ U P_n,

where U is the unitary operator defined by (<ref>). Thus, for any vector
ψ∈ℓ^2(𝒱), we have

    ⟨  A (-L + k + 1 - z)^-1 B^∗φ,ψ⟩
        = ∑_n  ≥  0⟨ U (-L_n + k + 1 - z)^-1 U^∗ U P_n
    B^∗φ,U P_n A^∗ψ⟩_L^2_ odd,

⟨·,·⟩_L^2_ odd being the inner product defined by
(<ref>). Together with Lemma <ref>, this gives

    ⟨ A (-L + k + 1 - z)^-1 B^∗φ,ψ⟩
        = ∑_n  ≥  0∫_𝕋( -2√(k)cos (θ) + k + 1 - z )^-1⟨ U P_n B^∗φ (θ),U P_n 
    A^∗ψ (θ) ⟩  dθ.

From (<ref>), it follows that for any ϕ∈ℓ^2(𝒱) and any bounded
operator W on ℓ^2(𝒱), we have

    U P_n W ϕ (θ) = ∑_j  ≥  0∑_0 ≤ m ≤ N_j⟨ϕ,W^∗ E_m^n,n+j⟩ U E_m^n,n+j (θ).

Then, combining (<ref>) and (<ref>), we obtain

    ⟨    A (-L + k + 1 - z)^-1 B^∗φ,ψ⟩
        = ∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ⟨φ,B E_m^n,n+j⟩⟨ψ,A E_q^n,n+ℓ⟩
       ×∫_𝕋( -2√(k)cos (θ) + k + 1 - z )^-1⟨ U E_m^n,n+j (θ),U E_q^n,n+ℓ (θ) 
    ⟩  dθ
        = ∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ⟨⟨φ,B E_m^n,n+j⟩ 
    A E_q^n,n+ℓ,ψ⟩
       ×∫_𝕋( -2√(k)cos (θ) + k + 1 - z )^-1⟨ U E_m^n,n+j (θ),U E_q^n,n+ℓ (θ) 
    ⟩  dθ.

According to the construction of the unitary operator U and (<ref>), for 
p = m, q and p' = j, ℓ respectively, we have

    U E_p^n,n+p' (θ) = 1/√(π)χ_p^n,n+p'sin( (p' + 1) θ).

Putting this together with (<ref>), we obtain

    ⟨ A (-L + k + 1 - z)^-1 B^∗φ,ψ⟩
        = 1/π∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ⟨⟨φ,B E_m^n,n+j⟩⟨χ_m^n,n+j,χ_q^n,n+ℓ⟩ A E_q^n,n+ℓ,ψ⟩
       ×∫_𝕋( -2√(k)cos (θ) + k + 1 - z )^-1sin( (j + 1) θ) sin( (ℓ + 1) θ)   dθ.

Now, (<ref>) implies that the action of the operator  
A (-L + k + 1 - z)^-1 B^∗ on ℓ^2(𝒱)
can be described by

    A (-L + k + 1 - z)^-1 B^∗φ (v) 
        = 1/π∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ⟨φ,B E_m^n,n+j⟩( A E_q^n,n+ℓ) (v) 
    ⟨χ_m^n,n+j,χ_q^n,n+ℓ⟩
       ×∫_𝕋( -2√(k)cos (θ) + k + 1 - z )^-1sin( (j + 1) θ) sin( (ℓ + 1) θ)   dθ.

Since we have

    ⟨φ,W E_m^n,n+j⟩ = 
    ∑_v' ∈𝒱φ (v') ( W E_m^n,n+j) (v'),

W being as above, then it follows from (<ref>) that

    A (-L + k + 1 - z)^-1 B^∗φ (v) = ∑_v' ∈𝒱 K(v,v') φ (v'),

with

    K(v,v') :=  
       1/π∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ( B E_m^n,n+j) (v')( A E_q^n,n+ℓ) (v) 
    ⟨χ_m^n,n+j,χ_q^n,n+ℓ⟩
       ×∫_𝕋( -2√(k)cos (θ) + k + 1 - z )^-1sin( (j + 1) θ) sin( (ℓ + 1) θ)   dθ.

Then, to complete the proof of the theorem, it remains only to show that

    1/π∫_𝕋   ( -2√(k)cos (θ) + k + 1 - z )^-1sin( (j + 1) θ) sin( (ℓ + 1) θ)   dθ
        = 1/2√(k)√(2/π)( - i e^i(j + ℓ + 2)Φ/√(u)√(4 - u) + i e^i| j - ℓ|Φ/√(u)√(4 - u)),

where the relation between z, u and Φ is given by (<ref>). To do this, 
we have to deal with the discrete Fourier transform ℱ : ℓ^2(,) 
→ L^2(), defined for any x ∈ℓ^2(,) and f ∈ L^2() by

    (ℱx)(θ) := (2π)^-1/2∑_n  ∈ 
    e^-inθ x(n),   ( ℱ^-1 f )(n) := (2π)^-1/2∫_ e^inθ f(θ)   dθ.

Let u ∈∖ [0,4] and introduce the following change of variables

    u = 4 sin^2 ( Φ/2),   (Φ) > 0.

Then, it can be proved (cf. e.g. <cit.>) that we have

    ( ℱ^-1( 2 - 2 cos (·) - u )^-1)(n)
    = ie^i | n |Φ/2 sin(Φ) = 
    ie^i | n |Φ/√(u)√(4 - u),

or equivalently

    1/π∫_( 2 - 2 cos(θ) - u )^-1 e^inθ  dθ
    = √(2/π)ie^i | n |Φ/√(u)√(4 - u).

Now, (<ref>) with the help of the transformations 
sin(nθ) = 1/2i( e^inθ - e^-inθ) and

    ( -2√(k)cos(θ) + k + 1 - z )^-1 = 1/√(k)( 2 - 2 cos(θ) - u )^-1,

where u = z + 2√(k) - (k + 1)/√(k) give immediately (<ref>).
This completes the proof of the theorem.









§ RESONANCES NEAR Z = T_±(K)





 §.§ Definition of the resonances


In this subsection, we define the resonances of the operator _M 
near the spectral thresholds t_±(k) given by (<ref>). As preparation, 
preliminary lemmas will be proved firstly. 




From now on, the potential perturbation M is assumed to satisfy Assumption (A). 
Moreover, the following determination of the complex square root

    ∖ (-∞,0] √(·)⟶^+ := { z ∈ : (z) > 0 }

will be adopted throughout this paper. For ε > 0 such that 
0 < ε < δ/4, we let D(0,ε)^∗
be the punctured neighborhood of 0 defined by

    D(0,ε)^∗ := {λ∈ : 0 < |λ| < 
    ε}.

Thanks to the first change of variables in (<ref>), to define and to study the 
resonances of the operator _M near the spectral thresholds t_±(k), 
it suffices to define and to study them respectively near u = 0 and u = 4. However, 
in practice, there is a simple way (see the comments just after Definition <ref>) 
allowing to reduce the analysis of the resonances near the second threshold t_+(k) to 
that of the first one t_-(k). For further use, let e_± be the multiplication 
operators by the functions

    v ⟼ e_± (v) := e^±δ/2| v |.

We have the following lemma:


Let z_t_-(λ) be the parametrization defined by (<ref>). Then, there exists 
0 < ε_0 ≤δ/8 small enough such that the operator-valued function

    λ↦ e_- ( -L + k + 1 - z_t_-(λ) )^-1 e_-,

admits an extension from D(0,ε_0)^∗∩^+
to D(0,ε_0)^∗, with values in ( ℓ^2(𝒱) 
) the set of compact linear operators on ℓ^2(𝒱). Moreover,
this extension is holomorphic.




By Theorem <ref>, for λ∈ D(0,ε)^∗∩^+,
0 < ε < δ/4 small enough, the operator 

    e_- ( -L + k + 1 - z_t_-(λ) )^-1 e_-

admits the kernel

    K(λ,v,v') = 1/2 √(k)√(2/π)( K_1(λ,v,v') 
    + K_2(λ,v,v') ),

where 

    K_1(λ,v,v') : = ∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ 
    e^-δ/2| v' | E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩ 
    f_1(j,ℓ,λ) e^-δ/2| v | E_q^n,n + ℓ (v),

and

    K_2(λ,v,v') := ∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ
    e^-δ/2| v' | E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩ 
    f_2(j,ℓ,λ) e^-δ/2| v | E_q^n,n + ℓ (v),

with 

    f_1(j,ℓ,λ) := - i e^i(j + ℓ + 2) 2arcsinλ/2/λ√(4 - λ^2) ,      
    f_2(j,ℓ,λ) := i e^i| j - ℓ| 2arcsinλ/2/λ√(4 - λ^2).


a) We want to prove the convergence of 
∑_v, v' ∈ 𝒱| K(λ,v,v') |^2 for 
λ∈ D(0,ε_0)^∗ for some
0 < ε_0 ≤δ/8 small enough.




We point out that constants are generic, i.e. can change from an estimate to another. By 
(<ref>)–(<ref>), we have

    ∑_v, v' ∈ 𝒱| K(λ,v,v') |^2     = 1/2kπ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'| K_1(λ,v,v') 
    + K_2(λ,v,v') |^2 
       ≤1/kπ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'( | 
    K_1(λ,v,v') |^2 + | K_2(λ,v,v') |^2 ).

Let us first prove that ∑_r  ≥  0 
 r' ≥  0∑_v  ∈  S_r 
 v' ∈  S_r'| K_1(λ,v,v') |^2 
converges accordingly to the above claim. Thanks to (<ref>), the properties 
(in Section <ref>) of the vectors E_m^n,n+j, χ_m^n,n+j
and (<ref>), we have 

    ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'| K_1(λ,v,v') |^2 
    = ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'|∑_n = 0^min(r,r')∑_j : n + j = r' 
    ℓ : n + ℓ = r∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ 
    e^-δ/2 r'
       × E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩( - i e^i(j + ℓ + 2) 2arcsinλ/2/λ√(4 - λ^2)) e^-δ/2r E_q^n,n + ℓ (v) |^2 
       ≤ C ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'( ∑_n = 0^min(r,r')∑_j : n + j = r' 
    ℓ : n + ℓ = r∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ 
    e^-δ/2 r'e^-(j + ℓ + 2) ( 2arcsinλ/2)/|λ√(4 - λ^2)| e^-δ/2r)^2 
       ≤ C ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'( ∑_n = 0^min(r,r')∑_j : n + j = r' 
    ℓ : n + ℓ = r 
    k^n + j k^n + ℓ e^-δ/2 r'e^-(j + ℓ + 2) ( 2arcsinλ/2)/|λ√(4 - λ^2)| e^-δ/2r)^2 
       ≤ C ∑_r  ≥  0 
     r' ≥  0 k^3r k^3r' e^-δ r 
    e^-δ r'( ∑_n = 0^min(r,r')∑_j : n + j = r' 
    ℓ : n + ℓ = re^-(j + ℓ + 2) ( 2arcsinλ/2)/|λ√(4 - λ^2)|)^2,

for some constant C > 0. Since for 0 < |λ|≪ 1 we have 
2 arcsinλ/2 = λ + o(|λ|), then there exists 
0 < ε_0 ≤δ/8 small enough such that for each 
0 < |λ|≤ε_0, we have

    e^-(j + ℓ + 2) ( 2arcsinλ/2)/|λ√(4 - λ^2)|≤e^-(  (λ) - δ/8) (j + ℓ + 2)/|λ√(4 - λ^2)|.

Then, it follows from (<ref>) that for each λ∈ D(0,ε_0)^∗, 
we have

    ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'| K_1(λ,v,v') |^2 
       ≤ C ∑_r  ≥  0 
     r' ≥  0 k^3r k^3r' 
    e^-δ r e^-δ r'( ∑_n = 0^min(r,r')∑_j : n + j = r' 
    ℓ : n + ℓ = re^-(  (λ) - δ/8) (j + ℓ + 2)/|λ√(4 - λ^2)|)^2.

Clearly, if F = F(j,ℓ) is a function of the variables
j and ℓ, then 

    ∑_n = 0^min(r,r')∑_j : n + j = r' 
    ℓ : n + ℓ = r F(j,ℓ)
    = ∑_n = 0^min(r,r') F(r'-n,r-n).

This together with (<ref>) imply that

    ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'| K_1(λ,v,v') |^2  
       ≤ C ∑_r  ≥  0 
     r' ≥  0 k^3r k^3r' 
    e^-δ r e^-δ r'( ∑_n = 0^min(r,r')e^-(  (λ) 
    - δ/8) (r + r' + 2) e^2( (λ) - δ/8)n /|λ√(4 - λ^2)|)^2.

Since (λ) - δ/8≤ 0, then it follows from (<ref>) that

    ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'| K_1(λ,v,v') |^2 
       ≤ C ∑_r  ≥  0 
     r' ≥  0 k^3r k^3r' 
    e^-δ r e^-δ r'( ∑_n = 0^min(r,r')e^-(  (λ) 
    - δ/8) (r + r' + 2)/|λ√(4 - λ^2)|)^2 
       ≤ C ∑_r  ≥  0 
     r' ≥  0 k^3r k^3r' e^-δ r 
    e^-δ r'e^-2(  (λ) - δ/8) (r + r' + 2)/|λ^2 (4 - λ^2) | (r + 1)(r' + 1) 
        = C ∑_r  ≥  0 
     r' ≥  0 (r + 1) k^3r e^-δ re^-2(  (λ) - δ/8) (r + 1) e^-2(  (λ) - δ/8) (r' + 1)/|λ^2 (4 - λ^2) | 
    (r' + 1) k^3r' e^-δ r'
       ≤ C ∑_r  ≥  0 
     r' ≥  0 (r + 1) k^3(r+1) e^-δ (r+1)e^-2(  (λ) - δ/8) (r + 1) e^-2(  (λ) - δ/8) (r' + 1)/|λ^2 (4 - λ^2) |
       × (r' + 1) k^3(r'+1) e^-δ (r'+1)
        = C ∑_r  ≥  0 
     r' ≥  0 (r + 1) e^-( δ/2 
    - 3 ln(k) ) (r+1)e^-2 ( (λ) + δ/8) 
    (r + 1) e^-2( (λ) + δ/8) (r' + 1)/|λ^2 (4 - λ^2) |
       × (r' + 1) e^-( δ/2 - 3 ln(k) ) (r'+1).

Assumption (A) implies that δ/2 - 3 ln(k) ≥ 0. 
Thus, the r.h.s. and then the l.h.s. of (<ref>) is convergent for any
λ∈ D(0,ε_0)^∗.



Similarly let us prove that ∑_r  ≥  0 
 r' ≥  0∑_v  ∈  S_r 
 v' ∈  S_r'| K_2(λ,v,v') |^2 
converges. As in (<ref>), we can show that 

    ∑_r  ≥  0 
     r' ≥  0   ∑_v  ∈  S_r 
     v' ∈  S_r'| K_2(λ,v,v') |^2 
    = ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'|∑_n = 0^min(r,r')∑_j : n + j = r' 
    ℓ : n + ℓ = r∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ
         e^-δ/2 r' E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩( i e^i | j - ℓ| 2arcsinλ/2/λ√(4 - λ^2)) e^-δ/2r E_q^n,n + ℓ (v) |^2 
       ≤ C ∑_r  ≥  0 
     r' ≥  0 k^3r k^3r' e^-δ r 
    e^-δ r'( ∑_n = 0^min(r,r')∑_j : n + j = r' 
    ℓ 
    : n + ℓ = re^- | j - ℓ|( 2arcsinλ/2)/|λ√(4 - λ^2)|)^2.

Thus, similarly to (<ref>), for each λ∈ D(0,ε_0)^∗, we have

    ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'| K_2(λ,v,v') |^2 
       ≤ C ∑_r  ≥  0 
     r' ≥  0 k^3r k^3r' 
    e^-δ r e^-δ r'( ∑_n = 0^min(r,r')∑_j : n + j = r' 
    ℓ : n + ℓ = re^-(  (λ) - δ/8) | j - ℓ|/|λ√(4 - λ^2)|)^2.

In this case, we use (<ref>) to write 

    ∑_r  ≥  0 
     r' ≥  0∑_v  ∈  S_r 
     v' ∈  S_r'| K_2(λ,v,v') |^2 
       ≤ C ∑_r  ≥  0 
     r' ≥  0 k^3r k^3r' 
    e^-δ r e^-δ r'( ∑_n = 0^min(r,r')e^-(  (λ) 
    - δ/8) | r - r' |/|λ√(4 - λ^2)|)^2 
       ≤ C ∑_r  ≥  0 
     r' ≥  0 k^3r k^3r' e^-δ r 
    e^-δ r'e^-2(  (λ) - δ/8) | r - r' |/|λ^2 (4 - λ^2) | (r + 1)(r' + 1) 
        = C ∑_r  ≥  0 
     r' ≥  0 (r + 1) e^-( δ/2 
    - 3 ln(k) )re^- δ/2r e^-2(  (λ) - δ/8) | r - r' |) e^- δ/2r'/|λ^2 (4 - λ^2) |
       × (r' + 1) e^-( δ/2 - 3 ln(k) )r'.

Assumption (A) implies that δ/2 - 3 ln(k) ≥ 0. Thus, the r.h.s. 
and then the l.h.s. of (<ref>) is convergent for any
λ∈ D(0,ε_0)^∗.



Now, since ∑_v, v' ∈ 𝒱| K(λ,v,v') |^2 
is convergent for λ∈ D(0,ε_0)^∗, 
then the operator given by (<ref>) belongs in
( ℓ^2(𝒱) ) for λ∈ D(0,ε_0)^∗, the class of Hilbert-Schmidt operators 
on ℓ^2(𝒱). Consequently, the operator-valued function defined by 
(<ref>) can be extended from D(0,ε_0)^∗∩^+ to 
D(0,ε_0)^∗, with values in ( ℓ^2(𝒱) ). 
It remains to prove that this extension is holomorphic.



b) To simply notation, let us denote this extension by 

    D(0,ε_0)^∗∋λ↦ T(λ).

Since the kernel of the operator T(λ) is given by K(λ,v,v') defined
by (<ref>), then to show the claim, it is sufficient to prove it for the
maps 

    D(0,ε_0)^∗∋λ↦ T_s(λ),

where for s = 1, 2, T_s(λ) is the operator with kernel given by 
K_s(λ,v,v') in (<ref>) and (<ref>). We give the proof only
for the case s = 1, the case s = 2 being treated in a similar way.
So, for λ∈ D(0,ε_0)^∗, let f_1(j,ℓ,λ) be the
function defined by (<ref>) and D_1(λ) be the operator whose kernel is

    ∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ 
    e^-δ/2| v' | E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩∂_λ f_1(j,ℓ,λ)
    e^-δ/2| v | E_q^n,n + ℓ (v) 
        = ∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ 
    e^-δ/2| v' | E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩
       ×( - i e^i(j + ℓ + 2) 2arcsinλ/2/λ^2
    (4 - λ^2)) e^-δ/2| v | E_q^n,n + ℓ (v)
    ( 2i (j + ℓ + 2) - 4 - 2λ^2/√(4 - λ^2)).

As in a) above, we can show that D_1(λ) ∈( ℓ^2(𝒱) ).
Therefore, for λ_0 ∈ D(0,ε_0)^∗, the kernel of the
Hilbert-Schmidt operator T_1(λ) - T_1(λ_0)/λ - λ_0 
- D_1(λ_0) is given by

    K_1(λ,λ_0,v,v') := ∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ 
    e^-δ/2| v' | E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩
       ×( f_1(j,ℓ,λ) - f_1(j,ℓ,λ_0)/λ - λ_0 - 
    ∂_λ f_1(j,ℓ,λ_0) ) e^-δ/2| v | E_q^n,n + ℓ (v).

Thus, to conclude the proof of the lemma, we have just to justify that

    ‖T_1(λ) - T_1(λ_0)/λ - λ_0 
    - D_1(λ_0) ‖_(ℓ^2(𝒱))⟶ 0
 
as λ→λ_0. Since we have

    ‖T_1(λ) - T_1(λ_0)/λ - λ_0 
    - D_1(λ_0) ‖_(ℓ^2(𝒱))≤∑_v, v' ∈ 𝒱| K_1(λ,λ_0,v,v') |^2,

then it suffices to prove that the r.h.s. of (<ref>) tends to zero
as λ→λ_0. The Taylor-Lagrange formula applied to the 
function 

    [0,1] ∋ t ↦ g(t) := f_1 (j,ℓ,tλ + (1-t)λ_0 )

asserts there exists θ∈ (0,1) such that

    f_1(j,ℓ,λ) = f_1(j,ℓ,λ_0) + (λ - λ_0) 
    ∂_λ f_1(j,ℓ,λ_0) + (λ - λ_0)^2/2∂_λ^(2) f_1 (j,ℓ, θλ + (1-θ)λ_0 ).

Then, it follows from (<ref>) and (<ref>) that K_1(λ,λ_0,v,v') 
can be represented as

    K_1(λ,λ_0,v,v') = λ - λ_0/2∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ 
    e^-δ/2| v' | E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩
       ×∂_λ^(2) f_1 (j,ℓ, θλ + (1-θ)λ_0 ) e^-δ/2| v | E_q^n,n + ℓ (v).

Now, easy but fastidious computations allow to see that there exists a family 
of holomorphic functions F_p,q, 0 ≤ p ≤ q, on D(0,ε_0)^∗ such that

    ∂_λ^(q) f_1(j,ℓ,λ) = ie^i(j+ℓ+2) 2arcsinλ/2∑_p=0^q F_p,q(λ) (j + ℓ + 2)^q-p.

In particular, for q = 2, we have

    ∂_λ^(2) f_1(j,ℓ,λ) = ie^i(j+ℓ+2) 2arcsinλ/2( F_0,2(λ)(j+ℓ+2)^2 + F_1,2(λ) (j+ℓ+2) + F_2,2(λ) ).

Putting this together with (<ref>), we obtain

    | K_1(λ,λ_0,v,v') |
       ≤|λ - λ_0 |/2| F_0,2( θλ + (1-θ)λ_0 ) |∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ| e^-δ/2| v' | E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩
       × e^i(j+ℓ+2) 2arcsinθλ + (1-θ)λ_0/2
    (j+ℓ+2)^2 e^-δ/2| v | E_q^n,n + ℓ (v) |
        + |λ - λ_0 |/2| F_1,2( θλ + (1-θ)λ_0 ) ||∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ| e^-δ/2| v' | E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩
       × e^i(j+ℓ+2) 2arcsinθλ + (1-θ)λ_0/2
    (j+ℓ+2) e^-δ/2| v | E_q^n,n + ℓ (v) |
        + |λ - λ_0 |/2| F_2,2( θλ + (1-θ)λ_0 ) |∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ| e^-δ/2| v' | E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩
       × e^i(j+ℓ+2) 2arcsinθλ + (1-θ)λ_0/2
     e^-δ/2| v | E_q^n,n + ℓ (v) |
        =: ∑_p=0^2 Q_p(λ,λ_0,v,v').

Thus, we have

    ∑_v, v' ∈ 𝒱| K_1(λ,λ_0,v,v') |^2 
       ≤ Const.( ∑_v, v' ∈ 𝒱 Q_1(λ,λ_0,v,v')^2
    + ∑_v, v' ∈ 𝒱 Q_2(λ,λ_0,v,v')^2 + 
    ∑_v, v' ∈ 𝒱 Q_3(λ,λ_0,v,v')^2 ).

For p ∈{ 0,1,2 }, let us show that ∑_v, v' ∈ 𝒱 Q_p(λ,λ_0,v,v')^2 ⟶ 0 as λ→λ_0. 
Since  θλ + (1-θ)λ_0 belongs in D(0,ε_0)^∗ for 
λ, λ_0 ∈ D(0,ε_0)^∗, then similarly to (<ref>) we have

    e^-(j + ℓ + 2) ( 2arcsinθλ + (1-θ)λ_0/2)   ≤ e^-(  (θλ + (1-θ)λ_0) - δ/8) (j + ℓ + 2)
       ≤ e^δ/4(j + ℓ + 2).

Therefore, by arguing as in a) above, we obtain that there exists a uniform 
constant Const. in λ and λ_0 such that for q ∈{ 0,1,2 },

    ∑_v, v' ∈ 𝒱 Q_p(λ,λ_0,v,v')^2 ≤ Const.|λ - λ_0 |^2/4| F_q,2(θλ + (1-θ)λ_0) |^2 λ→λ_0⟶ 0,

implying by (<ref>) and (<ref>) that ‖T_1(λ) - T_1(λ_0)/λ - λ_0 - D_1(λ_0) ‖_(ℓ^2(𝒱))⟶ 0 as λ→λ_0. Thus, the operator-valued function 
D(0,ε_0)^∗∋λ↦ T_1(λ) is holomorphic with derivative 
∂_λ T_1(λ) = D_1(λ). Similarly, 
D(0,ε_0)^∗∋λ↦ T_2(λ) is holomorphic, and then D(0,ε_0)^∗∋λ↦ T(λ). This concludes the proof of the lemma.



It follows from the identities

    ( _±M - z )^-1( I ±M (-L + k + 1 - z)^-1) = (-L + k + 1 - z)^-1,
 
that

    e_- (    _±M - z )^-1 e_- 
        = e_- (-L + k + 1 - z)^-1 e_- ( I ± e_+ M(-L + k + 1 - z)^-1 
    e_- )^-1.

Assumption (A) on the potential perturbation M implies that 

    e_+ M = ℳ e_-

for some bounded operator ℳ on ℓ^2(𝒱). Thus, combining (<ref>) and Lemma <ref>, we obtain that the operator-valued functions

    λ⟼± e_+ M( -L + k + 1 - z_t_-(λ) )^-1 e_-

are holomorphic in D(0,ε)^∗, with values in ( ℓ^2(𝒱) 
). Therefore, by the analytic Fredholm extension theorem, the operator-valued functions

    λ⟼( I ± e_+ M( -L + k + 1 - z_t_-(λ) 
    )^-1 e_- )^-1

admit meromorphic extensions from D(0,ε_0)^∗∩^+ to 
D(0,ε_0)^∗. Defining the Banach spaces 

    ℓ_±δ^2(𝒱) := e^±δ/2| v |ℓ^2(𝒱),

we then get the following proposition:
 
The operator-valued functions

    λ⟼( _±M - z_t_-(λ) )^-1∈ℒ( ℓ_-δ^2(𝒱),ℓ_δ^2(𝒱) ),
 
admit meromorphic extensions from D(0,ε_0)^∗∩^+ to 
D(0,ε_0)^∗. These extensions will be denoted by R_±M( z_t_-(λ) ) respectively.


As in (<ref>), Assumption (A) on M implies that there exists a bounded operator 
ℬ on ℓ^2(𝒱) such that √(|M|) = ℬ e_-.
Together with Lemma <ref>, this gives the following lemma:

 
Let J be defined by the polar decomposition M = J |M| 
of the potential perturbation M. Then, the operator-valued functions

    λ⟼𝒯_±M( z_t_-(λ) ) := ± J 
    √(|M|)( -L + k + 1 - z_t_-(λ) )^-1√(|M|),
 
admit holomorphic extensions from D(0,ε_0)^∗∩^+
to D(0,ε_0)^∗, with values in ( ℓ^2(𝒱) ).


We are now in position to define the resonances of the operator _M near 
the spectral thresholds z = t_±(k). Note that in the next definitions, the quantity 
Ind_γ(·) is defined in the appendix by (<ref>).

 
We define the resonances of the operator _M near t_-(k) as the poles 
of the meromorphic extension R_M(z), of the resolvent 
( _M - z )^-1 in ℒ( ℓ_-δ^2(𝒱),ℓ_δ^2(𝒱) ). 
The multiplicity of a resonance 

    z_t_- := z_t_-(λ) = t_-(k) + λ^2√(k),
 
is defined by 

    ( z_t_-) := Ind_γ ( I + 
    𝒯_M( z_t_-(·) ) ),

where γ is a small contour positively oriented containing λ as the only 
point satisfying that z_t_-(λ) is a resonance of _M.



As mentioned previously, to define the resonances of the operator _M near 
t_+(k), there exists a specific reduction which exploits a simple relation between the 
two thresholds t_±(k). Indeed, define the self-adjoint unitary operator Θ 
on ℓ^2(𝒱) by 

    (Θφ)(v) := (-1)^| v |φ(v).

We thus have


  * Θ^2 = I,

  * Θ L Θ^-1 = -L,

  * ΘMΘ^-1 = M.

In the last point, we have used the fact that M is the multiplication operator 
by the function M. Thus, it can be easily verified that we have

    Θ( -L + k + 1 + M - z ) Θ^-1 = 
    -(-L + k + 1) + M + 2(k + 1) - z,

so that

    Θ e_- ( _M - z )^-1 e_- Θ^-1
    = - e_- ( _-M - ( 2(k + 1) - z ) )^-1 e_-.

Set 

    ω := 2(k + 1) - z.

Since ω is near t_-(k) for z near t_+(k), then using relation (<ref>), 
we can define the resonances of the operator _M near t_+(k) as 
the poles of the meromorphic extension of the resolvent

    - ( _-M - ω)^-1 : ℓ_-δ^2(𝒱) 
    →ℓ_δ^2(𝒱),

near ω = t_-(k), similarly to Definition <ref>. More precisely, we have the 
following definition:


 
We define the resonances of the operator _M near t_+(k) as the poles 
of the meromorphic extension R_-M(ω), of the resolvent 
( _-M - ω)^-1 in ℒ( ℓ_-δ^2(𝒱),ℓ_δ^2(𝒱) ), for ω given by (<ref>) near t_-(k). 
The multiplicity of a resonance 

    z_t_+ := z_t_+(λ) = 2(k+ 1) - ( t_-(k) + λ^2√(k)) 
    = t_+(k) - λ^2√(k),
 
is defined by 

    (z_t_+) := Ind_γ ( I + 
    𝒯_-M( 2(k + 1) - z_t_+(·) ) ),

where γ is a small contour positively oriented containing λ as the only 
point satisfying that 2(k + 1) - z_t_+(λ) is a pole of R_-M(ω).



 
Notice that the resonances z_t_±(λ) near the spectral thresholds t_±(k) 
are defined in some two-sheets Riemann surfaces ℳ_t_± respectively.
Otherwise, the discrete eigenvalues of the operator _M near t_±(k) 
are resonances. Moreover, the algebraic multiplicity (<ref>) of a discrete eigenvalue 
coincides with its multiplicity as a resonance near t_±(k) respectively given by 
(<ref>) and (<ref>). Let us give the proof only for the equality 
(<ref>) = (<ref>), the equality (<ref>) = (<ref>) could be 
treated in a similar fashion. Let z_t_- := z_t_-(λ) ∈∖ 
[t_-(k),t_+(k)] be a discrete eigenvalue of _M near t_-(k). 
Firstly, observe that Assumption (A) on M implies that M is of 
trace-class. In this case, it is is well know (see e.g. <cit.>) that
z_t_-∈σ_( _M) if and 
only if h(z_t_-) = 0, where for z ∈∖ [t_-(k),t_+(k)], h is the 
holomorphic function defined by

    h(z) := ( I + M( -L + k + 1 - z )^-1) 
    = ( I + J √(|M|)( -L + k + 1 - z )^-1√(|M|)).

Moreover, the algebraic multiplicity (<ref>) of z_t_- is equal to its 
order as zero of the function h. Namely, by the residues theorem,

    (z_t_-) = ind_γ' h := 
    1/2iπ∫_γ'h'(z)/h(z) dz,

where γ' is a small circle positively oriented containing z_t_- as the only 
zero of h. Then, the claim follows directly from the equality 

    ind_γ' h = Ind_γ ( I + 𝒯_M( z_t_-(·) ) ),

see for instance <cit.> for more details. 
 










 §.§ Characterization of the resonances



In this subsection, we give a simple characterization of resonances of 
_M near the spectral thresholds t_±(k). 
The first one concerns the resonances near z = t_-(k).


The following assertions are equivalent:



  (a) z_t_- = z_t_-(λ) ∈ℳ_t_- is a resonance,

  (b) z_t_- is a pole of R_M(z), 

  (c) -1 is an eigenvalue of 𝒯_M( z_t_-(λ) ).




(a) ⟺ (b) is just the Definition <ref>, while 
(b) ⟺ (c) is a consequence of the identity

    ( I + J √(|M|) (-L + k + 1 - z)^-1√(|M|)) 
    ( I - J √(|M|)( _M - z )^-1√(|M|)) = I,

coming from the resolvent equation.


Similarly, we have the following proposition:


The following assertions are equivalent:



  (a) z_t_+ = z_t_+(λ) ∈ℳ_t_+ is a resonance,

  (b) 2(k + 1) - z_t_+(λ) is a pole of R_-M(ω) 
for ω given by (<ref>) near t_-(λ),

  (c) -1 is an eigenvalue of 
𝒯_-M( 2(k + 1) - z_t_+(λ) ).












§ PROOF OF THEOREM <REF>




This section is devoted to the proof of Theorem <ref>. It will be divided into 
tree steps.







 §.§ A preliminary result

The first step consists on refining the representations of the sandwiched resolvents
𝒯_M( z_t_-(λ) ) and 
𝒯_-M( 2(k + 1) - z_t_+(λ) )
near the spectral thresholds z = t_±(k). 
Notice that

    2(k + 1) - z_t_+(λ) = z_t_-(λ),
 
so that our analysis will be just reduced to the operators 
𝒯_±M( z_t_-(λ) ). 



Recall that 𝒯_±M( z_t_-(λ) ) = ± J 
√(|M|)( -L + k + 1 - z_t_-(λ) )^-1√(|M|), and let us set

    γ(λ) := ( e^i (j + ℓ + 2) 2arcsinλ/2 
    - 1 )/λ√(4 - λ^2)  and  β(λ) := ( e^i | j - ℓ| 
    2arcsinλ/2 - 1 )/λ√(4 - λ^2),   
    j, ℓ≥ 0.

By construction, as shows the proof of Lemma <ref>, for 
λ∈ D(0,ε_0)^∗ the operator 

    √(|M|)( -L + k + 1 - z_t_-(λ) )^-1√(|M|)

admits the integral kernel 

    ±1/2 √(k)√(2/π)( 𝒦_1^(λ)(v,v') + 𝒦_2^(λ)(v,v') ),

where 

    𝒦_1^(λ)(v,v') : =    ∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ√(|M|) (v')  E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩
       × i ( - γ(λ) - 1/λ√(4 - λ^2)) √(|M|) (v) E_q^n,n + ℓ (v),

and

    𝒦_2^(λ)(v,v') : =    ∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ√(|M|) (v')  E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩
       × i ( β(λ) + 1/λ√(4 - λ^2)) √(|M|) (v) E_q^n,n + ℓ (v).

Since γ and β can be extended to holomorphic functions on the open disk 
D(0,ε_0)^∗∪{ 0 }, then by combining identities 
(<ref>)-(<ref>), we get the following result:

 
For λ∈ D(0,ε_0)^∗∪{ 0 }, we have

    𝒯_±M( z_t_-(λ) ) = 
    ± J √(2/π)  Hol(λ),

where Hol(λ) defines a holomorphic operator on 
D(0,ε_0)^∗∪{ 0 } with values in 
( ℓ^2(𝒱) ), and with kernel given by

    i/2 √(k)∑_n  ≥  0∑_j  ≥  0 
    ℓ ≥  0∑_0 ≤ m ≤ N_j 
     0 ≤ q ≤ N_ℓ√(|M|) (v')  E_m^n,n+j (v')⟨χ_m^n,n + j,χ_q^n,n + ℓ⟩
       ×( β(λ) - γ(λ) ) √(|M|) (v) E_q^n,n + ℓ (v).













 §.§ Reformulation of the problem




Let ℋ be a separable Hilbert space, 𝒟⊆ be a 
domain containing 0, and (ℋ) denote the set of compact 
linear operators in ℋ. For a holomorphic operator-valued function 

    K : 𝒟∖{ 0 }⟶(ℋ),

and a subset Ω⊆𝒟∖{ 0 }, a complex 
number λ∈Ω is said to be a characteristic value of the 
operator-valued function

    λ⟼ I + K(λ),
 
if the operator I + K(λ) is not invertible (cf. Section <ref> for more 
details about the concept of characteristic value). By abuse of language,
we shall sometimes say that λ is a characteristic value of the operator I + K(λ). 
Once there exists λ_0∈Ω such that I + K(λ_0) is invertible, 
then by the analytic Fredholm theorem, the set of characteristic values 
λ∈Ω of I + K(·) is discrete. Moreover, according to Definition
<ref> and (<ref>), the multiplicity of a characteristic value λ is 
defined by

    (λ) := Ind_γ( I + K(·) ),

γ being a small contour positively oriented which contains λ as the only 
point satisfying I + K(z) is not invertible, and with I + K(·) not vanishing 
on γ. We then can reformulate Propositions <ref> and <ref> in the
following way:


For λ∈ D(0,ε_0)^∗, the following assertions are 
equivalent:



  (a) z_t_- = z_t_-(λ) ∈ℳ_t_- is a resonance,

  (b) λ is a characteristic value of  
I + 𝒯_M( z_t_-(·) ). 

Moreover, thanks to (<ref>), the multiplicity of the resonance z_t_-(λ) 
coincides with that of the characteristic value λ.





For λ∈ D(0,ε_0)^∗, the following assertions are 
equivalent:



  (a) z_t_+ = z_t_+(λ) ∈ℳ_t_+ is a resonance,

  (b) λ is a characteristic value of 
I + 𝒯_-M( 2(k + 1) - z_t_+(·) ).

Moreover, thanks to (<ref>), the multiplicity of the resonance z_t_+(λ) 
coincides with that of the characteristic value λ.












 §.§ End of the proof of Theorem <ref>




From Propositions <ref>, <ref> and <ref> together with the identity 
(<ref>), it follows that z_t_±(λ) is a resonance of _M 
near t_±(k) if and only if λ is a characteristic value of 

    I + 𝒯_±M( z_t_-(λ) ) = I ± 
    J √(2/π)  Hol(λ).

Since the operator Hol(λ) is holomorphic in the open disk 
D(0,ε_0)^∗∪{ 0 } with values in ( ℓ^2(𝒱) 
), then Theorem <ref> holds by applying Proposition <ref> with 


  * 𝒟 = Ω_r^∗∪{ 0 }, Ω_r^∗⊆ D(0,ε_0)^∗,

  * Z = { 0 },

  * F = I + 𝒯_±M( z_t_-(·) ).

This concludes the proof of Theorem <ref>.









§ APPENDIX





We recall some tools we need on characteristic values of
finite meromorphic operator-valued functions. For more details on the subject, we refer 
for instance to <cit.> and the book <cit.>. The content of this 
section follows <cit.>.




Let ℋ be separable Hilbert space, and let ℒ(ℋ) 
(resp. GL(ℋ)) denote the set of bounded (resp. invertible) linear 
operators in ℋ.


Let 𝒰 be a neighborhood of a fixed point w ∈, and 
F : 𝒰∖{ w }⟶ℒ(ℋ) 
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) = ∑_n = m^+∞ (z - w)^n A_n,    m > - ∞,

where (if m < 0) the operators A_m, …, A_-1 are of finite rank.
Moreover, if A_0 is a Fredholm operator, then the function F is said to be Fredholm 
at w. In that case, the Fredholm index of A_0 is called the Fredholm index of F 
at w.


We have the following proposition:

<cit.>
Let 𝒟⊆ℂ be a connected open set, Z ⊆𝒟 
be a closed and discrete subset of 𝒟, and F : 𝒟⟶ℒ(ℋ) be a holomorphic operator-valued function in 𝒟\ 
Z. Assume that:


  * F is finite meromorphic on 𝒟 (i.e. it is finite meromorphic near each 
point of Z),

  * F is Fredholm at each point of 𝒟,

  * there exists w_0 ∈𝒟\ Z such that F(w_0) is invertible. 

Then, there exists a closed and discrete subset Z' of 𝒟 such that:


  * Z ⊆ Z',

  * F(z) is invertible for each z ∈𝒟\ Z',

  * F^-1 : 𝒟\ Z' ⟶ GL(ℋ) is finite
meromorphic and Fredholm at each point of 𝒟.



In the setting of Proposition <ref>, we define the characteristic values
of F and their multiplicities as follows:


The points of Z' where the function F or F^-1 is not holomorphic are called the
characteristic values of F. The multiplicity of a characteristic value w_0 is 
defined by

    mult(w_0) := 1/2iπ∮_| w - w_0 | = ρ 
    F'(z)F(z)^-1 dz,

where ρ > 0 is chosen small enough so that { w ∈ : | w - 
w_0 |≤ρ}∩ Z' = { w_0 }.


According to Definition <ref>, if the function F is holomorphic in 𝒟,
then the characteristic values of F are just the complex numbers w where the operator 
F(w) is not invertible. Then, results of <cit.> and <cit.> imply that 
mult(w) is an integer.



Let Ω⊆𝒟 be a connected domain with boundary ∂Ω 
not intersecting Z'. The sum of the multiplicities of the characteristic values of
the function F lying in Ω is called the index of F with respect to the 
contour ∂Ω and is defined by 

    Ind_∂Ω F := 1/2iπ∮_∂Ω F'(z)F(z)^-1 dz = 1/2iπ∮_∂Ω F(z)^-1 F'(z) dz.
 




Acknowledgements: O. Bourget is supported by the Chilean Fondecyt 
Grant 1161732. D. Sambou is supported by the Chilean Fondecyt Grant 3170411. 


The authors express their gratitude to S. Golénia for bringing to their attention 
the paper <cit.>, and to V. Bruneau and S. Kupin for their helpful discussions 
and valuable suggestions.


















































































































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