On the spectral properties of non-selfadjoint discrete Schrödinger operators

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On the spectral properties of non-selfadjoint discrete Schrödinger operators

Olivier Bourget, Diomba Sambou and Amal Taarabt

Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, San- tiago, Chile

E-mails: bourget@mat.uc.cl, disambou@mat.uc.cl, amtaarabt@mat.uc.cl Abstract

Let H0 be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space andV be a compact perturbation. We relate the regularity properties of V to various spectral properties of the perturbed operatorH0+V. The struc- tures of the discrete spectrum and the embedded eigenvalues are analysed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results scattered throughout the literature are recovered and extended.

For illustrative purposes, the case of the one-dimensional discrete Laplacian is emphasized.

Mathematics subject classification 2010: 47B37,47B47,47A10,47A11,47A55,47A56.

Keywords: Discrete spectrum, Resonances, Limiting Absorption Principle, Complex scaling.

arXiv:1807.01282v4 [math.SP] 21 May 2020

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5 On positive commutators 22

5.1 Abstract results . . . 23

5.2 Proof of Lemma5.1 . . . 25

5.3 Deformed resolvents and first estimates . . . 25

5.4 Differential Inequalities . . . 30

5.5 Last step to the proof of Proposition5.1 . . . 34

5.6 Proof of Corollary5.1 . . . 35

5.7 Proof of Corollary5.2 . . . 35

5.8 Proof of Theorems2.6and2.7 . . . 36

6 Regularity classes 36 6.1 TheA(A0)class . . . 36

6.1.1 General considerations . . . 36

6.1.2 Applications . . . 39

6.2 TheC^k(A)andC^s,p(A)classes . . . 41

1 Introduction

The spectral theory of non-selfadjoint perturbations of selfadjoint operators has made significant advances in the last decade, partly due to its impact and applications in the physical sciences.

For an historical panorama on the relationhips between non-selfadjoint operators and quantum mechanics, we refer the reader to e.g. [4] and references therein. The existence of Limiting Absorption Principles (LAP) and the distributional properties of the discrete spectrum have been two of the main issues considered in this field, among the many results obtained so far.

LAP have been evidenced in various contexts, mostly based on the existence of some positive commutators. [35, 36] and [7] have developed respectively non-selfadjoint versions of the regular Mourre theory and the weak Mourre theory. The analysis of the discrete spectrum has been carried out from a qualitative point of view in [32,33,19,20,9], where the main focus is set on the existence and structure of the set of limit points. Quantitative approaches based on Lieb-Thirring inequalities and on the identification of the distribution law for the discrete spectrum around the limit points, have also been developed, see e.g. [10,12,22,6,14,21,23,26,31,41] and references therein. These results have been established mostly for specific models like Schrödinger operators and Jacobi matrices. Extensions to non-selfadjoint perturbations of magnetic Hamiltonians, Dirac and fractional Schrödinger operators have also been studied in [37, 38] and [11] respectively.

Specific developments concerning non-selfadjoint rank one perturbations are considered in [3,18, 30]. In general, the techniques used depend strongly on the model considered.

The present paper is an attempt to analyze more systematically the issue of the spectral properties of non-selfadjoint compact perturbations of Hamiltonians exhibiting some absolutely continuous spectrum. In particular, we show various relationships between the regularity of the perturbation and, first the properties of the discrete spectrum, second the existence of some LAP. With a view towards discretized models of non-selfadjoint operators [4], we have focused our discussion on non-selfadjoint compact perturbations of the discrete Schrödinger operator in dimension one. For a study of non-selfadjoint discrete Schrödinger operator in random regimes, we refer the reader to [16,17].

We have articulated this paper on the two following axes.

Firstly, for highly regular compact perturbations, we adapt various complex scaling arguments to our non-selfadjoint setting and show that the limit points of the discrete spectrum are necessarily contained in the set of thresholds of the unperturbed Hamiltonian (Theorem 2.2). We also exhibit some LAP on some suitable subintervals of the essential spectrum away from these thresholds (Theorem2.3). Since for perturbations displaying some exponential decay, the method of characteristic values allows to establish the absence of resonance in some neighborhood of the thresholds (Theorem2.4), we conclude on the finiteness of the discrete spectrum in this case (The- orem2.5). Let us mention that a variation of this result was previously obtained within the more

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restricted framework of Jacobi matrices with slightly weaker decay assumptions (see [19, Theorem 1]). Our approach extends that result in two ways: it applies to perturbations exhibiting a full off-diagonal structure and also proves the existence of a LAP, which seems to be new.

Secondly, for mildly regular compact perturbations, we show that the set of embedded eigenvalues away from the thresholds is finite and exhibits more restricted versions of LAP (Theorems2.6 and2.7). It is actually an application of a non-selfadjoint version of the Mourre theory developed under optimal regularity condition (Theorem5.1and Corollary5.2).

The paper is structured as follows. The main concepts and main results are introduced in Section 2 jointly with the model. The complex scaling arguments for non-selfadjoint operators are developed in Section 3 and culminate with the proof of Theorems 2.2 and 2.3. Section 4 is focused on the method of characteristic values and the proof of Theorem2.4. The development of a Mourre theory for non-selfadjoint operators under optimal regularity conditions is exposed in Section 5, ending with the proofs of Theorem5.1, Corollary 5.2, Theorems 2.6 and2.7. It is self-contained and can be read independently. Finally, various results related to the concepts of regularity used throughout the text are summarized and illustrated in Section6.

Notations: Throughout this paper,Z,Z+andNdenote the sets of integral numbers, non-negative and positive integral numbers respectively. Forδ≥0, we define the weighted Hilbert spaces

`²_±δ(Z) :=

x∈C^Z:X

n∈Z

e^±δ|n||x(n)|²<∞ ,

and observe the following inclusions: `²_δ(Z)⊂`²(Z)⊂`²_−δ(Z). In particular, `²(Z) = `²₀(Z). For δ > 0, we define the multiplication operators Wδ : `²_δ(Z)→ `²(Z)by (Wδx) (n) := e^(δ/2)|n|x(n), andW_−δ :`²(Z)→`²_−δ(Z)by(W_−δx) (n) := e^−(δ/2)|n|x(n). We denote by (en)_n∈Z the canonical orthonormal basis of`²(Z).

H will denote a separable Hilbert space, B(H)and GL(H) the algebras of bounded linear operators and boundedly invertible linear operators acting onH. S∞(H)and Sp(H), p≥1, stand for the ideal of compact operators and the Schatten classes. In particular, S2(H) is the ideal of Hilbert-Schmidt operators acting on H. For any operator H ∈ B(H), we denote its numerical range byN(H) :=

hHψ, ψi;ψ∈H,kψk= 1 , its spectrum byσ(H), its resolvent set byρ(H), the set of its eigenvalues byE_p(H). We also write

Re(H) = 1

2(H+H^∗) , Im(H) := 1

2i(H−H^∗).

We define its point spectrum as the closure of the set of its eigenvalues and write it σpp(H) = Ep(H). Finally, if Ais a selfadjoint operator acting onH, we writehAi:=√

A²+ 1.

For two subsets∆1 and ∆2 of R, we denote as a subset of C, ∆1+i∆2:=

z∈C: Re(z)∈

∆1,Im(z)∈∆2 . ForR >0 andζ0∈C, we setDR(ζ0) :=

z∈C:|z−ζ0|< R andD^∗_R(ζ0) :=

DR(ζ0)\ {ζ0}. In particular,D =D1(0) denotes the open unit disk of the complex plane. For Ω⊆Can open domain andBa Banach space,Hol(Ω,B)denotes the set of holomorphic functions from Ω with values in B. We will adopt the following principal determination of the complex square root: √

·:C\(−∞,0]−→

z∈C: Im(z)≥0 and we setC⁺:=

z∈C: Im(z)>0 . By 0<|k|<<1, we mean thatk∈C\ {0}is sufficiently close to0.

The discrete Fourier transform F : `²(Z) → L²(T), where T := R/2πZ, is defined for any x∈`²(Z)andf ∈L²(T)by

(Fx)(α) := (2π)⁻¹² X

n∈Z

e^−inαx(n), (F⁻¹f)(n) := (2π)⁻¹² Z

T

e^inαf(α)dα. (1.1) The operator F is unitary. For any bounded (resp. selfadjoint) operator L acting on `²(Z), we define the bounded (resp. selfadjoint) operatorLb acting onL²(T)by

Lb:=FLF⁻¹. (1.2)

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2 Model and Main results

2.1 The model

The unperturbed Hamiltonian. We denote byH₀the one-dimensional Schrödinger operator defined on`²(Z)by

(H0x)(n) := 2x(n)−x(n+ 1)−x(n−1). (2.1) H₀ is a bounded selfadjoint operator. We deduce that Hc₀ = FH₀F⁻¹ is the multiplication operator onL²(T)by the functionf where

f(α) := 2−2 cosα= 4 sin²α

2, α∈[−π, π]. (2.2)

It follows that σ(H0) = σac(H0) = [0,4], where {0,4} are the thresholds. For z ∈C\[0,4], we writeR0(z) = (H0−z)⁻¹. One deduces that for anyx∈`²(Z)

(FR₀(z)x)(α) = (Fx)(α)

f(α)−z, z∈C\[0,4].

Forz∈C\[0,4]small enough, we can introduce the change of variables z= 4 sin²φ

2, Im(φ)>0.

In this case, the resolventR₀(z)is represented by the convolution with the function R₀(z, n) = ie^iφ|n|

2 sinφ = iei|n|2 arcsin

√z

√ 2

z√

4−z , arcsin

√z

2 ∼

z=0

√z

2 . (2.3)

The perturbation. For any bounded operator V acting on `²(Z), we define the perturbed operator

HV :=H0+V. (2.4)

If

V(n, m) _(n,m)∈

Z² denotes the matrix representation of the operator V in the canonical orthonormal basis of`²(Z), forx∈`²(Z)the sequenceV x∈`²(Z)is given by

(V x)(n) = X

m∈Z

V(n, m)x(m) for anyn∈Z. (2.5)

IfV is represented by a diagonal matrix (i.e. V(n, m) =V(n, m)δnm), we writeV(n) :=V(n, n) andV is just the multiplicative operator defined by(V x)(n) =V(n)x(n),x∈`²(Z),n∈Z.

For further use, let us conclude this paragraph with the following observation (see Subsection 2.3 for more details). Let J : `²(Z) → `²(Z) be the unitary operator defined by (J x)(n) :=

(−1)ⁿx(n),x∈`²(Z), and define

V_J:=J V J⁻¹. (2.6)

Then,J²=Iand the matrix representation of the operatorVJin the canonical orthonormal basis of`²(Z)satisfies

VJ(n, m) = (−1)^n+mV(n, m), (n, m)∈Z².

In particular, ifV is a diagonal matrix, thenVJ =V. Calculations yield (see e.g. [28, Eq. (A.1)]):

J HVJ⁻¹=−H_−V_J+ 4so that

J(HV −z)⁻¹J⁻¹=−(H−V_J−(4−z))⁻¹. (2.7)

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The conjugate operator. Finally, let us define the auxiliary operatorA0, conjugate to H0 in the sense of the Mourre theory and acting on`²(Z)by

A0:=F⁻¹Ab0F, (2.8)

where the operator Ab₀ is (abusing notation) the unique selfadjoint extension of the symmetric operator

sinα(−i∂_α) + (−i∂_α) sinα defined onC^∞(T).

Remark 2.1 Define the position operator X and shift operator S on span {en : n ∈ Z} by (Xx)(n) = nx(n) and (Sx)(n) = x(n+ 1). A0 is also the unique selfadjoint extension of the symmetric operator Im(S)X+XIm(S)defined on span{en :n∈Z}.

2.2 Essential and discrete spectra

LetL be a closed operator acting on a Hilbert spaceH. If z is an isolated point ofσ(L), let γ be a small positively oriented circle centered atzand separating zfrom the other components of σ(L). The pointz is said to be a discrete eigenvalue ofLif its algebraic multiplicity

m(z) :=rank 1

2iπ Z

γ

(L−ζ)⁻¹dζ

(2.9) is finite. Note thatm (z)≥dim (Ker(L−z)), the geometric multiplicity ofz. Equality holds ifL is normal (see e.g. [29]). We define the discrete spectrum ofL by

σdisc(L) :=

z∈σ(L) :zis a discrete eigenvalue ofL . (2.10) We recall that a closed linear operator is a Fredholm operator if its range is closed, and both its kernel and cokernel are finite-dimensional. We define the essential spectrum ofLby

σess(L) :=

z∈C:L−zis not a Fredholm operator . (2.11) It is a closed subset ofσ(L).

Remark 2.2 If L is selfadjoint, σ(L) can be decomposed always as a disjoint union: σ(L) = σ_ess(L)Fσ_disc(L). If L is not selfadjoint, this property is not necessarily true. Indeed, consider for instance the shift operatorS:`²(Z+)→`²(Z+)defined by(Sx)(n) :=x(n+ 1). We have

σ(S) =

z∈C:|z| ≤1 , σ_ess(S) =

z∈C:|z|= 1 , σ_disc(S) =∅.

However, in the case of the operator HV, we have the following result:

Theorem 2.1 LetV belongs toS_∞(`²(Z)). Then,σ(HV) =σess(HV)Fσdisc(HV), whereσess(HV) = σess(H0) = [0,4]. The possible limit points ofσdisc(HV)are contained inσess(HV).

Proof. It follows from Weyl’s criterion on the invariance of the essential spectrum under compact perturbations and from [25, Theorem 2.1, p. 373]. See also [34, Corollary 2, p.113].

The reader will note that if V is compact and selfadjoint, thenσdisc(HV)⊂(−∞,0)∪(4,∞) and the set of limit points ofσdisc(HV)is necessarily contained in {0,4}. If V is non-selfadjoint, then σ_disc(H_V) may contain non-real numbers and the set of limit points may be considerably bigger, see e.g. [9] for the case of Laplace operators. However, we show in Theorem2.2that this cannot be the case ifV satisfies some additional regularity conditions.

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Definition 2.1 LetH be a Hilbert space,R >0andAbe a selfadjoint operator defined onH. An operator B∈ B(H)belongs to the classAR(A)if the map θ7→e^iθABe^−iθA, defined for θ∈Rex- tends holomorphically onDR(0)so that the corresponding extension belongs toHol(DR(0),B(H)).

In this case, we writeB∈ AR(A)andA(A) :=∪R>0AR(A)is the collection of bounded operators for which a complex scaling w.r.t. A can be performed.

Remark 2.3 (a) IfB∈S_∞(H)∩AR(A)for some selfadjoint operatorAand someR >0, then the holomorphic extension of the mapθ7→e^iθABe^−iθAbelongs actually toHol(DR(0),S_∞(H)), see e.g. [34, Lemma 5, Section XIII.5].

(b) The main properties of the classes AR(A) are recalled in Section 6. In particular, in our case, we have V ∈ AR(A₀)if and only if Vb ∈ AR(Ab₀).

We have:

Theorem 2.2 If V ∈ S_∞(`²(Z))∩ A(A0), then the possible limit points of σ_disc(H_V) belong to {0,4}.

Our next result, Theorem 2.3, provides additional informations on the essential spectrum σess(HV) = [0,4]. We recall that:

Definition 2.2 LetAbe a self-adjoint operator defined on the Hilbert spaceH. A vectorϕ∈H is analytic w.r.t. A ifϕ∈ ∩_k∈ND(A^k)and for someR >0, the power series

∞

X

k=0

|θ|^k k!

A^kϕ

converges for anyθ∈DR(0).

Later on, we will distinguish among the vectors which are analytic w.r.t. A₀, the vectors of the linear subspace span{e_n;n∈Z} (see e.g. Lemma6.2) and more generally, the vectors belonging to`²_δ(Z)for someδ >0 (see Corollary6.1).

Theorem 2.3 Let V ∈S∞(`²(Z))∩ A(A0). Then, there exists a discrete subsetD ⊂(0,4)whose only possible limits points belong to {0,4} and for which the following holds: given any relatively compact interval∆₀,∆₀⊂(0,4)\ D, there existsδ₀>0 such that for any analytic vectorsϕand ψ w.r.t. A₀,

sup

z∈∆0+i(−δ0,0)

|hϕ,(z−HV)⁻¹ψi|<∞, sup

z∈∆0+i(0,δ₀)

|hϕ,(z−HV)⁻¹ψi|<∞.

Remark 2.4 (a) For any subset∆ such that∆⊂(0,4),D ∩∆ is finite.

(b) If V is selfadjoint, then HV = HV∗ and D coincides with the set of eigenvalues of HV

embedded in (0,4) i.e. D = Ep(HV)∩(0,4). In the non-selfadjoint case HV 6= HV∗, we expect that Ep(HV)∩(0,4)⊂ D.

The proofs of Theorems2.2and2.3are postponed to Section3. Among perturbationsV which satisfy the hypotheses of Theorem2.2and2.3, we may consider:

• those which satisfy Assumption 2.1below,

• V = |ψihϕ|, where ϕ and ψ are analytic vectors for A0. We refer to Subsection 6.1.2 for more details and further examples.

For V ∈S_∞(`²(Z))∩ A(A0), Theorem2.2states that the points of(0,4) cannot be the limit points of any sequence of discrete eigenvalues. In the next section, we consider more specifically the case of the thresholds{0,4}.

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2.3 Resonances

In this section, we show how to control the distribution of resonances, in particular the eigenvalues ofH_V around the thresholds{0,4}. We perform this analysis by means of the characteristic values method. First, we formulate a new assumption and recall some basic facts on resonances.

Assumption 2.1 There existsδ >0 such that sup

(n,m)∈Z²

e^δ(|n|+|m|)

V(n, m) <∞,

where(V(n, m))_(n,m)∈_Z² is the matrix representation of the operatorV in the canonical orthonormal basis of`²(Z).

Remark 2.5 IfV satisfies Assumption2.1, then (a) V ∈S1(`²(Z))⊂S_∞(`²(Z)),

(b) V ∈ A(A0)(see Proposition6.6).

We also note that Assumption2.1holds forV if and only if it holds for VJ defined by (2.6).

As a preliminary, we have the following result, whose proof is contained in Section 4.

Proposition 2.1 Let Assumption 2.1holds. Set z(k) =k². Then, there exists 0< ε₀≤ ^δ₈, small enough, such that forV∈ {V,−V_J}, the operator-valued function with values inB(`²_δ(Z);`²_−δ(Z)),

k7−→(HV−z(k))⁻¹,

admits a meromorphic extension from D^∗_ε₀(0)∩C⁺ to D^∗_ε₀(0). This extension will be denoted R_V(z).

Now, we define the resonances of the operatorH_V nearz= 0andz= 4. We follow essentially the terminology and characterization of the resonances used in [5, Sect. 2 and 6] and refer to Section 4 for more details. In particular, the quantityIndγ(·), which denotes the index w.r.t. a contourγ, is defined by (4.11) and the weighted resolventsTV(·)are defined by (4.10).

Definition 2.3 The resonances of the operatorHV near 0 are the points z, which are the poles of the meromorphic extension of the resolvent RV(z), as introduced in Proposition2.1.

Fork∈D^∗_ε

0(0), setz₀(k) :=k². Givenk₁∈D^∗_ε

0(0), letz₁=z₀(k₁). By Proposition4.2, z₁is a resonance ofH_V near0if and only if k₁ is a characteristic value ofI+T_V(z₀(·)). We define the multiplicity of such a resonance as follows:

Definition 2.4 The multiplicity of a resonance z1 =z0(k1) is defined as the multiplicity of the characteristic valuek1, namely:

mult(z1) :=Indγ(I+TV(z0(·))), (2.12) where γ is a positively oriented circle centered at k₁, chosen sufficiently small so that k₁ is the only characteristic value enclosed inγ.

There exists a simple way to define the resonances of HV near the threshold z = 4. Indeed, relation (2.7) implies that

J W−δ(HV −z)⁻¹W−δJ⁻¹=−W−δ(H−V_J−(4−z))⁻¹W−δ. (2.13) Therefore, the analysis of the resonances ofH_V near the second threshold4is reduced to that of the first one 0(up to a sign and a change of variable). Combining (2.13) and Definition 2.3, we have:

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Definition 2.5 The resonances of the operatorHV near 4 are the points z= 4−uwhereuare the poles of the meromorphic extension of the resolventR_−V_J(u), as introduced in Proposition2.1.

For k∈D^∗_ε₀(0), set z4(k) := 4−k². Givenk1 ∈D^∗_ε₀(0), let z1=z4(k1). By Proposition4.3, z1is a resonance ofHV near4 if and only ifk1is a characteristic value ofI+T−V_J(4−z4(·)). As in Definition2.4, one has:

Definition 2.6 The multiplicity of a resonance z₁ =z₄(k₁) is defined as the multiplicity of the characteristic valuek₁, namely:

mult(z₁) :=Ind_γ(I+T_−V_J(4−z₄(·))), (2.14) where γ is a positively oriented circle centered at k1, chosen sufficiently small so that k1 is the only characteristic value enclosed inγ.

We denote byResµ(HV)the resonances set ofHV near the thresholdµ∈ {0,4}. The discrete eigenvalues of the operator HV near 0 (resp. near 4) are resonances. Moreover, the algebraic multiplicity (2.9) of a discrete eigenvalue coincides with its multiplicity as a resonance near 0 (resp. near 4), defined by (2.12) (resp. (2.14)). Let us justify it briefly in the case of the discrete eigenvalues near0 (the case concerning those near 4 can be treated similarly). Letz1= z0(k1)∈ C\[0,4] be a discrete eigenvalue ofHV near 0. According to [40, Chap. 9] and since V = W−δVW−δ is of trace class, V being defined by (4.9), this is equivalent to the property f(z₁) = 0, where forz∈C\[0,4], f is the holomorphic function defined by

f(z) := det(I+V(H₀−z)⁻¹) = det(I+VW_−δ(H₀−z)⁻¹W_−δ).

Moreover, the algebraic multiplicity (2.9) of z₁ is equal to its order as zero of the function f. Residue Theorem yields:

m(z1) =indγ⁰f := 1 2iπ

Z

γ⁰

f⁰(z) f(z)dz,

γ⁰ being a small circle positively oriented containingz₁ as the only zero off. The claim follows immediately from the equality

indγ⁰f =Indγ(I+TV(z0(·))), see for instance [5, Eq. (6)] for more details.

Remark 2.6 (a) The resonanceszµ(k)are defined in some two-sheets Riemann surfaces Mµ, µ∈ {0,4}, respectively.

(b) The discrete spectrum and the embedded eigenvalues of HV near µ belong to the set of resonances

zµ(k)∈ Mµ, Im(k)≥0.

The above considerations lead us to the following result:

Theorem 2.4 Let Assumption 2.1hold. Let ε₀ be as defined in Proposition 2.1. Then, we can chooseε⁰₀∈(0, ε₀]in such a way that H_V has no resonancez_µ(k), for µ∈ {0,4} andk∈D^∗_ε0

0(0).

According to Theorem 2.4, HV has no resonance in a punctured neighborhood of µ, in the two-sheets Riemann surface Mµ (where the resonances are defined). Figure 2.1 illustrates this fact.

Since the discrete eigenvalues of HV near µ are part of the set of resonances zµ(k) ∈ Mµ, Im(k)≥0, we conclude from Theorems2.2,2.4and Proposition 6.6that:

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Re(k) Im(k)

ε⁰₀

×

× ×

Absence of resonances

Thephysical plane

Thenon physical plane

(the second sheet of the Riemann surfaceMµ)

Figure 2.1: Resonances near the thresholdµ∈ {0,4}in variablek.

Theorem 2.5 Let Assumption 2.1 holds. Then, σ_disc(H_V) has no limit points in [0,4], hence is finite. There exists also a finite subset D⁰ ⊂ (0,4) for which the following holds: given any relatively compact interval ∆₀, ∆₀ ⊂(0,4)\ D⁰, there exists δ₀ >0 such that for any vectors ϕ andψ in`²_δ(Z),

sup

z∈∆0+i(−δ0,0)

|hϕ,(z−HV)⁻¹ψi|<∞, sup

z∈∆0+i(0,δ0)

|hϕ,(z−H_V)⁻¹ψi|<∞.

As mentioned in the introduction, the first conclusion of Theorem 2.5can be produced under weaker decay conditions along the diagonal provided that the perturbed operator HV is still a Jacobi operator, see e.g. [19, Theorem 1]. By contrast, Theorem2.5allows to handle perturbations displaying a full off-diagonal structure and exhibits some LAP.

If V is selfadjoint and satisfies Assumption 2.1, it follows from Proposition 2.1, Theorem2.5 and the usual complex scaling arguments (see e.g. [39]) that:

Corollary 2.1 Let Assumption2.1hold. If the perturbation V is selfadjoint, then:

• σ_ess(H_V) = [0,4]andσ_disc(H_V)is finite.

• There is at most a finite number of eigenvalues embedded in[0,4]. Each eigenvalue embedded in (0,4)has finite multiplicity.

• The singular continuous spectrum σsc(HV) = ∅ and the following LAP holds: given any relatively compact interval ∆0, ∆0 ⊂(0,4)\ Ep(HV), there exists δ0>0 such that for any vectors ϕandψin `²_δ(Z),

sup

z∈∆0+i(0,δ0)

|hϕ,(z−H_V)⁻¹ψi|<∞, sup

z∈∆0+i(−δ0,0)

|hϕ,(z−H_V)⁻¹ψi|<∞.

2.4 Embedded eigenvalues and Limiting Absorption Principles

For less regular perturbation V, we can still take advantage of the existence of some positive commutation relations to control some spectral properties of H_V. Let us define the regularity conditions involved in the statement of Theorem2.6below.

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Definition 2.7 Let H be a Hilbert space and A be a selfadjoint operator defined on H. Let k∈N. An operator B ∈ B(H) belongs to the classC^k(A), if the map WA :θ7→e^iθABe^−iθA is k-times strongly continuously differentiable onR. We also denote C^∞(A) =∩k∈NC^k(A).

Remark 2.7 A bounded operatorB belongs toC¹(A)if and only if the sesquilinear form defined on D(A)× D(A) by (ϕ, ψ) 7→ hAϕ, Bψi − hϕ, BAψi, extends continuously to a bounded form on H ×H. The (unique) bounded linear operator associated to the extension is denoted by adAB= adA(B) = [A, B]and we have (∂θWA)(0) =iadA(B).

Following [1], we also consider fractional order regularities:

Definition 2.8 Let H be a Hilbert space and A be a selfadjoint operator defined on H. An operator B∈ B(H)belongs toC^1,1(A) if

Z 1 0

e^iAθBe^−iAθ+e^−iAθBe^iAθ−2B

dθ θ² <∞.

Note thatB∈ C^1,1(A)if and only ifB can be suitably approximated by a family of operators in C²(A) ; we refer to Section5.5 for more details. Actually,C^1,1(A)is a linear subspace ofB(H), stable under adjunction∗. It is also known thatC²(A)⊂ C^1,1(A)⊂C¹(A), see e.g. inclusions (5.2.19) in [1].

Remark 2.8 Examples of operators belonging to the classC^1,1(A0)are given in Section6.2below.

We recall that the operator A0 is defined by (2.8). If we assume that Im(V) has a sign, we obtain Theorem2.6below. Mind that a statement involving the symbol ±has to be understood as two independent statements.

Theorem 2.6 Let V ∈S_∞(`²(Z))and assume ±Im(V)≥0. Fix any open interval∆ such that

∆⊂(0,4).

1. AssumeV ∈C¹(A0)andadA₀(Re(V))also belongs to S_∞(`²(Z)). Then, E_p(H_V)∩∆⊂σ_pp(Re(H_V))∩∆.

The set Ep(HV)∩∆ is finite and its eigenvalues have finite geometric multiplicity.

2. Assume thatV ∈ C^1,1(A0). Given any open interval ∆0such that∆0⊂∆\σpp(Re(H))and any s >1/2, the following LAP holds:

sup

∓Im(z)>0,Re(z)∈∆0

khA0i^−s(z−HV)⁻¹hA0i^−sk<∞.

Remark 2.9 (a) IfV ∈S_∞(H), thenV^∗,Re(V)andIm(V)also belong to S_∞(H).

(b) IfV belongs toC¹(A)(resp. C^1,1(A)), then, Re(V)and Im(V) also belong toC¹(A) (resp.

C^1,1(A)). In particular, if V belongs to C^1,1(A) andRe(V)∈S_∞(H), then adA(Re(V))∈ S_∞(H), see Remark (ii) in the proof of Theorem 7.2.9 in [1].

Finally, we also have:

Theorem 2.7 Let V ∈S_∞(`²(Z))and assume thatV ∈ C^1,1(A₀).

1. IfIm(V)>0andiad_A₀(Re(V)) +β₋Im(V)≥0for someβ₋≥0, then for any open interval

∆ such that ∆⊂(0,4) and anys >1/2, the following LAP holds:

sup

−Im(z)>0,Re(z)∈∆

khA0i^−s(z−HV)⁻¹hA0i^−sk<∞.

2. IfIm(V)<0andiadA₀(Re(V))−β+Im(V)≥0for someβ+≥0, then for any open interval

∆ such that ∆⊂(0,4) and anys >1/2, the following LAP holds:

sup

+ Im(z)>0,Re(z)∈∆

khA₀i^−s(z−H_V)⁻¹hA₀i^−sk<∞.

The proofs of Theorems 2.6 and 2.7 are direct applications of the abstract Mourre theory developed in Section5. See Section5.8for the details.

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3 Complex scaling

In this section, we use a complex scaling approach to study σ(HV) for compact perturbations V ∈ A(A0). Since the spectral properties ofHV andHdV coincide, we reduce our analysis to those ofHdV.

3.1 Before perturbation

Following the general principles exposed in [39], we first describe the scaling process for the unperturbed operator Hc₀, which is the multiplication operator by the function f (see (2.2)). In what follows, we have summarized the main results. When no confusion can arise, the operator Hc₀ is identified with the function f.

We consider the unitary group(eîθÂ^b⁰)_θ∈_R, so that forψ∈L²(T), one has (eîθÂ^b⁰ψ)(α) =ψ(ϕθ(α))p

J(ϕθ)(α), (3.1)

where

• (ϕθ)_θ∈_Ris the flow solution of the equation (∂θϕθ(α) = 2 sin(ϕθ(α)),

ϕ₀(α) = id_T(α) =α for each α∈T,

• J(ϕθ)(α)denotes the Jacobian of the transformationα7→ϕθ(α).

Existence and uniqueness of the solution follow from standard ODE results. Explicitly, ϕθ(α) =±arccos

−th(2θ) + cosα 1−th(2θ) cosα

for ±α∈T.

Using (3.1) and the fact that ϕθ₁◦ϕθ₂ =ϕθ₁+θ₂ for all(θ1, θ2)∈R², one has for allθ∈R

eîθÂ^b⁰Hc₀e^−iθÂ^b⁰ψ

(α) =f(ϕ_θ(α))ψ(α).

LetT :C→C,T(z) := 2(1−z). Note that the mapT is bijective with T⁻¹(z) = 1−z

2,

and maps [−1,1] onto [0,4]. The points T(−1) = 4 and T(1) = 0 are the thresholds of H₀ and Hc0. Note also that f = T ◦cos. Consider for θ ∈ R, the function Gθ defined on [0,4] by G_θ:=T◦F_θ◦T⁻¹ with

Fθ(λ) := λ−th(2θ)

1−λth(2θ), λ∈[−1,1]. (3.2) Then, for allθ∈R,

eîθÂ^b⁰Hc₀e^−iθÂ^b⁰ψ=G_θ(Hc₀)ψ. (3.3) In other words, for anyθ∈R,eîθÂ^b⁰Hc₀e^−iθÂ^b⁰ is the multiplication operator by the function(G_θ◦f), whereG_θ◦f = (T ◦F_θ◦cos) = (G_θ◦T◦cos).

Remark 3.1 In (3.2), the denominator does not vanish since

th(2θ)λ

<1 forλ∈[−1,1].

We summarise the following properties, whose verification is left to the reader.

Proposition 3.1 Let D:=D₁(0)denote the open unit disk of the complex plane C. Then, (a) For anyλ∈[−1,1], the map θ7→F_θ(λ)is holomorphic inD^π

4(0).

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(b) For θ ∈ C such that |θ| < ^π₈, the map λ 7→ Fθ(λ) is a homographic transformation with F_θ⁻¹=F−θ. In particular, forθ∈R,Fθ(D) =DandFθ([−1,1]) = [−1,1].

(c) Forθ∈Csuch that 0<|θ|<^π₈, the unique fixed points ofFθ are±1.

(d) Forθ₁,θ₂∈Cwith|θ1|,|θ2|<^π₈, we have that: F_θ₁◦F_θ₂=F_θ₁_+θ₂. From Proposition3.1, statements (a) and (b), we deduce:

Proposition 3.2 The bounded operator valued-function

θ7→eîθÂ^b⁰Hc0e^−iθÂ^b⁰∈ B(L²(T)), admits a holomorphic extension from (−^π₈,^π₈) to D^π

8(0), with extension operator given for θ ∈ D^π

8(0) by Gθ(Hc0), which is the multiplication operator by the functionGθ◦f = (T ◦Fθ◦cos) = (G_θ◦T ◦cos). In the sequel, this extension will be denoted Hc₀(θ).

Combining the continuous functional calculus, Proposition 3.2 and unitary equivalence properties, we get forθ∈D^π

8(0),

σ(Hc₀(θ)) =σ(G_θ(Hc₀)) =G_θ(σ(Hc₀)) =G_θ(σ(H₀)) =G_θ([0,4]).

Thus, forθ∈D^π

8(0),σ(Hc₀(θ))is a smooth parametrized curve given by σ(Hc₀(θ)) =n

T ◦F_θ(λ) =G_θ◦T(λ) :λ∈[−1,1]o

. (3.4)

More precisely, we have:

Proposition 3.3 Consider the family of bounded operators(Hc0(θ))_θ∈Dπ

8(0)defined in Proposition 3.2. Then, it holds:

(a) Forθ1,θ2∈D^π

8(0) such thatIm(θ1) = Im(θ2), we have

σ(Hc0(θ1)) =σ(Hc0(θ2)). (3.5) The curve σ(Hc0(θ))does not depend on the choice of Re(θ).

(b) For±Im(θ)>0,θ∈D^π

8(0), the curveσ(Hc₀(θ))lies inC±. (c) Let θ∈D^π

8(0). IfIm(θ)6= 0, the curveσ(cH₀(θ))is an arc of a circle, which contains the points0 and4. IfIm(θ) = 0,σ(Hc₀(θ)) = [0,4].

We refer to Figure3.1below for a graphic illustration.

Proof. Statement (a) can be derived from Proposition6.4(withHc0(·)in the role of the mapB(·)).

We give a direct proof here. Let θ1, θ2 ∈ D^π

8(0) with Im(θ1) = Im(θ2). Thanks to (3.4), it is enough to show that

n

Fθ₁(λ) :λ∈[−1,1]o

=n

Fθ₂(λ) :λ∈[−1,1]o

, (3.6)

to prove (3.5). So, letz=F_θ₁(λ), for someλ∈[−1,1]. Let us show that there exists λ⁰ ∈[−1,1]

such thatz=F_θ₂(λ⁰). By Proposition3.1, statements (b) and (d), we can write z=Fθ₂(F_θ⁻¹

2 (Fθ₁(λ))) =Fθ₂(F_−θ₂(Fθ₁(λ))) =Fθ₂(Fθ₁−θ2(λ)).

IfIm(θ1) = Im(θ2), thenθ1−θ2= Re(θ1−θ2)and it follows that:

z=F_θ₂(F_Re(θ₁_−θ₂₎(λ)).

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Since th(2 Re(θ1−θ2))< 1, then one has

F_Re(θ₁_−θ₂₎(λ)

≤1 for λ ∈ [−1,1]. By setting λ⁰ = F_Re(θ₁_−θ₂₎(λ), one gets z=Fθ₂(λ⁰)withλ⁰ ∈[−1,1]. This proves the inclusion

n

Fθ₁(λ) :λ∈[−1,1]o

⊂n

Fθ₂(λ) :λ∈[−1,1]o .

The opposite inclusion can be justified similarly once permuted the roles ofθ₁andθ₂. This proves the first claim. Statement (b) follows by direct calculations. Let us prove the last part. According to statement (a),θcan be chosen withRe(θ) = 0, sayθ=iy withy∈R. The caseIm(θ) = 0(i.e.

y = 0) is immediate since σ(Hc₀(θ)) =σ(cH₀(0)) = σ(cH₀) = σ(H₀) = [0,4]. Now, suppose that y6= 0. Observe that the mapT is a composition of a translation and a homothety. To prove that σ(Hc₀(θ))is an arc of a circle, it is enough to observe that {F_iy(λ) :λ∈[−1,1]} is a continuous parametrised curve and that forλ∈[−1,1], the real and imaginary parts ofFiy(λ)satisfy a circle equation. Indeed, denotingth(2θ) =itan(2y) =:it, one has

Fiy(λ) = λ(1 +t²)

1 +λ²t² +i(λ²−1)t

1 +λ²t² =:X+iY, and

X²+ Y −1−t² 2t

!²

= 1 +t² 2t

!²

. (3.7)

Remark 3.2 Forθ∈D^π

8(0), Im(θ)6= 0, Equation (3.7)allows to recover the centercθ∈Cand the radius Rθ>0 of the circle supportingσ(Hc0(θ)).

3.2 After perturbation

Now, we focus on the complex scaling of the perturbationVb together with the perturbed operator HdV =Hc0+Vb. ForVb ∈ AR(Ab0),R >0, and for allθ∈DR(0), we set

Vb(θ) := eîθÂ^b⁰Vb e^−iθÂ^b⁰. The following lemma holds:

Lemma 3.1 Let Vb ∈S_∞(L²(T))∩ AR(Ab0),R >0. Then, for allθ∈DR(0),Vb(θ)is compact.

Proof. This follows from [34, Lemma 5, Section 5], since Vb(·)is the analytic continuation of a bounded operator-valued function with compact values on the real axis.

Now, for allθ∈D2R⁰(0)with2R⁰:= min(R,^π₈), we consider

dH_V(θ) :=Hc₀(θ) +Vb(θ), Vb ∈ A_R(Ab₀). (3.8) We obtain:

Proposition 3.4 Let Vb ∈ AR(Ab0),R >0. Then,

(a) Hd_V(θ) is a holomorphic family of bounded operators onD_2R⁰(0).

(b) For anyθ⁰ ∈Rsuch that|θ⁰|< R⁰, we have

HdV(θ+θ⁰) = eîθ⁰Â^b⁰HdV(θ) e^−iθ⁰Â^b⁰, for all θ∈DR⁰(0).

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Proof. Statement (a) follows from Proposition3.2. Now, we prove (b). We fixθ⁰∈(−R⁰, R⁰)and observe thatDR⁰(0)⊂DR(0)∩DR(−θ⁰). The maps

θ7−→eîθ⁰Â^b⁰Hd_V(θ) e^−iθ⁰Â^b⁰ and θ7−→dH_V(θ+θ⁰),

are bounded operator-valued and holomorphic onD_R⁰(0). Moreover, they coincide onR∩D_R⁰(0) =

(−R⁰, R⁰). Hence, they also coincide onD_R⁰(0).

The next proposition gives the key to the proofs of Theorems 2.2and2.3.

Proposition 3.5 Let R > 0 and Vb ∈ S_∞(L²(T))∩ AR(Ab0), and let R⁰ > 0 such that 2R⁰ = min(R,^π₈). Then, for anyθ∈DR⁰(0), we have

(a) σ(dH_V(θ))depends only on Im(θ).

(b) It holds: σess(dHV(θ)) =σess(Hc0(θ)) =σ(Hc0(θ))and σ(dH_V(θ)) =σ_disc(dH_V(θ))G

σ_ess(Hc₀(θ)), where the possible limit points of σdisc(dHV(θ))lie inσess(Hc0(θ)).

Proof. Statement (a) is a consequence of the unitary equivalence established in Proposition3.4 (b). Statement (b) follows from Lemma3.1, the Weyl criterion on the invariance of the essential spectrum and [25, Theorem 2.1, p. 373] (see also [34, Corollary 2, p.113]).

3.3 Proof of Theorem 2.2

The proof of Theorem2.2follows from Proposition3.6below as an adaptation of the usual complex scaling arguments to our non-selfadjoint setting (see e.g. [34, Theorem XIII.36], [27]).

For anyθ∈D^π

8(0),cθ∈CandRθ>0 stand respectively for the center and the radius of the circle supportingσ(Hc₀(θ)). For±Im(θ)≥0, we define the open domainsS_θ^∓ :=C\A^±_θ where

A^±_θ :=

z∈C: Re(z)∈[0,4],±Im(z)≥0,|z−c_θ| ≥R_θ , ±Im(θ)>0, and A^±₀ =

z ∈C: Re(z)∈[0,4],±Im(z)≥0 . According to Proposition 3.3, the domainsS^±_θ depend only on Im(θ). In addition, if0≤Im(θ⁰)<Im(θ), S_θ⁻0 (S_θ⁻ and if Im(θ)<Im(θ⁰)≤0, S_θ⁺0 (S_θ⁺.

Proposition 3.6 Let R >0,Vb ∈ AR(Ab0)and R⁰ >0 such that2R⁰ = min(R,^π₈). Let (θ, θ⁰)∈ DR⁰(0)×DR⁰(0). Then:

(a) If0≤Im(θ⁰)<Im(θ), we haveσdisc(dHV(θ⁰))∩S_θ⁻0 =σdisc(dHV(θ))∩S_θ⁻0 ⊂σdisc(dHV(θ))∩S_θ⁻. In particular,

σdisc(dHV(θ))∩S₀⁻=σpp(dHV)∩S₀⁻.

(b) IfIm(θ)<Im(θ⁰)≤0, we haveσdisc(dHV(θ⁰))∩S_θ⁺0 =σdisc(dHV(θ))∩S_θ⁺0 ⊂σdisc(dHV(θ))∩S_θ⁺. In particular,

σ_disc(dH_V(θ))∩S₀⁺=σ_pp(dH_V)∩S₀⁺.

As a consequence, the discrete spectrum ofHd_V (andH_V) can only accumulate at0 and4.

Proof. We focus our attention on case (a). For a moment, fix θ0∈DR⁰(0) such that Im(θ0)>0 and suppose that λ ∈ σ_disc(dH_V(θ₀)). Since the map θ 7−→ dH_V(θ) is analytic, there exist open neighborhoodsV_θ₀ andW_λ ofθ₀and λrespectively such that [29,34]:

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Figure 3.1: Spectral structure of the operatordHV(θ)forθ∈D^π

8(0)andIm(θ)≤0.

• For all θ ∈ Vθ0, the operator Hd_V(θ) has a finite number of eigenvalues in Wλ denoted by λ_j(θ)∈ {1, . . . , n}, all of finite multiplicity.

• These nearby eigenvalues are given by the branches of a finite number of holomorphic functions inVθ₀ with at worst algebraic branch point nearθ0.

Ifϕ=θ−θ₀∈Rforθ∈ V_θ₀, then dH_V(θ₀+ϕ)andHd_V(θ₀)are unitarily equivalent according to Proposition3.4. So, the only eigenvalue ofdHV(θ0+ϕ)nearλinWλisλ. Therefore,λj(θ) =λfor anyj∈ {1, . . . , n}and forθ∈ Vθ0 withθ−θ₀∈R. By analyticity, we deduce that for allθ∈ Vθ0

and allj ∈ {1, . . . , n}one hasλ_j(θ) =λ. Finally, we have proved that givenθ₀∈D_R⁰(0)andλ∈ σdisc(dHV(θ0)), there exists a neighborhoodVθ₀ ofθ0, such that for allθ∈ Vθ₀, λ∈σdisc(dHV(θ)).

Now, following [34, Problem 76, Section XIII], ifγ ∈ C⁰([0,1], DR⁰(0)) is a continuous curve and λ∈ σ_disc(dH_V(γ(0))), then either λ∈ σ_disc(dH_V(γ(1))) or λ∈ σ_ess(dH_V(γ(t))) for some t ∈ (0,1].

We apply this observation twice keeping in mind Proposition3.5.

Fixθ∈DR⁰(0)withIm(θ)>0. Starting withλ∈σdisc(dHV)∩S₀⁻=σpp(dHV)∩S₀⁻, considering the continuous curveγ⁺: [0,1]→DR⁰(0)

γ⁺(t) =tθ,

and applying the above observation yields: σdisc(dHV)∩S⁻₀ ⊂σdisc(dHV(θ))∩S₀⁻. Now starting withλ∈σdisc(dHV(θ))∩S⁻₀, considering the opposite continuous curveγ⁻: [0,1]→DR⁰(0)

γ⁻(t) = (1−t)θ,

and applying the above observation yields the opposite inclusion. So, we have proven that given θ∈D_R⁰(0)withIm(θ)>0,λ∈S₀⁻, thenλ∈σ_disc(dH_V(θ))∩S₀⁻if and only ifλ∈σ_disc(dH_V)∩S₀⁻= σpp(dHV)∩S₀⁻.

The proof of the identityσ_disc(dH_V(θ⁰))∩S_θ⁻0 =σ_disc(dH_V(θ))∩S_θ⁻0 for any0≤Im(θ⁰)<Im(θ)is similar. The inclusionσ_disc(dH_V(θ))∩S_θ⁻0 ⊂σ_disc(dH_V(θ))∩S_θ⁻follows from the inclusionS_θ⁻0 (S_θ⁻, which allows us to conclude on case (a).

The proof of case (b) is analog.

Once proven Statements (a) and (b), we conclude as follows. From Proposition 3.5, we know that the limit points ofσ_disc(dH_V)belong necessarily toσ_ess(dH_V) = [0,4]. Pick one of these points,

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sayλ0. It is necessarily the limit point of a subsequence of eitherσdisc(dHV)∩S⁻₀ orσdisc(dHV)∩S⁺₀. Without any loss of generality, assume there exists a subsequence ofσ_disc(dH_V)∩S₀⁻which converges to λ0 ∈[0,4]. Sinceσdisc(dHV)∩S₀⁻ =σdisc(dHV(θ))∩S₀⁻ for any Im(θ)>0,θ∈DR⁰(0), λ0 also belongs toσ_ess(dH_V(θ)). Sinceσ_ess(dH_V(θ))∩σ_ess(dH_V) ={0,4} for anyIm(θ)>0,λ₀∈ {0,4}and

the last part follows.

3.4 Proof of Theorem 2.3

Let R > 0, Vb ∈ AR(Ab0) and R⁰ > 0 such that 2R⁰ = min(R,^π₈). Observe first that for any θ∈D_R⁰(0)withIm(θ)>0,D₊ :=σ_disc(dH_V(θ))∩(0,4)is discrete and its possible accumulation points belong to{0,4}. Letϕandψbe two analytic vectors forAb0with convergence radiusR0>0 and denote byϕ:θ7→ϕ(θ) andψ:θ 7→ψ(θ)their analytic extension on DR₀(0). Consider the functionF(z, θ) =hψ(¯θ),(HV(θ)−z)⁻¹ϕ(θ)iwhenever it exists. Forθ∈D_min(R₀_,R⁰₎(0),F(·, θ)is analytic inC\σ(dHV(θ))and meromorphic inC\σess(dHV(θ)). Now, let us fixz∈S₀⁻\σdisc(dHV).

ThenF(z,·)is analytic on some regionDmin(R₀,R⁰)(0)∩ {θ∈C:−z <Im(θ)}, for somez >0.

Since for any η ∈Dmin(R₀,R⁰)(0)∩R, direct calculations yieldF(z, η) =F(z,0), we conclude by analyticity thatF(z,·)is constant inD_min(R₀_,R0)(0)∩ {θ∈C:−z<Im(θ)}. In particular, given θ₀ ∈D_min(R₀_,R0)(0) with Im(θ₀)> 0, F(·, θ0) provides an analytic continuation to the function F(·,0) from S₀⁻ \ σ_disc(dH_V) to S_θ⁻

0 \σ_disc(dH_V(θ₀)). In particular, for any relatively compact interval∆0, ∆0⊂(0,4)\ D+, the map hψ,(HV −z)⁻¹ϕiextends continuously from some region

∆0−i(0, δ₀⁻], for someδ₀⁻>0, to∆0−i[0, δ⁻₀]. Note thatδ⁻₀ can be chosen as:

δ⁻₀ =dist(∆0,min{Im(z);z∈σdisc(dHV)∩S⁻₀})>0.

This proves the first statement.

The proof of the other case is similar (with θ ∈ DR⁰(0) and Im(θ)< 0). Once proven both cases, we can setD=D+∪ D− and the proposition follows.

4 Resonances for exponentially decaying perturbations

Throughout this section, the perturbation V is assumed to satisfy Assumption 2.1. We aim at defining and characterizing the resonances of the operatorH_V near the thresholds0and4.

4.1 Preliminaries

The purpose of this first paragraph is to prove the following result:

Lemma 4.1 Set z(k) :=k². Then, there exists 0< ε₀≤ ^δ₈ small enough such that the operator- valued function with values in S_∞(`²(Z))

k7→W_−δ(H0−z(k))⁻¹W_−δ, admits a holomorphic extension fromD_ε^∗₀(0)∩C⁺ toD^∗_ε₀(0).

Proof. For k ∈ D^∗_ε(0)∩C⁺, 0 < ε < ^δ₄ small enough, and x ∈ `²(Z), the operator W_−δ(H₀− z(k))⁻¹W_−δ satisfies

W−δ(H0−z(k))⁻¹W−δx

(n) = X

m∈Z

e⁻^δ²^|n|R0(z(k), n−m)e⁻^δ²^|m|x(m), (4.1) R₀(z,·)being the function defined by (2.3). Thus, for any(n, m)∈Z², we have

e⁻^δ²^|n|R0(z(k), n−m)e⁻^δ²^|m|= e⁻^δ²^|n|ie^2i|n−m|^arcsin^k² k√

4−k² e⁻^δ²^|m|=:f(k). (4.2)

On the spectral properties of non-selfadjoint discrete Schrödinger operators