Maha Aafarani

LARGE TIME BEHAVIOR OF SOLUTIONS TO SCHR ÖDINGER EQUATION WITH COMPLEX-VALUED POTENTIAL

Keywords: Mathematics Subject Classification. 35P05, 35Q41, 47A10 Non-selfadjoint operator, Schrödinger operator, positive resonances, zero eigenvalue, zero resonance. 1

Large time behavior solutions to Schrödinger equation with complex-valued potential

Maha Aafarani

Introduction

In this work, we are interested in the large-time behavior of the solution u(t) = e -itH u 0 as t → +∞ of the time-dependent Schrödinger equation

i∂ t u(t, x) = Hu(t, x) , x ∈ R 3 , t > 0 u(0, x) = u 0 (x), (1.1)
where H = -∆ + V is a perturbation of -∆ by a complexe-valued potential supposed to satisfy the decay condition

|V (x)| ≤ C v x -ρ , ∀x ∈ R 3 , (1.2)
where ρ > 2 and x = (1

+ |x| 2 ) 1/2 .
This equation is a fundamental dynamical equation for the wave-function u(t, x) describing the motion of particles in non relativistic quantum mechanics. An interesting model in nuclear physics where this non-seladjoint operator arises, is the optical nuclear model [START_REF] Feshbach | Model for Nuclear Reactions with Neutrons[END_REF]. This model describes the dynamic of a compound elastic neutron scattering from a heavy nucleus. In this example, the interaction between the neutron and the nucleus is modeled by a complex-valued potential with negative imaginary part.

It turns out that the behavior of non-selfadjoint Schrödinger operators may differ from selfadjoint ones (see [START_REF] Schwartz | Some non-selfadjoint operators[END_REF][START_REF] Davies | Non-selfadjoint differential operators[END_REF]). Previously, Jensen and Kato [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF] have studied the three dimensional selfadjoint operator H = -∆ + V with rapidly decreasing real potential V (V satisfies the decay condition (1.2)). Obviously, in the selfadjoint case the decay in time of the solution is strongly linked to the analysis at low energies part of the resolvent depending on the presence of the zero eigenvalue or/and zero resonance. In [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF] the low energy asymptotics of the resolvent was obtained. Also local decay in time of the solution to the time-dependent Schrodinger equation is found. In an extension work [START_REF] Komech | Dispersive decay for the magnetic schrödinger equation[END_REF] of the latter, a similar results has been obtained for Schrödinger operator with a magnetic potential under the assumption that zero is a regular point i.e. it is not an eigenvalue nor a resonance in the sense of [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF][START_REF] Newton | Noncentral potentials: The generalized levinson theorem and the structure of the spectrum[END_REF]. For studies of the non-selfadjoint Schrödinger operator, we refer for example to [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF] on Gevrey estimates of the resolvent for slowly decreasing potentials and sub-exponential time-decay of solutions, to [START_REF] Wang | Time-decay of semigroups generated by dissipative schrödinger operators[END_REF][START_REF] Zhu | Large-time Behavior of the Solutions to Dissipative Schrödinger Equation[END_REF] for time-decay of solutions to dissipative Schrödinger equation and to [START_REF] Goldberg | Dispersive bounds for the three-dimensional schrödinger equation with almost critical potentials[END_REF][START_REF] Goldberg | A dispersive bound for three-dimensional schrödinger operators with zero energy eigenvalues[END_REF] for dispersive estimates.

Our goal is to extend the results of [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF] to non-selfadjoint Schrödinger operator with a rapidly decreasing complex-valued potential V . We are interested in the spectral analysis of the operator H and the large time behavior of e -itH for some model operator having real resonances. Here, we mean by the latter a real number λ 0 ≥ 0 for which the equation -∆u + V u -λ 0 u = 0 has a non trivial solution ψ ∈ L 2,-s (R 3 ) \ L 2 (R 3 ), ∀s > 1/2. In particular, for λ 0 > 0 it can be seen that this solution satisfies the Sommerfeld radiation condition

ψ(x) = e ±i √ λ0|x| |x| w( x |x| ) + o( 1 |x| ), as |x| → +∞, (1.3)
where w ∈ L 2 (S 2 ), w = 0, with S 2 = {x ∈ R 3 : |x| = 1}.

It is well known that selfadjoint Schrödinger operator can have zero resonance only but no positive resonances (see [START_REF] Agmon | Spectral properties of schrödinger operators and scattering theory[END_REF][START_REF] Ikebe | Limiting absorption method and absolute continuity for the schrödinger operator[END_REF][START_REF] Komech | Dispersive decay for the magnetic schrödinger equation[END_REF]). In [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF] positive resonances are absent and intermediate energies do not contribute to the large time behavior of e -itH . While these numbers are the main difficulty to study the spectral properties of non selfadjoint operators near the positive real axis. Wang in [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF] has included the presence of positive resonances for a compactly supported perturbation of the Schrödinger operator with a slowly decaying potential satisfying a condition of analyticity. However, usually these real numbers are supposed to be absent (see [START_REF] Goldberg | A dispersive bound for three-dimensional schrödinger operators with zero energy eigenvalues[END_REF]). See [START_REF] Wang | Time-decay of semigroups generated by dissipative schrödinger operators[END_REF] for an example of a positive resonance of a dissipative operator in dimension three and [START_REF] Pavlov | The nonself-adjoint schroedinger operator[END_REF] in dimension one.

Real resonances are responsible for most remarkable physical phenomena and many problems arising from analysis of spectral properties for non selfadjoint operators. Effectively, the zero resonance is responsible for Efimov effect for N-body quantum systems (see [START_REF] Amado | There is no efimov effect for four or more particles[END_REF][START_REF] Tamura | The efimov effect of three-body schrödinger operators[END_REF][START_REF] Tamura | The efimov effect of three-body schrödinger operators: asymptotics for the number of negative eigenvalues[END_REF][START_REF] Wang | On the existence of the n-body efimov effect[END_REF] and see also the physical review [START_REF] Naidon | Efimov physics: a review[END_REF]). In addition, the presence of positive resonances affects essentially the asymptotic completeness of wave operators (cf. [START_REF] Faupin | Asymptotic completeness in dissipative scattering theory[END_REF]). They are further the only points to which complex eigenvalues may eventually accumulate, as well as the boundary values of the resolvent on the cut along the continuous spectrum, known as limiting absorption principle, does not exist globally (cf. [START_REF] Royer | Limiting absorption principle for the dissipative helmholtz equation[END_REF][START_REF] Saito | The principle of limiting absorption for the nonselfadjoint schrödinger operator in R N (N = 2)[END_REF][START_REF] Wang | Time-decay of semigroups generated by dissipative schrödinger operators[END_REF]).

In order to obtain the large-time behavior of e -itH , we establish the expansions of R(z) at threshold zero and positive resonances in the sense of bounded operators between two suitable weighted Sobolev spaces (see Section 2). The novelties in our results are the following. We establish the asymptotic expansions of R(z) near positive resonances (Theorem 2.5) which are assumed to be singularities of finite order with a specific hypothesis (H3) on the behavior of solutions defined by (1.3) (see [START_REF] Faupin | Asymptotic completeness in dissipative scattering theory[END_REF][START_REF] Schwartz | Some non-selfadjoint operators[END_REF] for more hypotheses concerning positive resonances). In addition, in theorems 2.2-2.4 we obtain the asymptotic expansions of the resolvent at low energies for non-selfadjoint operator. In particular, explicit calculation of lower order coefficients has been performed. The difficulty is that the algebraic and geometric eigenspaces need not coincide, i.e not only eigenvectors but also Jordan chain may occur. Our main approach extends the method of Lidskii [START_REF] Lidskii | Perturbation theory of non-conjugate operators[END_REF] to study the giant matrices representation that we find, where there exist many Jordan blocks corresponding to the eigenvectors associated with zero eigenvalue. Under our hypothesis (H1) and (H2) we get same singularities (negative powers of z 1/2 ) that have appeared in [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF] because the presence of zero eigenvalue or/and zero resonance. More recent results can be followed in [START_REF] Goldberg | A dispersive bound for three-dimensional schrödinger operators with zero energy eigenvalues[END_REF][START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF].

The obtained results are useful for the study of the asymptotic behavior in time of wave functions that would depend on the nature of the threshold energy and the characteristic of positive resonances, as well as the asymptotic expansions of the resolvent and the semigroup e -itH as t → +∞ have many applications in the scattering theory (see for example [START_REF] Faupin | Asymptotic completeness in dissipative scattering theory[END_REF][START_REF] Faupin | Scattering matrices for dissipative quantum systems[END_REF]). This paper is organized as follows. In Section 2, we introduce our hypothesis and we state the main results. In Section 3 we study the asymptotic expansions of the resolvent at zero energy. We prove Theorem 2.3 when zero is an eigenvalue of arbitrary geometric multiplicity, then we prove Theorem 2.2 in the case of zero resonance and Theorem 2.4 in the more complicated case when zero is both an eigenvalue and a resonance of H. Section 4 is devoted to the study of outgoing positive resonances, we establish the proof of Theorem 2.5 by assuming that the set of positive resonances is finite and their associated eigenvectors satisfy an appropriate assumption. Moreover, we obtain another result (Theorem 2.5) in more general situation. Finally, in Section 5 we establish a representation formula for the semigroup e -itH as t → +∞ which allows us to prove Theorem 2.6.

Notation. Let X and X be two Banach spaces. We denote B(X , X ) the set of linear bounded operators from X to X . For simplicity, B(X ) = B(X , X ). For all m, m , s, s ∈ R, we denote by H m,s the weighted Sobolev space on R 3 H m,s = {u ∈ S (R 3 ) : u m,s = x s (1 -∆) m/2 u L 2 < ∞}, such that for m < 0, H m,s is defined as the dual of H -m,-s with dual product identified with the scalar product of L 2 •, • . The index s is omitted for standard Sobolev spaces, i.e. H m denotes H m,0 . In particular H 0 = L 2 with the associated norm • 0 . Let B(m, s, m , s ) = B H m,s , H m ,s . For linear operator T , we denote by Ran T the range of T and by rank T its rank. We also define the following subsets:

R + = [0, +∞[, R -=] -∞, 0], C ± = {z ∈ C, ±Im (z) > 0} and C± = {z ∈ C, ±Im (z) ≥ 0}.

Assumptions and formulation of the main results

2.1. The operator. We consider the Schrödinger operator H = -∆ + V in R 3 , where ∆ denotes the Laplacian and V is a complex-valued potential which will be assumed to satisfy the following decay condition

|V (x)| ≤ C v x -ρ , ∀x ∈ R 3 , (2.1)
where ρ > 2 throughout the paper and will depend on the results to obtain. Under the previous assumptions, H is a closed non-selfadjoint operator on L 2 with domain the standard Sobolev space H 2 . Moreover, the condition (2.1) implies that the operator V of multiplication by V (x) is relatively compact with respect to -∆. It is then known that the essential spectrum of H denoted by σ e (H) coincides with that of the non perturbed operator -∆ (cf. [START_REF] Hislop | Relatively Compact Operators and the Weyl Theorem[END_REF]). Thus σ e (H) covers the positive real axis [0, +∞[. In addition, the operator H has no eigenvalues along the half real axis ]0, +∞[ (cf. [START_REF] Kato | Growth properties of solutions of the reduced wave equation with a variable coefficient[END_REF]). Hence, the spectrum of H denoted by σ(H) is the disjoint union of σ e (H) and a countable set denoted by σ d (H), with

σ d (H) := {z ∈ C \ [0, +∞[: ∃ 0 = u ∈ D(H), Hu = zu} (2.2)
consisting of discrete eigenvalues with finite algebraic multiplicities. For z ∈ σ d (H), the associated Riesz projection of H is defined by

Π z = - 1 2iπ |w-z|= (H -wId) -1 dw, (2.3)
for > 0 small enough. These eigenvalues can accumulate only on the half axis [0, +∞[ at zero or at positive resonances (see Definition 2.1).

It should be noted that in this work it is sufficient to assume the existence of some constants ρ > 2 and C v , R > 0 such that the assumption (2.1) on the potential V is replaced by the following:

   |V (x)| ≤ C v x -ρ , x ∈ R 3 with |x| > R, V is -∆ -compact.
Resolvent. Denote R 0 (z) = (-∆-zId) -1 for z ∈ C\R + and R(z) = (H -zId) -1 for z ∈ C\σ(H). In order to obtain the asymptotic expansion of R(z), we use the following relation between the resolvents

R(z) = (Id + R 0 (z)V ) -1 R 0 (z), ∀z / ∈ σ(H), (2.4)
and we need to recall some well-known facts about R 0 (z).

For

z ∈ C \ R + , R 0 (z) is a convolution operator from L 2 to itself with integral kernel R 0 (z)(x, y) = e +i √ z|x-y| 4π|x -y| , Im √ z > 0.
Here the branch of

√ z is holomorphic in C \ R + such that lim →0 + √ λ ± i = ± √ λ, ∀λ > 0. Moreover,
the boundary values of the free resolvent on R + are defined by the following limits

R ± 0 (λ) := s -lim →0 R 0 (λ ± i ), for λ > 0, (2.5)
which exist in the uniform operator topology of B(0, s, 0, -s ), for s, s > 1/2 (see [START_REF] Agmon | Spectral properties of schrödinger operators and scattering theory[END_REF]Theorem 4.1]).

In addition, for all

∈ N R 0 (z) = j=0 (i √ z) j G j + o(|z| /2 ), (2.6)
where G j is an integral operator with integral kernel

G j (x, y) = |x -y| j-1 /4πj!, j = 0, 1, • • • , , such that G 0 ∈ B(-1, s, 1, -s ), s, s > 1/2, s + s > 2, (2.7)
G j ∈ B(-1, s, 1, -s ), s, s > j + 1/2, j = 0, 1, • • • , . (2.8)
In particular, G 0 := s -lim z→0,z∈C\R+ R 0 (z) is formally inverse to -∆. See [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF]Section 2.]. We will often work with the variable a square root, η, z = η 2 with Im η > 0, in order to use analyticity arguments.

Let C \ R + z → K(z) := R 0 (z)V : L 2 -→ L 2
be an analytic operator valued function. As mentioned before, we see that K(z) is a compact operator for all z ∈ C \ R + , and that {Id + K(z), z ∈ C \ R + } is a holomorphic family of Fredholm operators ( [START_REF] Dyatlov | Mathematical theory of scattering resonances[END_REF], Annexe C.2). By (2.5), the latter can be continuously extended to a family of operators in B(L 2,-s ) for 1/2 < s < ρ -1/2 in the two closed half-planes C± . Therefore, applying analytic Fredholm theory with respect to z, it follows that

(Id + R 0 (z)V ) -1 is a meromorphic operator valued function in C \ R + with values in B(L 2,-s ), whose poles are discreet eigenvalues of H in C \ R + . Moreover, for λ > 0, the limits lim →0 + (Id + R 0 (λ ± i )V ) -1 = (Id + R ± 0 (λ)V ) -1 ,
exist in B(0, -s, 0, -s), for every 1/2 < s < ρ -1/2, if and only if Id + R ± 0 (λ)V is one to one. In other terms, the above limits do not exist if there exists a non trivial solution ψ ∈ H 1,-s , ∀s > 1/2, of R 0 (λ ± i0)V g = -g. And, it can be easily proved that ψ ∈ H 1,-s , ∀s > 1/2, is a solution of R 0 (λ ± i0)V g = -g if and only if (H -λ)ψ = 0 and ψ satisfies the radiation condition (1.3). Similarly, in view of (2.6), we see that lim z∈C\R+,z→0

(Id + R 0 (z)V ) -1 = (Id + G 0 V ) -1 ,
exists in B(0, -s, 0, -s), for 1/2 < s < ρ -1/2, if and only if Id + G 0 V is one to one. In the following we shall use the notations

K + (λ) := R + 0 (λ)V and K 0 := G 0 V . Definition 2.1. A positive number λ 0 > 0 is called an outgoing positive resonance of H if -1 ∈ σ(K + (λ 0 )) and it is called an incoming positive resonance if -1 ∈ σ(K -(λ 0 )). Moreover, zero is said to be a resonance of H if Ker L 2,-s (Id + K 0 )/Ker L 2 (Id + K 0 ) = {0}, ∀s > 1/2. Let σ + r (H) denotes the set of all outgoing positive resonances.
This work is mainly concerned with the singularities of the resolvent at zero and at outgoing positive resonances.

Note that zero may be an embedded eigenvalue or/and a resonance of the non-selfadjoint operator H if the decay condition ρ > 2 is satisfied. In the selfadjoint case, it is known that the resonance at zero, if it occurs, is simple, i.e. dim Ker H 1,-s H/Ker L 2 H = 1, ∀s > 1/2 (cf. [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF]Theorem 3.6]). Similarly, in the non-selfadjoint case we can show that zero resonance is geometrically simple. Indeed, let s = 1/2 + , ρ = 2 + 0 , 0 < < 0 and ψ ∈ H 1,-s such that (Id + G 0 V )ψ = 0. Then ψ(x) behaves as |x| → +∞ like

ψ(x) ∼ C |x| + 1 |x| 1+ 0-φ(x) with C = -1 4π R 3 V (y)ψ(y)dy, (2.9)
where φ is some bounded function on {x ∈ R 3 , |x| > 1}. The same argument must be reiterated as necessary in showing that

R 3 V (y)ψ(y)dy = 0 if and only if ψ ∈ L 2 .
(2.10) Furthermore, zero is an eigenvalue of H if and only if -1 is an eigenvalue of the compact operator K 0 on L 2,-s and the associated eigenfunctions belong to the orthogonal space of 1 defined by {ψ ∈ L 2,-s ; ψ, V 1 = 0} (see (2.10)). If this occurs, then their associated eigenspaces coincide. In particular, they have the same geometric multiplicity, i.e. dim Ker

L 2 (H) = dim Ker L 2 (Id + K 0 ). Let Π 1 : L 2,-s → L 2,-s (resp., Π λ 1 : L 2,-s → L 2,-s
) be the well defined Riesz projection associated with the eigenvalue -1 of the compact operator K 0 (resp., K + (λ)) (see (2.3)). Noting that the Riesz projection corresponding to the embedded eigenvalue 0 of H cannot be defined. We denote m := rank Π 1 the algebraic multiplicity of -1 as eigenvalue of K 0 .

Our study covers all the situations of zero energy. We will use the following terminology introduced in [START_REF] Newton | Noncentral potentials: The generalized levinson theorem and the structure of the spectrum[END_REF]: If zero is a resonance and not an embedded eigenvalue of H zero is said to be a singularity for H of the first kind. If zero is an embedded eigenvalue and not a resonance of H zero is said to be a singularity for H of the second kind. Finally, if zero is both an embedded eigenvalue and a resonance of H zero is said to be a singularity for H of the third kind. In the last two cases, we look at the eigenvalue -1 of K 0 of geometric multiplicity k ≥ 2. Since algebraic and geometric eigenspaces need not be equal, generalized eigenvectors occur in Ran Π 1 . We decompose Ran Π 1 to k invariant subspaces of

K 0 , E 1 , • • • , E k . The subspaces E i , 1 ≤ i ≤ k, are spanned by Jordan chains of length m i , i.e E i = Span{u (i) r = (Id + K 0 ) mi-r u (i) mi , 1 ≤ r ≤ m i } for some vectors u (i) mi ∈ Ker(Id + K 0 ) mi \ Ker(Id + K 0 ) mi-1 , such that dim Ker(Id + K 0 )| Ei = 1.
This decomposition yields the Jordan canonical form of the matrix Π 1 (Id + K 0 )Π 1 given in (3.8). See Lemma 3.1.

2.2.

Hypotheses. Our first two hypotheses are about zero energy: Hypothesis (H1): If zero is a singularity for H of the second kind with geometric multiplicity k ∈ N * , one assumes that there exists a basis {φ 1 ,

• • • , φ k } of Ker L 2 (Id + K 0 ) such that det( φ j , Jφ i ) 1≤i,j≤k = 0, (2.11)
where J : w → w is the complex conjugation.

Hypothesis (H2):

If zero is both an eigenvalue with geometric multiplicity k ∈ N * and a resonance of H, one assumes that:

(1) There exists 1

≤ i 0 ≤ k + 1 such that Ker(Id + K 0 )| Ei 0 = Ker L 2,-s (Id + K 0 )/Ker L 2 (Id + K 0 ) and Ker(Id + K 0 )| Ei ⊂ Ker L 2 (Id + K 0 ), ∀1 ≤ i ≤ k + 1, i = i 0 .
(2) There exists a basis {φ 1 , • • • , φ k } of Ker L 2 (Id + K 0 ) verifying the condition in (2.11).

Before stating the hypothesis on positive resonances we define for λ > 0 the symmetric bilinear form

B λ (•, •) on H -1,s × H -1,s by B λ (u, w) = R 3 ×R 3 e i √ λ|x-y| u(x)V (x)w(y)V (y) dx dy.
(2.12)

Hypothesis (H3): One assumes that H has a finite number of outgoing positive resonances, i.e.

σ + r (H) = {λ 1 , • • • , λ N }.
In addition, one supposes that for each λ j ∈ σ + r (H), there exist N j ∈ N * and a basis {ψ

(j) 1 , • • • , ψ (j) Nj } in L 2,-s of Ker (Id + K + (λ j )) such that det B λj (ψ (j) r , ψ (j) l )
1≤l,r≤Nj = 0.

(2.13)

In Section 3.3 (resp. Section 4) we find some numerical function d(z) such that (Id + K(z)) is invertible if and only if d(z) = 0 and if the condition (2.11) (resp., (2.13)) is satisfied, then -1 is an eigenvalue of K 0 on L 2 (resp. K + (λ 0 ) on L 2,-s ) with geometric multiplicity k if and only if 0 (resp. λ 0 ) is a zero of d(z) with multiplicity k. In [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF] the condition (2.11) was used in the case when -1 is a semi-simple eigenvalue of K 0 (i.e. geometric and algebraic multiplicities are equal) to expand the function d(z) in power of z 1/2 near 0. However, we use in the present work conditions (2.11) and (2.13) to compute in addition exactly the leading terms of resolvent expansions near 0 and positive resonances (see proofs of Theorem 2.3 and Theorem 2.5).

In particular, under conditions (2.13) and (2.1) for ρ > N + 1 with N ∈ N * , we can expand the function d(z) as follows

d(z) = ω N0 (z -λ 0 ) N0 + ω N0+1 (z -λ 0 ) N0+1 + • • • + O(|z -λ 0 | N0+N ) (2.14)
for z ∈ C + , |z -λ 0 | < δ, with some ω N0 = 0, where N 0 is given in (H3). Note that the expansion (2.14) can hold under the analyticity condition on the potential V . See [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF]Remark 6.1 ].

We mention that we find in [START_REF] Wang | Time-decay of semigroups generated by dissipative schrödinger operators[END_REF]Remark 5.4] an example of a resonance state ψ associated with an outgoing resonance λ 0 for a perturbation of -∆ by a compactly supported complex-valued potential V , where it can be checked that ψ satisfies

R 3 ×R 3 e i √ λ0|x-y| V (y)ψ(y)V (x)ψ(x) dxdy = 0.
However, in the general case it is not clear if the condition (2.13) can be satisfied. In section 4 we will study the resolvent expansion near λ 0 with the more general condition (2.14).

Main results.

As first result, we establish asymptotic expansions for R(z) near zero and positive resonances. For δ > 0 small, we denote

Ω δ := {z ∈ C \ R + : |z| < δ}. ( 2
R(z) = -2 j=-1 z j/2 R (1) j + R (1)
-2 (z), (2.16) where R

(1)

-1 : L 2,s → L 2,-s , u → i u, Jφ φ, with φ is a resonance state normalized by 1 2 √ π R 3 V (x)φ(x) dx = 1.
(2.17)

Furthermore, the remainder term R

-2 (z) is a C -2 operator-valued function of z from Ω δ to B(-1, s, 1, -s) and for 0 < λ < δ the limits lim

→0 + R (1) -2 (λ ± i ) := R (1) -2 (λ ± i0) (2.18)
exist in B(-1, s, 1, -s) and satisfy

d r dλ r R (1) -2 (λ ± i0) B(-1,s,1,-s) = o(|λ| -2 2 -r ), ∀λ ∈]0, δ[, r = 0, 1, • • • , -2. (2.19) If ρ > 2 and s > 1/2, we can obtain R(z) = z -1/2 R (1) -1 + o(|z| -1/2
). In Theorem 2.2, we have not used any implicit assumption on zero resonance. In particular, we do not know if zero resonance is algebraically simple.

Theorem 2.3. Suppose that zero is a singularity for H of the second kind and that (H1) holds. Let k ∈ N * be the geometric multiplicity of the eigenvalue zero. Assume ρ > 2 -3, ∈ N with ≥ 4. Let s > -3/2 and z ∈ Ω δ , δ > 0 small. We have

R(z) = -4 j=-2 z j/2 R (2) j + R (2) -4 (z), (2.20)
as operators in B(-1, s, 1, -s). Here R

(2)

-2 = -P (2) 0 , R (2)
-1 : L 2 → Ker(Id + K 0 ), P (2)
0 = k j=1 •, JZ (2) j Z
(2) j with Z

(2)

i , JZ (2)
j = δ ij , ∀1 ≤ i, j ≤ k,
where

{Z (2) 1 , • • • , Z (2)
k } is a basis of Ker(Id + K 0 ). Moreover, the remainder term R

-4 (z) is a C -4 operator-valued function of z from Ω δ to B(-1, s, 1, -s) and for 0 < λ < δ the limits R

(2) -4 (λ ± i0) (see (2.18)) exist in B(-1, s, 1, -s) and satisfy

d r dλ r R (2) -4 (λ ± i0) B(-1,s,1,-s) = o(|λ| 2 -2-r ), ∀λ ∈]0, δ[, r = 0, 1, • • • , -4.
(2.21)

If ρ > 4 and s > 3/2 we can obtain R(z) = z -1 R (2)
-2 + z -1/2 R (2)
-1 + o(|z| -1/2 ) and if ρ > 3 and s > 1/2, R(z) = z -1 R (2)
-2 + o(|z| -1 ). Note that although there is no natural spectral projection associated with the embedded eigenvalue 0 of H, our result shows that the leading term is still given by spectral projection P

(2) 0 . Theorem 2.4. Suppose that zero is a singularity for H of the third kind and (H2) holds. Assume ρ > 2 -3, ∈ N with ≥ 4. Then for s > -3/2, and z ∈ Ω δ , we have

R(z) = -4 j=-2 z j/2 R (3) j + R (3) -4 (z), (2.22)
in B(-1, s, 1, -s). Here R

-2 = -P

(3) 0 , R (3)
-1 = i •, Jψ ψ + S (3)
-1 , P

0 , S

-1 : L 2 → Ker L 2 (Id + K 0 ), P (3)
= k j=2 •, JZ (3) 0
(3) j with Z

i , JZ

j = δ ij , ∀2 ≤ i, j ≤ k, (3)
where

{Z (3) 2 , • • • , Z (3)
k } is a basis of Ker L 2 (Id + K 0 ) and ψ is a resonance state satisfying (2.17). In addition, the remainder term R The following theorem gives the resolvent expansion near an outgoing positive resonance λ 0 for z in a set Ω + δ given by Ω + δ := {z ∈ C + : |z -λ 0 | < δ}.

(2.23)

Theorem 2.5. Suppose that (H3) holds. Let λ 0 ∈ σ + r (H). Assume ρ > 2 -1, ∈ N with ≥ 2. Then, for s > -1/2 and z ∈ Ω + δ , we have

R(z) = P(λ 0 ) z -λ 0 + -2 j=0 (z -λ 0 ) j R j (λ 0 ) + R -2 (z -λ 0 ), (2.24)
in B(-1, s, 1, -s), where

P(λ 0 ) = N0 j=1 •, Jψ (λ0) j ψ (λ0) j with 1 i8π √ λ 0 B λ0 (ψ (λ0) i , ψ (λ0) j ) = δ ij , such that {ψ (λ0) 1 , • • • , ψ (λ0)
N0 } is a basis of Ker(Id + R + 0 (λ 0 )V ), and B λ0 is the bilinear form defined in (2.12). The remainder term R -2 (z -λ 0 ) is analytic in Ω + δ and for λ > 0 with |λ -λ 0 | < δ the limit lim

→0 + R -2 (λ -λ 0 + i ) = R -2 (λ -λ 0 + i0)
exists in the norm of B(-1, s, 1, -s) and satisfies

d r dλ r R -2 (λ -λ 0 + i0) B(-1,s,1,-s) = o(|λ -λ 0 | -2-r ), r = 0, 1, • • • , -2.
(2.25)

If ρ > 2 and s > 1/2, we can obtain R(z) = R -1 (λ 0 ) z -λ 0 + o(|z -λ 0 | -1 ).
Using the preceding results, we show that under the assumption ρ > 2, H has at most a finite number of discrete eigenvalues located in the closed upper half-plane. However, if zero is an eigenvalue of H we need the stronger assumption ρ > 3. See Proposition 5.1.

We obtain asymptotic expansions in time of the strongly continuous Schrödinger semigroup (e -itH ) t≥0 , as t → +∞, if zero is a resonance or an eigenvalue of H taking into account the presence of outgoing positive resonances. Our main result is the following: Theorem 2.6. Assume that (H3) holds.

(a) Suppose that zero is a singularity for H of the first kind. If ρ > 5 then for s > 5/2 we have

e -itH - p j=1 e -itH Π zj + N j=1 e -itλi R -1 (λ j ) = (2.26) (iπ) -1/2 •, φ φ t -1 2 + o(t -1/2 ), t → +∞,
in B(0, s, 0 -s), where φ (resp. R -1 (λ j )) is given by Theorem 2.2 (resp. Theorem 2.5).

(b) Assume ρ > 7. Suppose that zero is a singularity for H of the second kind and that (H1) holds. Then, for s > 7/2 the expansion at the right hand side of (2.26) has the following form

P (2) 0 -i(iπ) -1/2 R (2) -1 t -1 2 -(4iπ) -1/2 R (2) 1 t -3 2 + o(t -3/2 ),
where P

(2) 0 and R

(2) s for s = -1, 1 are given by Theorem 2.3. If ρ > 5 and s > 5/2, then (2.26) holds with the right hand side replaced by

P (2) 0 -i(iπ) -1/2 R (2) -1 t -1 2 + o(t -1/2 ).
In the above expansions Π zj denote the Riesz projections associated with the discrete eigenvalues z j located in the closed upper half-plane.

Note that (a) holds if zero is both an eigenvalue and a resonance of H. To obtain the above expansions we find some curve Γ ν (η), for some η, ν > 0 small, which does not intersect the real axis at zero or at points in σ + r (H), such that above this curve, H has a finite number of eigenvalues (see Figure 1. in Section 5). Then the expansions are deduced by representing e -itH as a sum of some residue terms and a Dunford integral of R(z) on Γ ν (η).

Remark 2.7. With regard to the case zero is a regular point for H, i.e it is not an eigenvalue nor a resonance of H, we can obtain the same result obtained in [13, Theorem 6.1] for H = -∆ + V with real V . If ρ > 3 and s > 3/2, then for z ∈ Ω δ with δ > 0 small, we have

R(z) = R (0) 0 + z 1/2 R (0) 1 + o(|z| 1/2 ), (2.27) in B(-1, s, 1, -s), where R (0) 0 = (Id + G 0 V ) -1 , R (1) 0 = i(Id + G 0 V ) -1 G 1 (I -V (Id + G 0 V ) -1 G 0 ).
Here, G 0 , G 1 are given in (2.6) and Ω δ is defined in (2.15). The proof of [13, Theorem 6.1] can be done here because it is based on the expansion of (I + R 0 (z)V ) -1 in a Neumann series which works also for non-real V . However, in presence of positive resonances of H a stronger assumption on ρ is needed to obtain the expansion in time of e -itH with reminder o(t -3/2 ) even if zero is a regular point for H. We obtain the following result: Suppose that zero is a regular point for H and (H3) holds. If ρ > 7 and s > 7/2, we have

e -itH - p j=1 e -itH Π zj + N j=1 e -itλi R -1 (λ j ) = -(4iπ) -1/2 R (0) 1 t -3 2 + o(t -3/2 ), t → +∞,
in B(0, s, 0, -s). (See (2.26)).

Resolvent expansions at low energies

Consider H 0 = -∆ and the perturbed non-selfadjoint operator H = -∆ + V . In the following, we always assume that V satisfies (2.1), where a stronger assumption on ρ is needed for a high-order asymptotic expansion in z 1/2 of the resolvent. In this section we will use some tools developed in [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF]Section 5.4].

Riesz projection. Set E = Ran Π 1 , where Π 1 is the Riesz projection associated with the eigenvalue -1 of K 0 on L 2,-s for 1/2 < s < ρ -1/2 (see Section 2.1). Let J : f → f be the operation of complex conjugation. Then we have V * = JV J, H * = JHJ and the following relations

JV Π 1 = Π * 1 JV, Π 1 JG 0 = JG 0 Π * 1 , (3.1)
JV (Id + K 0 ) = (Id + K * 0 )JV, (3.2)
(Id + K 0 )JG 0 = JG 0 (Id + K * 0 ). (3.3) It follows that JV (resp. JG 0 ) : Ran Π 1 (resp. Ran Π * 1 ) → Ran Π * 1 (resp. Ran Π 1 ) is injective. We deduce that the bilinear form Θ(•, •) defined on E by Θ(u, v) = R 3 V (x)u(x)v(x)dx = u, v * (3.4)
is non-degenerate on E × E, where we denoted

u * = JV u. (3.5)
See the proof of Lemma 5.13 in [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF].

Using the above non-degenerate bilinear form, we obtain the following decomposition lemma:

Lemma 3.1. Assume that -1 is an eigenvalue of K 0 of geometric multiplicity k ≥ 1 and algebraic multiplicity m. Then there exist k invariant subspaces of K 0 denoted by E 1 , • • • , E k such that (1) E = E 1 ⊕ • • • ⊕ E k , where ∀i = j: E i ⊥ Θ E j . (2) ∀1 ≤ i ≤ k, there exists a basis U i := {u (i) 1 , • • • , u (i) mi } of E i such that u (i) r := (Id + K 0 ) mi-r u (i) mi , ∀1 ≤ r ≤ m i , (3.6)
u (i) mi ∈ Ker(Id + K 0 ) mi and Θ(u (i) 1 , u (i) mi ) = c i = 0.
(3) ∀1 ≤ j ≤ k, there exists a basis W j := {w

(j) 1 , • • • , w (j) mj } of E j such that w (j) r ∈ Ker(Id + K 0 ) mj +1-r , and Θ(u (i) , w (j) r ) = δ ij δ r . (3.7) (4) dim ker(Id + K 0 )| Ej = 1, ∀j = 1, • • • , k.
Moreover, the matrix of

Π 1 (Id + K 0 )Π 1 in the basis U := k i=1 U i of E is a m × m block diagonal matrix of the following form J m = diag(J m1 , J m2 , • • • , J m k ), (3.8
) where

J mj =          0 1 0 • • • 0 0 0 1 . . . . . . 0 0 . . . . . . 0 . . . . . . . . . . . . 1 0 0 • • • 0 0          mj ×mj
is a Jordan block. We have also denoted

m j = dim E j for j = 1, • • • , k, such that m = m 1 + • • • + m k .
The following statement is an immediate consequence of the previous lemma.

Corollary 3.2. The Riesz projection Π 1 has the following representation:

Π 1 = k j=1 mj r=1 •, (w (j) r ) * u (j) r .
See [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF]Corollary 5.16] for the proof of the corollary in the case k = 1. We now prove Lemma 3.1.

Proof. We proceed by induction on k = dim Ker(Id + K 0 ). Initially, for k = 1, we refer to [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF]Section 5.4] for the case of geometrically simple eigenvalue -1 of K 0 . For k = 2, we shall show the parts ( 1) and ( 2) as follows. First, note that K 0 is Θ-symmetric which implies that, if F is a stable subspace of

Id + K 0 its Θ-orthogonal F ⊥Θ is stable and if in addition Θ| F ×F is non degenerate then E = F ⊕ F ⊥Θ and Θ| F ⊥ Θ ×F ⊥ Θ is non degenerate. Second, If E(u µ ) = Span(u k = (Id + K 0 ) µ-k u µ , 1 ≤ k ≤ µ), with u µ ∈ Ker(Id+K 0 ) µ and Θ(u µ , u 1 ) = c = 0, then u µ / ∈Ker(Id+K 0 ) µ-1 , {u 1 , • • • , u µ } is a basis of E(u µ ), Θ| E(uµ)×E(uµ) is non degenerate, E = E(u µ ) ⊕ E(u µ ) ⊥Θ and Θ| E(uµ) ⊥ Θ ×E(uµ) ⊥ Θ is non degenerate. In addition, {u 1 , • • • , u µ } has Θ-dual basis {w 1 , • • • , w µ } with w µ = c -1 u 1 which
can be constructed as in [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF]Lemma 5.15] since Θ| E(uµ)×E(uµ) is non degenerate. Therefore it suffices to find one u m1 such that the second statement holds and to find in addition one u m2 such that E(u m1

) ⊥Θ = E(u m2 ). (i) Let m 1 , 1 ≤ m 1 ≤ m, be the smallest integer such that E =Ker(Id + K 0 ) m1 . If m 1 = 1, E =Ker(Id + K 0 )
. By non degeneracy of Θ, we can easily find u 1 , v 1 ∈ E such that Θ(u 1 , u 1 ) = 0, Θ(v 1 , v 1 ) = 0 and Θ(u 1 , v 1 ) = 0. This solves the problem for

m 1 = 1. If m 1 > 1, set Q = (Id + K 0 ) m1-1 which is Θ-symmetric. The bilinear form B(u, v) = Θ(Qu, v)
is symmetric on E and not identically zero. Otherwise Θ(w, v) = 0 for all v ∈ E and w ∈ Ran Q and because Θ is non degenerate it follows that E = (Ran Q) ⊥ =Ker Q ( it can be seen that Ker Q ⊂ (Ran Q) ⊥ and the equality comes from the non degenracy of Θ). This contradicts the definition of µ. By polarization identity a symmetric bilinear form B on E is the null bilinear form if and only if B(u, u) = 0 for all u ∈ E. Hence there exists

u m1 ∈ E such that Θ((Id + K 0 ) m1-1 u m1 , u m1 ) = 0. (ii) Consider the restrictions of (Id + K 0 ) and Θ to E(u m1 ) ⊥Θ . Let m 2 = m -m 1 . Since dim Ker((Id+K 0 )| E(um 1 ) ⊥ Θ ) = 1, then m 2 is the smallest integer such that E(u m1 ) ⊥Θ =Ker((Id+ K 0 )| E(um 1 ) ⊥ Θ ) m2
, so following (i) one finds u m2 .

Assume now that (1) and ( 2)

are true for k = -1, ∈ N, ≥ 2. For k = , E = E(u m1 )⊕E(u m1 ) ⊥Θ with dim Ker(Id + K 0 )| E(um 1 ) ⊥ Θ = -1 by (i).
Applying the inductive hypothesis on E(u m1 ) ⊥Θ we prove (1) and ( 2) for k = ∈ N * .

Finally, for the statement about the basis W once the basis [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF]Lemma 5.15]. We have established (1)-( 3) and (4) follows directly. Remark 3.3. By construction of E j in the previous lemma, we have

U = k j=1 U j is constructed, W j , j = 1, • • • , k, are given by
Θ(u (j) 1 , u (j) mj ) = c j = 0, ∀1 ≤ j ≤ k,
and Θ| Ej ×Ej is non degenerate. Then, applying Lemma 5.15 in [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF] to construct the basis {w

(j) 1 , • • • , w (j) mj } of {u (j) 1 , • • • , u (j)
mj }, we take w

(j) mj = c -1 j u (j) 1 , ∀1 ≤ j ≤ k.
In order to establish the asymptotic expansion of the resolvent R(z), we use the resolvent equation given in (2.4). We must establish the asymptotic expansion of (I + R 0 (z)V ) -1 . Our approach is to use the so called Grushin's method. To avoid repetition we will introduce a Grushin problem in the more general case when dim Ker L 2,-s (Id

+ K 0 ) = k, k ∈ N * . Set M(z) = Id + K(z). Given the decomposition of E in Lemma 3.1, we can identify E j with C mj and C m with C m1 ⊕ • • • ⊕ C m k to construct the following Grushin problem.
3.1. Grushin problem for the inverse of (I + R 0 (z)V ). We consider

P(z) := M(z) S T 0 , (3.9)
S :

k ⊕ j=1 C mj -→ Ran Π 1 ; ζ = k ⊕ j=1 (ζ (j) 1 , • • • , ζ (j) mj ) → Sζ := k j=1 mj i=1 ζ (j) i u (j) i , T : Ran Π 1 -→ k ⊕ j=1 C mj ; v → T v := k ⊕ j=1 ( v, (w (j) 1 ) * , • • • , v, (w (j) mj ) * ).
The operators S and T verify T S = I m and ST = Π 1 (see Corollary 3.2), where S and T are chosen so that the problem P(z) is invertible. Since Id + K 0 is injective on Ran Π 1 , where Π 1 = Id -Π 1 , by the alternative Fredholm theorem Π 1 (Id + K 0 )Π 1 is invertible on Ran Π 1 . Then, using an argument of perturbation for δ > 0 small enough Π 1 M(z)Π 1 is also invertible on Ran Π 1 for all z in Ω δ , with inverse

E(z) = (Π 1 M(z)Π 1 ) -1 Π 1 .
In view of (2.6), for

ρ > 2 + 1, + 1/2 < s < ρ --1/2, = 1, 2 • • • and z ∈ Ω δ , δ > 0 small, the expansion of E(z) in B(1, -s, 1, -s
) can be written as follows:

E(z) = j=0 z j/2 E j + E (z), (3.10)
where

E 0 = (Π 1 (Id + K 0 )Π 1 ) -1 Π 1 , E 1 = iE 0 G 1 V Π 1 E 0 and other terms E j , j = 2, • • • , can be computed directly.
Moreover, the remainder term E (z) satisfies

d r dz r E (z) B(H 1,-s ) = o(|z| 2 -r ), ∀z ∈ Ω δ , r = 0, 1, • • • , . (3.11)
In addition, for 0 < λ < δ, it follows from (2.5) that the limits

lim →0 E (λ ± i ) = E (λ ± i0) (3.12)
exist as operators in B(-1, s, 1, -s). By taking → 0 + in (3.11) with z = λ ± i we show that the above limits satisfy

d r dλ r E (λ ± i0) B(H 1,-s ) = o(|λ| 2 -r ), ∀ 0 < λ < δ, r = 0, 1, • • • , . (3.13)
Then the Grushin problem is invertible with inverse

M(z) S T 0 -1 = E(z) E + (z) E -(z) E -+ (z) : H 1,-s × C m -→ H 1,-s × C m , (3.14)
where

E + (z) = S -E(z)M(z)S E -(z) = T -T M(z)E(z) E -+ (z) = -T M(z)S + T M(z)E(z)M(z)S (3.15) Therefore, M(z) is invertible if and only if E -+ (z) is invertible, with M(z) -1 = E(z) -E + (z)E -+ (z) -1 E -(z) on H 1,-s . (3.16)
3.2. Zero singularity of the first kind. In this section, zero will be only a resonance and not an eigenvalue of H, in which case -1 is a geometrically simple eigenvalue of the compact operator K 0 on L 2,-s , ∀s > 1/2. The same construction made in Lemma 3.1 for a single subspace E i can be done for E = Ran Π 1 at the present case. By this lemma we find

U := {u 1 , • • • , u m } ⊂ L 2,-s a basis of E such that Θ(u 1 , u m ) = c = 0 and W = {w 1 , • • • , w m } its Θ-dual basis. In particular, u 1 ∈ Ker L 2,-s (Id + K 0 ) is a resonance state.
Let δ > 0 small, we denote

Ω δ = {z ∈ C \ R + , |z| < δ}. (3.17)
To calculate the singularity of (I + K(z)) -1 due to zero resonance, we must establish an asymptotic expansion of E -+ (z) -1 by using the above Grushin problem with dim Ker L 2 ,-s

(Id+K 0 ) = k = 1.
First, let ρ > + 1 with ∈ N * . For z ∈ Ω δ , δ > 0, introducing the expansions in z 1/2 of R 0 (z) and E(z) at order given in (2.6) and (3.10) respectively, we obtain

E -+ (z) = N + √ zA + j=2 z j/2 E -+,j + E -+, (z), (3.18)
where by using Lemma 3.1 (2) and w m = c -1 u 1 we have

N := -(Id + G 0 V )u r , JV w 1≤ ,r≤m = -(δ i+1j ) 1≤i,j≤m , A := -i G 1 V u r , JV w 1≤ ,r≤m , E -+,2 = (G 2 V -G 1 V E 0 G 1 V )u r , JV w 1≤ ,r≤m + (iE 1 G 1 V -E 0 G 2 V )u r , JV (Id + G 0 V )w 1≤ ,r≤m
.

In particular, the (m, 1)-th entry of the matrix A is non zero and is given by

a m1 = (ic) -1 G 1 V u 1 , JV u 1 = (i4π c) -1 u 1 , JV 1 2 . (3.19)
Moreover, the remainder E -+, (z) is a C matrix-valued function of z in Ω δ and for 0 < λ < δ the limits

lim →0 + E -+, (λ ± i ) = E -+, (λ ± i0) (3.20)
exist and satisfy

d r dλ r E -+, (λ ± i0) = o(|λ| 2 -r ), r = 0, 1, • • • , . (3.21)
Also, E -+,j , j = 3, • • • , can be obtained explicitly but they become even more complicated. The condition ρ > + 1 is necessary to obtain the expansion of E -+ (z) up to order z /2 . Let us check this for some terms appearing in the computation of E -+, . We must obtain the terms

G V u r , JV w l , (Id + G 0 V )E (Id + G 0 V )u r , JV w l , (Id + G 0 V )E 0 G V u r , JV w l , and G V E 0 (Id + G 0 V )u r , JV w l . Consider G V u r . We have u r ∈ H 1,-1 2 -, 0 < < 1/2, so V u r ∈ H -1,ρ-1 2 -and G V u r ∈ H 1,--1 2 - because ρ -1/2 > + 1/2, where G is given in (2.8)). Thus G V u r , JV w l makes sense since V w l ∈ H -1,ρ-1 2 + ⊂ H -1, + 1 2 + . Consider now (Id + G 0 V )E (Id + G 0 V )u r . (Id + G 0 V )u r ∈ H 1,-1 2 -, to apply E we must have ρ -1/2 > + 1/2, and it maps to H 1,--1 2 -(see (3.10)). Finally, since (Id + G 0 V ) ∈ B(1, -s, 1, -s) for 1/2 < s < ρ -1/2, then (Id + G 0 V )E (Id + G 0 V )u r ∈ H 1,--1 2 - because + 1 2 < ρ -1/2.
For the properties of the remainder term E -+, , see the argument used for E (z) in (3.10).

Set E -+, (z) = N + √ zA. Then for ρ > 2 and z ∈ Ω δ , δ > 0, E -+ (z) = E -+,1 (z) + o(|z| 1/2 ). This yields to det E -+ (z) = det E -+,1 (z) + o( √ z) = √ z a m1 + o(|z| 1/2 ) (3.22)
where a m1 is the non-zero constant given in (3.19).

Proof of Theorem 2.2. It follows from the previous paragraph that E -+,1 (z) is invertible for z ∈ Ω δ , δ > 0 small, and we can easily check that

E -+,1 (z) -1 = t ComE -+,1 (z) detE -+,1 (z) = 1 √ z A + O(1), (3.23)
where

A = 1 a m1      0 • • • 0 1 0 • • • 0 0 . . . . . . . . . 0 • • • 0 0      m×m .
Then, if ρ > + 1 and z ∈ Ω δ , δ > 0 small, E -+ (z) -1 exists, with

E -+ (z) -1 = (I + E -+,1 (z) -1 j=2 z j/2 E -+,j ) -1 E -+,1 (z) -1 = 1 √ z A + -2 j=0 z j/2 F (2) j + E (-1) -+, -2 (z),
where A is the above matrix, F

j , j = 1, • • • , -2 (2)
, can be computed explicitly. Moreover, for 0 < λ < δ the limits of the remainder term E (-1)

-+, -2 (λ ± i0) exist and satisfy d r dλ r E (-1) -+, -2 (λ ± i0) = o(|λ| l 2 -1-r ), λ ∈]0, δ[, r = 0, 1, • • • , -2. ( 3.24)
Next, using the formula (3.16) we can verify that if

-1/2 < s < ρ -+ 1/2, ∈ N with ≥ 2 (Id + R 0 (z)V ) -1 = -2 j=-1 z j/2 B (2) j + B (2)
-2 (z), (3.25) in B(1, -s, 1, -s) where

B ( 2
)
-1 = -S AT = - 1 ca m1 •, JV u 1 u 1
and the remainder B

(2)

-2 (z) is a C -function from Ω δ to B(1, -s, 1, -s) such that for 0 < λ < δ the limits B (2) -2 (λ ± i0) exist and satisfy d r dλ r B (2) -2 (λ ± i0) B(1,-s,1,-s) = o(|λ| 2 -1-r ), λ ∈]0, δ[, r = 0, 1, • • • , -2. ( 3.26)
To make this computation we introduce (3.23) in (3.16) together with the expansion (3.10) of E(z) up to order -1. We obtain the expansion of

E + (z)E -+ (z) -1 E -(z) up to order -2 with remainder o(|z| - 2
2 ) which requires the assumption -1/2 < s < ρ -+ 1/2. In addition, estimates (3.26) can be checked from (3.13) and (3.24).

Consequently, for ρ > 2 -1 and s > -1/2, ∈ N with ≥ 2, using the equation (2.4), (3.25) and (2.6) the expansion of R(z) in B(-1, s, 1, -s), can be written as follows:

R(z) = -2 j=-1 z j/2 R (2) j + R (2) l-2 (z), where R (2) -1 = - 1 ca m1 •, JG 0 V u 1 u 1 = 1 c a m1 •, Ju 1 u 1 = i 4π u 1 , JV 1 2 •, Ju 1 u 1 . Finally, let φ = 2 √ π u 1 , JV 1 u 1 .
Then φ is a resonance state of H satisfying (2.17) and R

-1 = i •, φ φ. Moreover, the estimate (2.19) follows from (3.26).

3.3.

Zero singularity of the second kind. In this case we assume that -1 is an eigenvalue of the operator K 0 on L 2,-s , 1/2 < s < ρ -1/2, with geometric multiplicity k ≥ 1. Indeed the case k = 1 could be treated using the similar method used in [START_REF] Wang | Gevrey estimates of the resolvent and sub-exponential time-decay for the heat and Schrödinger semigroups[END_REF] to study the situation of geometrically simple zero eigenvalue for a compactly supported perturbation of the non-selfadjoint Schrödinger operator H 0 = -∆+V 0 (x) with a slowly decreasing potential V 0 . In the latter case, the matrix of Π 1 (Id+K 0 )Π 1 on Ran Π 1 is consisting of one Jordan block. Therefore, the usual tools can be used to compute the singularity of the resolvent at threshold zero. This work is concerned with the more interesting case

k ≥ 2. Assume from now that k ≥ 2. Let U = ∪ k j=1 {u (j) 1 , • • • , u (j) mj } and W = ∪ k j=1 {w (j) 1 , • • • , w (j)
mj } be the basis constructed in Lemma 3.1. In particular {u

(1) 1 , • • • , u (k) 1 } is a basis of Ker L 2 (Id + K 0 ).
In order to prove Theorem 2.3, we use the Grushin's method introduced in Section 3.1 for arbitrary Jordan structure. We begin by computing the asymptotic expansion of the matrix E -+ (z). To study this matrix we need to decompose it conformally with the block decomposition of the matrix J m in (3.8). More precisely, using formula (3.15) and basis U j and W i , we have

E -+ (z) = E (ij) -+ (z) 1≤i,j≤k with E (ij) -+ (z) = ( E -+ (z)u (j) r , JV w (i) l ) 1≤l≤mi,1≤r≤mj , where E (ij)
-+ (z) denotes the m i × m j block entry of E -+ (z) located in the same row as J mi and in the same column as J mj .

Thus, if ρ > + 1, ∈ N with ≥ 3, using (2.6), (3.10) and (3.15) we obtain the following expansion of E -+ (z) for z ∈ Ω δ , δ > 0 small:

E -+ (z) = E -+,2 (z) + j=3 z j/2 E -+,j + E -+, (z), (3.27)
where ij) , such that for all 1 ≤ ≤ m i and 1 ≤ r ≤ m j we have

E (ij) -+,2 (z) = N (ij) + z 1/2 A (ij) + z B (
N (ij) r = -(Id + G 0 V )u (j) r , JV w (i) , A (ij) r = -i G 1 V u (j) r , JV w (i) + i E 0 G 1 V u (j) r , JV (Id + G 0 V )w (i) , B (ij) r = (G 2 V -G 1 V E 0 G 1 V )u (j) r , JV w (i) + (iE 1 G 1 V -E 0 G 2 V )u (j) r , JV (Id + G 0 V )w (i) also E (ij)
-+,n , n = 3, • • • , , can be computed explicitly. Moreover, the remainder term E -+, (z) is a C matrix-valued function of z in Ω δ and for 0 < λ < δ the limits lim

→0 + E -+, (λ ± i ) = E -+, (λ ± i0) (3.28)
exist and satisfy

d r dλ r E -+, (λ ± i0) = o(|λ| 2 -r ), r = 0, 1, • • • , . (3.29)
We can further simplify the previous expression of the matrix E -+,2 (z) as follows: ∀1 ≤ j ≤ k, u (2.10). This implies that A

(j) 1 ∈ ker(Id + K 0 ) implies that N (ij) 1 = 0, ∀1 ≤ ≤ m i , while for all 2 ≤ r ≤ m j , N (ij) r = -u (j) r-1 , JV w (i) = -δ r-1 δ ij by (3.7). Moreover, since w (i) mi = c i u (i) 1 ∈ L 2 for some c i = 0 (see Remark 3.3) then G 1 V u (j) 1 = 0 = G 1 V w (i) mi , ∀1 ≤ i, j ≤ k, by
(ij) 1 = 0 = A (ij) mir , ∀1 ≤ ≤ m i , 1 ≤ r ≤ m j .
Summing up, we obtain E -+,2 (z) = N + z 1/2 A + z B, with

N (ij) =       0 -δ ij • • • 0 0 0 . . . . . . . . . . . . . . . -δ ij 0 0 • • • 0       mj ×mj , (3.30)
A (ij) =      0 0 A (ij) . . . 0 0 • • • 0      mi×mj , B (ij) =      * * • • • * . . . . . . . . . * * • • • * β ij * • • • *      mi×mj , ∀1 ≤ i, j ≤ k, (3.31)
where

β ij = -lim z∈Ω δ ,z→0 1 z (Id + R 0 (z)V )u (j) 1 , JV w (i) mi = -lim z∈Ω δ ,z→0 1 z { (Id + G 0 V )u (j) 1 , JV w (i) mi + z G 0 V u (j) 1 , JR 0 (z)V w (i) mi } = -c -1 i u (j) 1 , Ju (i) 1 , ∀1 ≤ i, j ≤ k. (3.32)
See the proof of (3.18) for more details.

Unfortunately, we have found in (3.27) a block matrix E -+ (z) of arbitrary block structure, where the usual methods of algebra are no longer practical to calculate its determinant and to explicitly develop its inverse matrix. To treat this matrix, we propose a method based on that of Lidskii developed in his original paper [START_REF] Lidskii | Perturbation theory of non-conjugate operators[END_REF], and used latter in [START_REF] Moro | On the lidskii-vishik-lyusternik perturbation theory for eigenvalues of matrices with arbitrary jordan structure[END_REF] for the problem of eigenvalues of matrices with arbitrary Jordan structure.

Remark 3.4. Set φ k = β ij 1≤i,j≤k , L k = u (j) 1 , Ju (i) 1 1≤i,j≤k , (3.33)
where β ij are the coefficients in (3.31). Then, it is seen from (3.32) and Remark 3.3 that

φ k = -C k L k with C k = diag(c -1 1 , • • • , c -1 k ).
(3.34) Lemma 3.5. Assume that zero is a singularity of the second kind of H and that (H1) holds. If ρ > 3, we have det

E -+ (z) = σz k + O(|z| k+ ), ∀z ∈ Ω δ ,
for some 0 < < 1/2, where σ = σ × det( u

1 ) 1≤i,j≤k with σ ∈ C * and k = dim Ker(Id + K 0 ). Proof. We begin by writing the expansion of the matrix E -+ (z) in (3.27) as follows:

E -+ (η) = E -+,2 (η) + O(|η| 2(1+ ) ), (3.35)
for some 0 < < 1/2. Let Z(η) = detE -+,2 (η). Then, we reduce the computation to that of Z(η) close to η = 0. To do it we introduce the following diagonal matrix L(η) partitioned conformally with the block structure of the matrix E -+ (η):

L(η) = diag L 1 (η), • • • , L k (η) , L i (η) = diag(1, • • • , 1, η -2 ), 1 ≤ i ≤ k, (3.36)
where η ∈ {z ∈ C + : |z| < δ}. We now define

E -+,2 (η) = L(η)E -+,2 (η), Z(η) = det E -+,2 (η). (3.37)
Then, by regularity of the matrix L(η) for η = 0, we see that Z(η) = 0 if and only if Z(η) = 0. Also, we can show that Z(η) is polynomial in η. Indeed, we write

E -+,2 (η) = L(η)(N + η A + η 2 B) := N (η) + A(η) + B(η),
where, by (3.30), (3.31) and (3.36) we see that

(i) N (η) = L(η)N = N. (ii) A(η) = ηL(η)A with A (ij) (η) =      0 . . . η A (ij) 0 • • • 0      mi×mj . (iii) B(η) = η 2 L(η)B with B (ij) (η) =       η 2 0 • • • 0 0 . . . . . . . . . . . . . . . η 2 0 0 • • • 0 1       mi×mi      * * • • • * . . . . . . . . . * * • • • * β ij * • • • *      mi×mj .
This shows that there is no negative powers of η in E -+,2 (η). We will then examine Z(0). It follows from (i) (resp. (ii)) that N (0) = N (resp. A(0) = 0). Moreover,

B (ij) (0) =      0 0 • • • 0 . . . . . . . . . 0 0 • • • 0 β ij * • • • *      mi×mj , ∀1 ≤ i, j ≤ k. (3.38)
Thus,

E (ij) -+,2 (0) =         0 -δ ij 0 • • • 0 . . . . . . . . . . . . . . . 0 . . . . . . 0 0 0 -δ ij β ij * • • • * 0         mi×mj , ∀1 ≤ i, j ≤ k. (3.39)
We can now calculate Z(0) which is the determinant of the above matrix E -+,2 (0). By expanding the determinant along the rows of E -+,2 (0) that are containing only -1, we obtain the following:

Z(0) = det (β ij ) 1≤i,j≤k , (3.40)
(see proof of Theorem 2.1 in [START_REF] Moro | On the lidskii-vishik-lyusternik perturbation theory for eigenvalues of matrices with arbitrary jordan structure[END_REF] for a specific example with 12×12 matrix that illustrates the strategy). Hence, there exist , η > 0 small such that for η ∈ {z ∈ C + : |z| < δ}

Z(η) = Z(0) + O(|η| 2 ) = detΦ k + O(|η| 2 ) (3.41)
where Φ k = (β ij ) 1≤i,j≤k is defined in (3.33). Then, it follows from (3.37) and (3.41) with detL(η) = η -2k

Z(η) = (detL(η)) -1 Z(η) = η 2k detΦ k + O(|η| 2(k+ ) ). ( 3
detE -+ (η) = η 2k detΦ k + O(|η| 2(k+ ) ),
where det(Φ k ) = σ × det( u j 1 , Ju i 1 ) 1≤i,j≤k with some σ = 0 by Remark 3.4. Now, we are able to prove Theorem 2.3.

Proof of Theorem 2.3. Firstly, it follows from Lemma 3.5 that E -+ (z) -1 exists under the hypothesis (H1). Then, we will show that for ρ > + 1, = 4, 5, . . . and z ∈ Ω δ , δ > 0 small, the expansion of E -+ (z) -1 has the following form :

E -+ (z) -1 = -4 j=-2 z j/2 F (1) j + E (-1)
-+, -4 (z), (3.43) where F

-2 is a matrix of rank k, whose blocks are of the form (F

-2

) (ij) = 1 detΦ k      0 • • • 0 γ ij 0 • • • 0 0 . . . . . . . . . 0 • • • 0 0      mi×mj , ∀1 ≤ i, j ≤ k, (3.44)
for some γ ij that will be determined during this proof and the matrix F ( )

-1 = -F (1)
-2 E -+,3 F

-2 has rank at most k. Moreover, for 0 < λ < δ, the limits

lim →0 + E (-1) -+, -4 (λ ± i ) = E (-1) -+, -4 (λ ± i0) (3.45)
exist and satisfy

d r dλ r E (-1) -+, -4 (λ ± i0) = o |λ| 2 -2-r , r = 0, 1, • • • , -4. (3.46)
In order to prove (3.43) we consider the same notations that we have just used in the previous proof. We see that for Im η > 0 and |η| < δ with δ > 0 small, E -+,2 (η) can be developed in powers of η as follows:

E -+,2 (η) = E -+,2 (0) + ηA + η 2 B 1 + o(|η| 2 ), (3.47)
where B 1 = B -B(0) and the matrices A, B and B(0) are given in (3.31) and (3.38).

In addition, E -+,2 (0) -1 exists by (3.40) under the condition (2.11) with

E -+,2 (0) -1 = t Com E -+,2 (0)
detΦ k = F (1)
-2 + E (1) , (3.48)
where F

-2 is the above matrix and E is a block matrix, with block entries of the form

(E (1) ) (ij) = 1 detΦ k          * * • • • * 0 α (ij) 2 0 • • • 0 0 0 . . . . . . . . . . . . . . . . . . 0 • • • 0 α (ij) ni 0          mi×mj ∀1 ≤ i, j ≤ k, (3.49)
where α

(ij) r
∈ C are minors of order m -1 of the matrix E -+,2 (0). In particular α

(ij) r = 0 if i = j.
Here, we have applied the same process used in the previous proof to calculate the minors of order m -1 of the matrix E -+,2 (0). For the rest of this proof we need to give more precision on the coefficients γ ij of the above matrix

F (1) -2 . Let |[M ] j i | :=det [M ] j
i denote the minors of a matrix M , where [M ] j i are the resulting matrices when the i-th row and the j-th column of M are deleted. Then

γ 1j =(-1) 1+m1+•••+mj |[ E -+,2 (0)] 1 m1+•••+mj | and (3.50) γ ij =(-1) µij |[ E -+,2 (0)] 1+m1+•••+mi-1 m1+•••+mj |, µ ij = 1 + i-1 l=1 m l + j r=1 m r (3.51)
for 2 ≤ i ≤ k and 1 ≤ j ≤ k. Then, for η ∈ {η ∈ C + ; |η| < δ} with δ small enough E -+,2 (η) in (3.47) is invertible, as well as E -+,2 (η) with E -+,2 (η) -1 = E -+,2 (η) -1 L(η). More precisely, using (3.37) we obtain

E -+,2 (η) -1 = E -+,2 (0) -1 + o(|η| 2 ) L(η) = F (2) -2 η 2 + F (2) 0 + O(|η|), ∀η ∈ Ω δ ∩ C + , (3.52)
where

F (2) 0 = E (2) -E (2) B 1 F (2)
-2 and we can check that

F (2) 0 L(η) = F (2) 0 . Let E I -+ (η) = E -+ (η) -E -+,2 (η). For η ∈ {η ∈ C + , |η| < δ}, we see that E -+,2 (η) -1 E I -+ (η) = O(|η|)
. Consequently, we deduce from (3.27) and (3.52) that E -+ (η) -1 exists for Im η > 0 and |η| < δ, δ > 0 small enough with

E -+ (η) -1 = F (2) -2 η 2 - F (2) -2 E -+,3 F (2) -2 η -F (2) 0 -F (2)
-2 E -+,4 F

(2) -2

(3.53)

+ -4 j=1 η j F (2) j + E (-1) -+, -4 (η),
where the estimates (3.46) follow from (3.29). See the proof of (3.28) for (3.45).

It is remaining to prove that F

-2 has rank k. We shall show that

Γ k := (γ ij ) 1≤i,j≤k = (det Φ k )Φ -1 k . ( 3
γ ij =                    (-1) µij (-1) mi-1+•••+mj -1 |[Φ k ] i j | if i < j, (-1) µij (-1) mi-1 |[Φ k ] i i | if i = j, (-1) µij |[Φ k ] i j | if i = j + 1, (-1) µij (-1) mj+1-1+•••+mi-1-1 |[Φ k ] i j | if i > j + 1, (3.55)
where |[Φ k ] i j | is the (j, i)-th minor of the invertible matrix Φ k defined in (3.33). Substituting the values of µ ij given in (3.51) yields

γ ij = (-1) i+j |[Φ k ] i j |, which is the (j, i)-th cofactor of the matrix Φ k for all 1 ≤ i, j ≤ k.
Secondly, if -3/2 < s < ρ --3/2, we obtain the expansion of (I + R 0 (η 2 )V ) -1 up to order -4 with remainder o(|η| -4 ) by using the formula (3.16), the expansion (3.10) up to order -2 and the expansion of E -1 -+ (η) in (3.53).

Finally, if ρ > 2 -3 and s > -3/2, using the identity R(z) = (Id+R 0 (z)V ) -1 R 0 (z), the expansion of the resolvent in B(-1, s, 1, -s) has the following form:

R(z) = -4 j=-2 z j/2 R (2) j + R (2) -4 (z), (3.56)
where R

(2)

-1 = -SF (2)
-1 T G 0 , R (2)
0 = E 0 G 0 + SF (2)
-2 T G 2 -SF 0 T G 0 , S and T are defined in (3.9) and for

f ∈ H -1,s R (2) -2 f = -SF (2) -2 T G 0 f = -1 detφ k k i=1 k j=1 γ ij f, JG 0 V w (j) mj u (i) 1 = 1 detφ k k i=1 k j=1 c -1 j γ ij f, Ju (j) 1 u (i) 1 = k i=1 f, Jv i u (i) 1 . (3.57) Let V =    v 1 . . . v k    , U =     u (1) 1 . . . u (k) 1     , Γ k = (γ ij ) 1≤i,j≤k .
Then, we have

V = 1 detΦ k Γ k C k U.
(3.58) Thus, using (3.54) we obtain

V = Φ -1 k C k U = -(C k L k ) -1 C k U = -L -1 k U = -t QQU, (3.59)
where

C k = diag(c -1 1 , • • • , c -1 k ), L k = ( u (j) 1 , Ju (i)
1 ) 1≤i,j≤k is an invertible complex symmetric matrix with (3.34) and Q = (q ij ) 1≤i,j≤k is the upper triangular matrix obtained by the Cholesky decomposition of the matrix L -1 k (cf. [START_REF] Rakotonirina | Matrices de commutation tensorielle: de l'équation de Dirac vers une application en physique des particules[END_REF]Proposition 25]). Thus by returning to (3.57), we get

R (2) -2 f = - k i,j=1 k =1 q i q j f, Ju (j) 1 u (i) 1 = -P (2) 0 f
where

P (2) 0 = k =1 •, JZ (2) Z (2) with Z (2) = k i=1 q i u (i) 1 ,
and we see that P

(2) 0 is a projection of rank k since for all 1 ≤ i, j ≤ k we have

Z (2) i , JZ (2)
j = k ,m q i q jm u ( ) 1 , Ju (m) 1 = k l=1 q i (QL k ) j = (QL k t Q) ji = δ ij .
Moreover, other terms R

(2)

j , j = 1, • • • , - 4
, can be obtained explicitly. Finally, the estimate (2.21) can be seen from (3.13) and (3.46), also (2.18) follows from (2.5), (3.12) and (3.45).

3.4.

Zero singularity of the third kind. In this section we discuss the case when zero is both an embedded eigenvalue and a resonance of H. We assume that the eigenvalue zero has geometric multiplicity k -1, k ∈ N with k ≥ 2. In this case, we have dim ker L 2,-s (Id

+ K 0 ) = k and dim ker L 2,-s (Id + K 0 )/ker L 2 (Id + K 0 ) = 1. Set m = rank Π 1 . Let V i = {v (i) 1 , • • • , v (i)
mi } be a basis of E i and

X i = {χ (i) 1 , • • • , χ (i)
mi } be its Θ-dual basis (see Lemma 3.1). Under the hypothesis (H2) one can consider that Ker L 2,-s (Id+K 0 )/Ker L 2 (Id+K 0 ) = Span{v

1 , • • • , v (k)
1 }. This means that the matrix of Π 1 (Id + K 0 )Π 1 in the basis X is a k × k block diagonal matrix with Jordan block on the diagonal such that only the first Jordan block corresponds to the resonant state and the other blocks correspond to the eigenvectors that are the solutions in L 2 of (Id + K 0 )g = 0.

Since -1 is an eigenvalue of the operator K 0 of geometric multiplicity k ≥ 1, the same computations made to develop E -+ (z) -1 in Section 3.3 can be done here with a slight difference. Indeed, (3.27) holds with some difference in the block entries of the matrix A as follows:

A (ij) mi1 = -i G 1 V v (j) 1 , JV χ (i) mi = (i4πc i ) -1 v (i) 1 , JV 1 v (j)
1 , JV 1 do vanish for all i, j except that for i = j = 1 (see (2.10)), so we have

A (11) =      * . . . A (11) * a * • • • *      m1×m1 , a = (i4πc 1 ) -1 v (1)
1 , JV 1 2 = 0, (3.60)

A (i1) , i = 2 • • • , k, (resp., A (1j) , j = 2, • • • , k
) are sub-matrices with last row zero (resp., first column), and

A (ij) =      0 . . . A (ij) 0 0 0 • • • 0      mi×mj , ∀2 ≤ i, j ≤ k.
We now check the invertibility of E -+ (z) by following the same steps as before. We define

Φ k-1 = (b ij ) 2≤i,j≤k .
Here

b ij := -c i v (j) 1 , v (i)
1 , 2 ≤ i, j ≤ k, are computed in the same way as coefficients

β ij given in (3.32)). Let L k-1 = ( v (j) 1 , Jv (i) 1 ) 2≤i,j≤k . Then, we have Φ k-1 = -C k-1 L k-1 with C k-1 = diag(c -1 2 , • • • , c -1 k ).
(3.61) Lemma 3.6. Assume ρ > 3. Suppose that zero is a singularity of the third kind of H such that the eigenvalue zero has geometric multiplicity k -1, k ∈ N with k ≥ 2. We assume in addition that (H2) holds. Then det

E -+ (z) = σ k z k-1/2 + o(|z| k-1/2 ), (3.62) for z ∈ Ω δ , δ > 0 small, where σ k = a × det Φ k-1 = 0 with a is the constant given in (3.60).
Proof. We proceed in the same way as in the proof of Lemma 3.5. To avoid repetition we omit details. First, we define

E -+,2 (η) = N + ηA + η 2 B, Z(η) = detE -+,2 (η), ( 3
.63) for Im η > 0 and |η| < δ. Now, we introduce the matrix

L(η) = diag( L 1 (η), • • • , L k (η)) with L 1 (η) = diag(1, • • • , 1, η -1 ) and L i (η) = diag(1, • • • , 1, η -2 ), i = 2, • • • , k. We denote E -+,2 (η) = L(η)E -+,2 (η) = N + A(η) + B(η) and Z(η) = det E -+,2 (η). (3.64)
Then, it is non difficult to see that

(i) L(η)N = N. (ii) A (ij) (η) =                       0 0 • • • 0 . . . . . . . . . 0 0 • • • 0 a δ ij * • • • *      +O(|η|) if i = 1, 1 ≤ j ≤ k, O(|η|) if 2 ≤ i ≤ k, 1 ≤ j ≤ k. (iii) B (ij) (η)) =                       0 0 • • • 0 . . . . . . . . . 0 0 • • • 0 b ij * • • • *      +O(|η| 2 ) if 2 ≤ i ≤ k, 1 ≤ j ≤ k, O(|η|) if i = 1, 1 ≤ j ≤ k.
Let us calculate Z(0). We see from (i), (ii) and (iii) that the block entries

E (ij)
-+,2 (0), 1 ≤ i, j ≤ k, of the block matrix E -+,2 (0) have the following forms

E (1j) -+,2 (0) =         0 -δ ij 0 • • • 0 . . . . . . . . . . . . . . . 0 . . . . . . 0 0 0 -δ ij aδ 1j * • • • * 0         m1×mj , 1 ≤ j ≤ k, (3.65) E (ij) -+,2 (0) =         0 -δ ij 0 • • • 0 . . . . . . . . . . . . . . . 0 . . . . . . 0 0 0 -δ ij b ij * • • • * 0         mi×mj , 2 ≤ i ≤ k, 1 ≤ j ≤ k. (3.66)
This yields

Z(0) = det      a 0 • • • 0 b 21 b 22 • • • b 2k . . . . . . . . . b k1 b k2 • • • b kk      = a × det(b ij ) 2≤i,j≤k = a × detΦ k-1 .
Thus Z(0) = 0 since a = 0 and in view of (3.61) detΦ k-1 does not vanish if the condition (2) of (H2) is satisfied. Finally, using detL(η) = η -2k+1 , we obtain

det E -+ (η) = det E -+,2 (η) + o(|η| 2k-1 ) = (a × det Φ k-1 ) η 2k-1 + o(|η| 2k-1 ). Lemma 3.7. Let ρ > + 1, ∈ N with ≥ 4.
Assume that the hypotheses in the previous lemma hold. Then

E -+ (z) -1 = -4 j=-2 z j/2 F (3) j + E (-1)
-+, -4 (z), (3.67)

for z ∈ Ω δ , δ > 0 small, where F

-2 is a matrix of rank k -1 with

(F (3)
-2

) (ij) =      0 • • • 0 α ij 0 • • • 0 0 . . . . . . . . . 0 • • • 0 0      , 1 ≤ i, j ≤ k, (3.68)
such that α 1j = α i1 = 0, ∀1 ≤ i, j ≤ k, and F

-1 is a matrix of rank at most k, where Proof. We have showed in the proof of Lemma 3.6 that E -+,2 (0) is invertible if the hypothesis (H2) is satisfied. We calculate that

(F (3) -1 ) (ij) =      0 • • • 0 µ ij 0 • • • 0 0 . . . . . . . . . 0 • • • 0 0      , ∀1 ≤ i, j ≤ k, ( 3
E -+,2 (0) -1 = t Com E -+,2 (0) det E -+,2 (0) = F (3) -2 + E (3) , (3.70)
where

( F (3) -2 ) (ij) =                   0 • • • 0 α ij 0 • • • 0 0 . . . . . . . . . 0 • • • 0 0      , i = j = 1 or 2 ≤ i ≤ k, 1 ≤ j ≤ k, 0 , i = 1, 2 ≤ j ≤ k.
In particular α 11 = a -1 . The matrix E (3) has the same form as E (2) in (3.48).

On the other hand, for Im η > 0, |η| < δ with δ > 0 small, the matrix E -+,2 (η) can be developed as follows:

E -+,2 (η) = E -+,2 (0) + η E -+,2 (0) + O(|η| 2 ), (3.71)
where the m i × m j block entries of E -+,2 (0) have the following form:

( E -+,2 (0)) (ij) =                                     * . . . A (1j) * b 1j * • • • *      , i = 1, 1 ≤ j ≤ k,      * . . . A (ij) * 0 0 • • • 0      , 2 ≤ i ≤ k, 1 ≤ j ≤ k. (3.72)
Then, it follows from (3.27), (3.64), (3.70) and the regularity of L(η) that for Im η > 0 and |η| < δ, δ > 0 small enough, E -+,2 (η) -1 exists with

E -+,2 (η) -1 = F (3) -2
For the proof of the theorem we start by the following lemma, where we refer to Section 3.3 for details.

Lemma 4.1. For ρ > l + 1, l = 2, 3, • • • , and z ∈ Ω + δ , δ > 0 small, we have the following expansion of E -+ (z):

E -+ (z) = N + (z -λ 0 )A(λ 0 ) + l j=2 (z -λ 0 ) j E -+,j (λ 0 ) + E -+,l (z -λ 0 ), (4.4)
where N is the block matrix as defined in (3.30) and

A (ij) (λ 0 ) =     A (ij) (λ 0 ) a ij (λ 0 ) * • • • *     mi×mj , ∀1 ≤ i, j ≤ N 0 , (4.5)
where

a ij (λ 0 ) = -G + 1 V u (λ0,j) 1 , JV w (λ0,i) mi = 1 i8π √ λ 0 c i (λ 0 ) R 6 e +i √ λ0|x-y| V (x)u (λ0,i) 1 (x)V (y)u (λ0,j) 1 (y) dxdy,
such that c i (λ 0 ) = 0 by Remark 3.3 and G + 1 is the integral operator defined in (4.3). Moreover E -+,l (z -λ 0 ) is analytic in Ω + δ and for λ > 0, |λ -λ 0 | < δ, the limit lim

→0 + E -+,l (λ -λ 0 + i ) exists and satisfies d r dλ r E -+,l (λ -λ 0 + i0) = o(|λ -λ 0 | l-r ), ∀ |λ -λ 0 | < δ, r = 0, 1, • • • , l.
The expansion (4.4) can be obtained directly by introducing in (3.15) the expansion (4.1) of R 0 (z) in B(0, s, 0, -s) (see Section (3.3) for the details).

Before proving Theorem 2.5, we establish the expansion of E -+ (z) -1 . Let ρ > + 1, = 2, 3, • • • , and assume that the hypothesis (H3) holds. We introduce the block diagonal matrix

L(ξ) = diag(L 1 (ξ), • • • , L N0 (ξ)), L i (ξ) = diag(1, • • • , 1, ξ -1 ), ∀1 ≤ i ≤ N 0 , (4.6)
for ξ ∈ {ξ ∈ C + : |ξ| < δ}. Then, by proceeding in the same way as in the proof of Lemma 3.5 we obtain det (N + ξA 0 (λ

0 )) = ξ N0 det (a ij (λ 0 )) 1≤i,j≤N0 + O(|ξ| N0+1 ), (4.7)
where a ij (λ 0 ), 1 ≤ i, j ≤ N 0 , are given above and N 0 = dim ker(Id + K + (λ 0 )). It follows that, if the condition (2.13) is satisfied then (4.4) together with (4.7) gives

Proposition 4.2. If ρ > + 1, ∈ N with ≥ 2, we have det E -+ (z) = j=0 a j (λ 0 )(z -λ 0 ) N0+j + o(|z -λ 0 | N0+ ),
for z ∈ Ω + δ , where a 0 (λ 0 ) = det (a ij (λ 0 )) 1≤i,j≤N0 = 0. Since E -+,1 (ξ, λ 0 ) := N + ξA 0 (λ 0 ) has the same form as E -+,2 (ξ) defined in the proof of Theorem 2.3, then the same computation can be done here. We obtain

E -+,1 (ξ, λ 0 ) -1 = t ComE -+,1 (ξ, λ 0 ) detE -+,1 (ξ, λ 0 ) = 1 ξ F -1 (λ 0 ) + F 0 (λ 0 ), (4.8)
where F -1 (λ 0 ) is a N 0 × N 0 block matrix of rank N 0 with block entries

F (ij) -1 (λ 0 ) = 1 a 0 (λ 0 )      0 • • • 0 b ij (λ 0 ) 0 • • • 0 0 . . . . . . . . . 0 • • • 0 0      mi×mj (4.9)
where b ij (λ 0 ) are the (j, i)-th cofactors of the invertible matrix (a ij (λ 0 )) 1≤i,j≤N0 (see Remark 3.55).

On the other hand, by (4.4), for ∈ N * , ρ > + 1 and z ∈ Ω + δ , we have

E -+ (ξ, λ 0 ) = E -+,1 (ξ, λ 0 )×   I + (E -+,1 (ξ, λ 0 )) -1   j=2 ξ j E -+,j (λ 0 ) + E -+, (ξ, λ 0 )     .
Then, by (4.8), for ξ ∈ C + and |ξ| < δ with δ > 0 small enough, E -+ (ξ, λ 0 ) -1 exists, with

E -+ (ξ, λ 0 ) -1 = 1 ξ F -1 (λ 0 ) + -2 j=0 ξ j E -+,j (λ 0 ) + E -+, -2 (ξ). (4.10)
where

E -+,0 (λ 0 ) = F 0 (λ 0 ) -F -1 (λ 0 )E -+,2 (λ 0 )F -1 (λ 0 ), the other terms E -+,j (λ 0 ), j = 1, • • • , -2,
can be also directly found and the remainder

E -+, -2 (ξ) is analytic in {ξ ∈ C + : |ξ| < δ} and for λ > 0, |λ -λ 0 | < δ d r dλ r E -+, -2 (λ -λ 0 + i0) = o(|λ -λ 0 | -2-r ), r = 0, 1, • • • , -2.
Remark 4.3.

(1) We see that the coefficients a ij (λ 0 ), 1 ≤ i, j ≤ N 0 , defined in Lemma 4.1 can be expressed in the following matrix:

(a ij (λ 0 )) 1≤i,j≤N0 = 1 i8π √ λ 0 C N0 A N0 (λ 0 ), (4.11)
where

A N0 (λ 0 ) = B λ0 (u (λ0,j) 1 , u (λ0,i) 1 ) 1≤i,j≤N0
,

C N0 = diag 1 c 1 (λ 0 ) , • • • , 1 c N0 (λ 0 )
, and the bilinear form B λ0 is defined in (2.12).

(2) The entries b ij (λ 0 ) of the matrix F -1 (λ 0 ) in (4.9) can be expressed as follows

(b ij (λ 0 )) 1≤i,j≤N0 = i8π λ 0 a 0 (λ 0 )A N0 (λ 0 ) -1 C -1 N0 . (4.12)
Proof of Theorem 2.5. Applying Grushin problem (3.14) to M(z)

:= Id + R 0 (z)V , it follows from (4.10) that M(z) is invertible for z ∈ Ω + δ , with M(z) -1 = E(z) -(I -E(z)M(z))SE -+ (z) -1 T (I -M(z)E(z)).
Moreover, for -1/2 < s < ρ -+ 1/2, we have the following expansion in B(H 1,-s )

(Id + R 0 (z)V ) -1 = - 1 z -λ 0 SF -1 (λ 0 )T + -2 j=0 (z -λ 0 ) j F j (λ 0 ) + F (z -λ 0 ) (4.13)
where F -1 (λ 0 ) is the matrix of rank N 0 given in (4.8), so that for g ∈ H 1,-s

SF -1 (λ 0 )T g = 1 a 0 (λ 0 ) k j=1 b ij (λ 0 ) g, JV w (λ0,j) mj u (λ0,j) 1 , (4.14)
and F 0 (λ 0 ) = -S E -+,0 (λ 0 )T + E(0) with

E(0) = (I -Π λ0 1 )(Id + G + 0 V )(I -Π λ0 1 ) -1 (I -Π λ0 1 ).
The remainder F (z -λ 0 ) is analytic in Ω + δ and continuous up to R + verifying the following estimates:

d r dz r F (z -λ 0 ) B(H 1,-s ) = o(|z -λ 0 | -2-r ), ∀z ∈ Ω + δ , r = 0, • • • , - 2. (4.15)
Moreover, for λ ∈ Ω

+ δ ∩ R + , the limit lim →0 + F (λ -λ 0 + i ) = F (λ -λ 0 + i0)
exists as operator in B(H 1,-s ) and still satisfy the estimates in (4.15). Consequently, if ρ > 2 -1 and s > -1/2 using R(z) = (Id + R 0 (z)V ) -1 R 0 (z) and expansions (2.6), (4.13) together with (4.12) and (4.14) , it follows that for f ∈ H

-1,s R(z)f = R -1 (λ 0 )f z -λ 0 + -2 j=0 R j (λ 0 )f + R -2 (z -λ 0 )f in H 1,-s , where R -1 (λ 0 )f = - 1 a 0 (λ 0 ) N0 i=1 N0 j=1 b ij (λ 0 ) c j (λ 0 ) f, JG + 0 V u (λ0,j) 1 u (λ0,i) 1 = 1 a 0 (λ 0 ) N0 i=1 N0 j=1 b ij (λ 0 ) c j (λ 0 ) f, Ju (λ0,j) 1 u (λ0,i) 1 = i8π λ 0 N0 i=1 f, Jφ i (λ 0 ) u (λ0,i) 1
, such that by Remark 4.3 and a simple computation we have

   φ 1 (λ 0 ) . . . φ N0 (λ 0 )    = A N0 (λ 0 ) -1     u (λ0,1) 1 . . . u (λ0,N0) 1     ,
where A N0 (λ 0 ) -1 is given in (4.12). Thus, to simplify the above sum we decompose

A N0 (λ 0 ) -1 = t Q(λ 0 )Q(λ 0 ) (see (3.59)
). Hence, R -1 (λ 0 ) can be written in the following form

R -1 (λ 0 )f = N0 i=1 f, Jψ i (λ 0 ) ψ i (λ 0 ), where    ψ 1 (λ 0 ) . . . ψ N0 (λ 0 )    = (i8π λ 0 ) 1/2 Q(λ 0 )     u (λ0 ,1) 1 . . . u (λ0,N0) 1 
   with 1 i8π √ λ 0 B λ0 (ψ i (λ 0 ), ψ j (λ 0 )) = δ ij , B λ0 (•, •)
is the bilinear form defined in (2.12). In addition,

R 0 (λ 0 )f = - N0 i=1 f, JG + 1 V ψ i (λ 0 ) u (λ0,i) 1 + F 0 (λ 0 )G 0 f.
Moreover, we see from (2.5) and (4.15) that R l-2 (z -λ 0 )f can be continuously extended to Ω + δ with estimates (2.25).

General situation. We end this section with a more general result if the condition (2.13) does not hold.

Theorem 4.4. Let λ 0 be an outgoing positive resonance of H. Suppose that there exists an integer µ 0 > 0 and a small δ > 0 such that

d(z) := det E -+ (z) = (z -λ 0 ) µ0 g(z), ∀z ∈ Ω + δ , (4.16)
where g is an analytic function on

Ω + δ such that g(λ 0 ) = 0. Assume ρ > 2 + 1, s > + 1/2 with = µ 0 -N 0 + q, q ∈ N * . We have R(z) = R(λ 0 ) (z -λ 0 ) µ0-N0+1 + q-1 j=-µ0+N0 (z -λ 0 ) j R j (λ 0 ) + R q (z -λ 0 ) (4.17)
in B(-1, s, 1, -s), where N 0 is the geometric multiplicity of -1 as eigenvalue of K + (λ 0 ) and

R(λ 0 ) : L 2,s → Ker(Id + K + (λ 0 )) ⊂ L 2,-s .
Moreover, the remainder term R q (z -λ 0 ) is analytic in Ω + δ and satisfies the estimates d r dλ r R q (λ -λ 0 + i0) B(-1,s,1,-s) = o(|λ -λ 0 | q-1-r ), (4.18) for |λ -λ 0 | < δ and r = 0, 1, • • • , q -1.

If ρ > 3 and s > 3/2, then we can obtain

R(z) = R(λ 0 ) (z -λ 0 ) µ0-N0+1 + O(|z -λ 0 | -µ0+N0 ), ∀z ∈ Ω + δ . (4.19)
Proof. Under the condition (4.16) there exists δ > 0 such that for z ∈ Ω + δ the m × m matrix E -+ (z) is invertible and

E -+ (z) -1 = M (z) d(z)
where the entries of the matrix M (z) := (a ij (z)) 1≤i,j≤m are polynomials of the entries of E -+ (z) which are analytic in z ∈ Ω + δ . Moreover, taking the expansion (4.4) up to order , ≥ µ 0 -N 0 + 1, we obtain

E -+ (z) -1 = 1 d(z)   j=0 (z -λ 0 ) j+N0-1 B j (λ 0 )   + 1 d(z) E -1 -+, (z -λ 0 ) = j=0 (z -λ 0 ) j-(µ0-N0+1) B j (λ 0 ) + E -1 -+, (z -λ 0 )
where N 0 =dim Ker (Id + K + (λ 0 )),

( B 0 (λ 0 )) ij =      0 • • • 0 β j i (λ 0 ) 0 • • • 0 0 . . . . . . . . . 0 • • • 0 0      , 1 ≤ i, j ≤ N 0 ,
such that β j i (λ 0 ) are polynomials of the entries of A(λ 0 ) and the remainder term

E -1 -+,q (z -λ 0 ) is analytic in z ∈ Ω + δ and satisfying d r dλ r E -1 -+, (λ -λ 0 + i0) = o(|λ -λ 0 | q -r ), (4.20)
for |λ -λ 0 | < δ, r = 0, 1, • • • , q and q = -(µ 0 -N 0 + 1).

In the rest of the proof we can proceed in the same way as in the previous proof. We obtain the leading term

R(λ 0 ) = N0 i=1 •, ψ 1 (λ 0 ) u (λ0,i) 1 on L 2,-s , where    ψ 1 (λ 0 ) . . . ψ N0 (λ 0 )    = B N0 (λ 0 )     u (λ0,1) 1 . . . u (λ0,N0) 1     , B N0 (λ 0 ) = (β ij (λ 0 )) 1≤i,j≤N0 C N0 ,
(see (4.11)). Moreover, the estimate (4.18) can be seen from (4.20).

5.

Large-time expansion of the semigroup (e -itH ) t≥0

In this section, we look at the asymptotic expansion in time, as t → +∞, of solutions to the Schrödinger equation (1.1). We use the preceding results for the resolvent behavior on a contour surrounding the positive resonances in the upper half-plane, encircling the origin and down to the lower half-plane.

First, since V satisfies the condition (1.2) for ρ > 2, we can check that for arbitrary small > 0, there exists R > 0 large enough such that the numerical range of H denoted by N (H) is included in an angular sector We deduce from our previous main results that H has a finite number of discrete eigenvalues in the closed upper half-plane. r (H), then to show that there is at most a finite number of these eigenvalues it suffices to prove that zero and the outgoing positive resonances of H are not accumulation points of σ + d (H) even though zero is an eigenvalue or/and a resonance. Indeed, in view of Theorem 2.5 for each λ j ∈ σ + r (H) there is small δ > 0 such that if ρ > 2 and s > 1/2 x -s R(z) x -s is uniformly bounded in z on every compact set in Ω + δ ⊂ C + given in (2.23). Then we have found a set Ω + δ ⊂ C + which does not contain any pole of x -s R(z) x -s . Thus Ω + δ ∩ σ d (H) = ∅. Hence λ j is not an accumulation point of σ d (H) ∩ C+ . Also, if zero is a resonance of H then it is seen from Theorem 2.2 that the same argument can be done. Moreover, if zero is an eigenvalue of H and the hypothesis (H1) holds, using the same argument, it follows from Theorem 2.3 with ρ > 3 that σ d (H) ∩ Ω δ = ∅ for some δ > 0 small enough, where Ω δ is given in (2.15). We can check also that zero is not an accumulation point of σ d (H) ∩ C+ if it is both an eigenvalue and a resonance of H under the hypothesis (H2).

N (H) ⊆ {z ∈ C : Re z ≥ -R , | arg(z + R )| ≤ 2 }. ( 5
In addition, we check the existence of the limiting absorption principle for the non-selfadjoint Schrödinger operator H on each subinterval of R + which does not contain any outgoing positive resonance. We also establish high energy estimates of the derivatives of the resolvent.

In the following, we denote by C j (Ω, F ) the set of all functions f : Ω ⊂ E -→ F that is of class C j on Ω, where E and F denote normed vector spaces. And for a > 0, we define the open set Λ a and its closure Λa by

Λ a = N j=0 {z ∈ C + : |z -λ j | > a}, Λa = N j=0 {z ∈ C+ : |z -λ j | > a},
where λ 0 = 0 and λ j ∈ σ + r (H), ∀1 ≤ j ≤ N .

The following proposition gives the high energy resolvent estimate as |z| → +∞, when R(z) is extended through the upper half-plane to Λa for a > 0. It can be proved in the same way as in [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF]Theorem 9.2] (see also [START_REF] Komech | Dispersive decay for the magnetic schrödinger equation[END_REF]Theorem 3.8]).

Proposition 5.2. Assume that (H3) holds. Let ∈ N. If ρ > + 1, then for s > + 1 2 and f, g ∈ L 2,s : Λ a z → R(z)f, g can be continuously extended to a function in C Λa ; C . Moreover, the boundary values R(λ + i0)f, g satisfy the following estimates:

| d dλ R(λ + i0)f, g | ≤ C a |λ| +1 2 f 0,s g 0,s , λ ∈ Λa ∩ R + , λ → +∞, (5.2)
for some constant C a > 0.

The existence of the above limit is a direct consequence of the two main theorems 2.4 and 2.5 and the following known results: for f ∈ L 2,-s , h ∈ L 2,s and 1/2 < s < ρ -1/2 the functions z → (Id + R 0 (z)V ) -1 f, h and z → R 0 (z)V f, h can be continuously extended to uniformly bounded functions on Λa (see [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF]Lemma 9.1]). In addition to the following estimates in [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF]Theorem 8.1]

| d r dλ r R 0 (λ ± i0)f, g | ≤ C |λ| r+1 2
f 0,s g 0,s , r ∈ N, as λ → +∞.

Before proving Theorem 5.3, we must establish a representation formula for the Schrodinger semigroup e -itH , as t → +∞, generated by the non-selfadjoint operator H. Our representation formula is based on the Dunford-Taylor integral (cf. [START_REF] Kato | Perturbation theory for linear operators[END_REF]Section IX.1.6]), which is valid for m-sectorial operator, which means whose numerical range is a subset of a sector {| arg z| ≤ θ < π 2 }. More precisely, we will find a curve Γ ν (η) such that Γ ν (η

) ∩ σ + d (H) ∪ σ + r (H) ∪ {0} = ∅ and Γ ν (η) := Γ ν -(η) ∪ Γ 0 (η) ∪ Γ 1 (η) ∪ Γ + , where Γ ν -(η) = {z = a(η) -λe iν , λ ≥ 0}, a(η) = 2η -iη sin η, σ -(η) = {z = λ -iη sin η, η cos η ≤ λ ≤ 2η}, Γ 0 (η) = {z = ηe i(2π-θ) , η < θ < 2π}, Γ 1 (η) = ∪ N j=1 (σ j (η) ∪ γ j (η)) , σ 1 (η) = {z = λ + i0, η ≤ λ ≤ λ 1 -η}, σ j (η) = {z = λ + i0, λ j-1 + η ≤ λ ≤ λ j -η}, j = 2, • • • , N, γ j (η) = {z = (λ j + ηe i(π-θ) ), 0 < θ < π}, j = 1, • • • , k, Γ + = {z = λ + i0, λ ≥ λ N + η},
for some η > 0 and ν ∈]0, π 2 [ chosen so that there are no eigenvalues of H between σ -(η) ∪ Γ 0 (η) ∪ Γ 1 (η) and the real axis, nor between Γ ν -(η) and the negative real axis. See Figure 1.

R iR σ1(η) > Γ ν -(η) > σ -(η) Γ0(η) > > γ2(η) λ 1 λ 2 λ N • × × × × × × × × Figure 1. The curve Γ ν (η). λ 1 , • • • , λ N are the outgoing positive resonances of H.
We now state the intermediate theorem Theorem 5.3. Let ρ > 3 and s > 5/2. Assume that hypothesis (H3) on positive resonances holds. If zero is an eigenvalue of H we assume in addition that (H1) or (H2) holds. Then, for f and g ∈ L 2,s , we have the following representation formula:

e -itH f, g = zj ∈σ + d (H) e -itH Π zj f, g (5.3)
+ 1 2iπ Γ ν (η) e -itz (H -z) -1 f, g dz, ∀t > 0,
where σ + d (H) is the finite set of discrete eigenvalues of H located in the closed upper half-plane with associated Riesz projections {Π zj } j and Γ ν (η) is the above curve (see Figure 1.).

Proof. We proceed in 3 steps: First step: Let > 0. Let P = i(e -i H -i R ). Then it can be seen from (5.1) that

N (P ) ⊆ {i(e -i z -i R ), Re z ≥ -R , | arg(z + R )| ≤ 2 } ⊂ Sθ ,
where S θ denotes the open sector with angle θ . Let θ = (π -arctan )/2 ∈]0, π/2[. Moreover, for λ := e i(π/2+ ) (λ + R ) ∈ C \ N (H) with λ > 0 large, it can be seen that (P + λ) = ie -i (H -λ ) is a bijection of H 2 into L 2 . This shows that P is m-sectorial with semi-angle θ . Hence, -P is the unique generator of the semigroup (e -tP ) t≥0 , which is bounded by e -tP ≤ 1 (cf. [START_REF] Kato | Perturbation theory for linear operators[END_REF]Theoreme IX.1.24]). Therefore, there exists a closed curve Γ, oriented in the anticlockwise sense, included in the resolvent set of -P and enclosing the numerical range of -P in its interior, where Γ = {λe -i(π-θ -δ) , λ ≥ 0} ∪ {λe i(π-θ -δ) , λ ≥ 0} for some 0 < δ < π 2 -θ , such that the semigroup integral representation is written:

e -tP u = 1 2iπ Γ e tz (P + z) -1 udz, ∀u ∈ L 2 , ∀t > 0, (5.4)
and we have the following estimate e -ite -i H u 0 ≤ e tR u 0 , t ≥ 0.

(5.5)

Denote by Π zj : L 2 → L 2 the Riesz Projection (2.3) associated with eigenvalue z j ∈ σ + d (H). σ + d (H) is a finite set by Proposition 5.1 and H j := HΠ zj defines a bounded operator on the finite dimensional subspace Ran Π zj , with σ(H j ) = {z j } (see [15, p. 178-179]). Let H = e -i H and z = e -i z. By analytic deformation in ρ(H) of the curve Γ, we can find a set of curves ∪ p j=1 Σ j around the eigenvalues z 1 , • • • , z p ∈ σ + d (H) located above a curve Γ ν (η, ) (defined below) oriented in the anti-clockwise sense such that

U (t)u := e -itH u = - 1 2iπ zj ∈σ + d (H) Σj e -itz (H j -z) -1 udz + 1 2iπ Γ ν (η, ) e -itz (H -z) -1 udz (5.6) = zj ∈σ + d (H) e -itH Π zj u + 1 2iπ Γ ν (η, ) e -itz (H -z) -1 u dz, ∀t > 0,
where Γ ν (η, ) is a closed curve oriented from -∞ to +∞ and Γ

ν (η, ) = Γ ν -(η) ∪ Γ 0 (η, ) ∪ Γ 1 (η, ) ∪ Γ + ( ): Γ ν -(η) = {z = a(η) -λe iν , λ ≥ 0}, a(η) = 2η -iη sin η, σ -(η) = {z = λ -iη sin η, η cos η ≤ λ ≤ 2η} Γ 0 (η, ) = {z = ηe i(2π-θ) , η < θ < 2π -η }, η = arcsin( /η), (5.7)
Γ 1 (η, ) = ∪ N j=1 (σ j (η, ) ∪ γ j (η, )) , σ 1 (η, ) = {z = λ + i , η cos η ≤ λ ≤ λ 1 -η}, σ j (η, ) = {z = λ + i , λ j-1 + η ≤ λ ≤ λ j -η}, j = 2, • • • , N, γ j (η, ) = {z = λ j + i + ηe i(π-θ) , 0 < θ < π}, j = 1, • • • , N, Γ + ( ) = {z = λ + i , λ ≥ λ N + η},
for some fixed 0 < ν < π 2 and η, > 0 small chosen so that Γ ν (η, ) ∩ σ(H) = ∅ and there is no eigenvalues of H between Γ 0 (η, ) ∪ Γ 1 (η, ) ∪ Γ + ( ) and the positive real axis, nor between Γ ν -(η) and the negative real axis.

Second step: Let f, g ∈ L 2,s . We define

U (t)f, g := zj ∈σ + d (H) e -itH Π zj f, g + 1 2iπ Γ ν (η)
e -itz (H -z) -1 f, g dz.

In this step we will show that

U (t)f, g = lim →0 + U (t)f, g , ∀f, g ∈ L 2,s , ∀t > 0.
(5.8)

Let us show the convergence of the integral in (5.6) as → 0 + by decomposing it onto two parts:

Γ ν (η, ) ∩ {|z| ≤ R 1 } and Γ ν (η, ) ∩ {|z| > R 1 }, where R 1 > λ N + 1 with λ N := max σ + r (H).
On one hand, if zero is an eigenvalue of H Theorem 2.3 gives the uniformly boundedness of the resolvent on Γ 0 (η) in B(0, s, 0, -s) if ρ > 3 and s > 3/2. In addition, Theorems 2.5 shows that the resolvent is uniformly bounded in z on each semicircle γ j (η) surrounding the singularity λ j on the positive real axis, where a weaker assumption is required, so that ρ > 2. Moreover, by Proposition 5.2 the integrand g (z, t) := e -ite -i z (H -z) -1 f, g is continuously extended to an uniformly bounded function in z on σ -(η) ∪ N j=1 σ j (η) for f, g ∈ L 2,s , 1/2 < s < ρ -1/2. Then in view of these results

Γ ν (η, )∩{|z|≤R1} g (z, t) dz -→ →0 + Γ ν (η)∩{|z|≤R1} e -itz (H -z) -1 f, g dz . ( 5.9)
On the other hand, we have

Γ ν (η, )∩{|z|>R1} = Γ ν -(η)∩{|z|>R1} + Γ+( )∩{|z|>R1}
.

(5.10)

Since g (z, t) is uniformly bounded in ∈]0, ν 2 [ on Γ ν -(η) ∩ {|z| > R 1 } with |e -ita(η)e -i e itλe i(ν-) (H -(a(η) -λe iν )) -1 f, g | ≤ C η,R1 f 0,s g 0,s e -tλ sin( ν 2 ) λ -1/2 , ∀λ ∈ [R 1
, +∞[, η > 0 and t > 0 where the function at the right-hand side is integrable on [R 1 , +∞[ for all η > 0 small and t > 0. Then we deduce by Lebesgue's dominated convergence theorem lim

→0 + Γ ν -(η)∩{|z|>R1} g (z, t) dz = Γ ν -(η)∩{|z|>R1} e -itz (H -z) -1 f, g dz. ( 5
| d 2 dλ 2 (H -(λ + i )) -1 f, g | ≤ C R1 λ 3/2 f 0,s g 0,s , ∀ > 0,
that requires ρ > 3 and s > 5/2. Then, for t > 0 fixed we integrate twice by parts to obtain

+∞ R1 (-ite -i ) -2 e -itλe -i d 2 dλ 2 (H -(λ + i )) -1 f, g dλ + O(t -2 |e -i(tR1e -i + ) |) f 0,s g 0,s := +∞ R1 f (t, λ)dλ + O(t -2 |e -i(tR1e -i + ) |) f 0,s g 0,s . (5.12)
Since for ρ > 3, s > 5/2 and t > 0, f (t, .) is uniformly bounded in small > 0 as

|f (t, λ)| ≤ C R1 t 2 1 λ 3/2 f 0,s g 0,s ,
then by Lebesgue's dominated convergence theorem +∞ R1 f (t, λ)dλ converges as → 0, also the second term at the right-hand side of (5.12) is uniformly bounded in > 0 by O(t -2 ) f 0,s g 0,s . This shows that the second integral at the right-hand side of (5.10) is uniformly convergent in > 0 for all t > 0. Consequently, this together with (5.9) and (5.11) implies

Γ ν (η, ) g (z, t) dz -→ →0 + Γ ν (η)
e -itz (H -z) -1 f, g dz, which must establish (5.8).

Finally, we will show at the third step that for all f and g ∈ L 2,s we have the following convergence e -itH f -e -itH f, g -→ →0 + 0, ∀t > 0.

Third step: Let φ be a test function in C ∞ 0 (R 3 ). We write e -itH φ -e -itH φ = t 0 d dr e -irH e -i(t-r)H φ dr = (-i)(e -i -1)

t 0 e -irH He -i(t-r)H φ dr. Now let t > 0 be fixed. By (5.5) and the previous equality, we see that e -itH φ -e -itH φ 0 ≤ Ce tR |e -i -1|

t 0 e -i(t-r)H Hφ 0 dr -→ →0 + 0, i.e. e -itH φ converges in L 2 norm to e -itH φ as → 0 for all φ ∈ C ∞ 0 (R 3 ). This by uniqueness of the weak limit in (5.8) gives

e -itH φ, ψ = U (t)φ, ψ , ∀φ, ψ ∈ C ∞ 0 (R 3 ), ∀t > 0. Finally, by density of C ∞ 0 (R 3 ) in L 2,s , we conclude that U (t)f, g = e -itH f, g , ∀f, g ∈ L 2,s , ∀t > 0,
which establishes the desired representation formula.

Next we quote a lemma for some generalized integrals given in [8, Section II.2.].

Lemma 5.4.

(1) The limit of the function λ → (λ + iµ) -1 as µ → 0 + , is the generalized function (λ + i0) -1 defined in the following sense: For every test function φ ∈ C 1 0 (R)

(x + i0) -1 , φ(x) = |x|≤1 φ(x) -φ(0) x dx + |x|>1 φ(x) x dx -iπφ(0).
Moreover, for t > 0 we have the following generalized integral R e -itλ λ + i0 dλ = -i2π.

(2) For t > 0 and j = -1, 0, 1,

• • • we have +∞ 0 λ j/2 e -itλ dλ = Γ( j 2 + 1)(-it) -j 2 -1 ,
where Γ( j 2 + 1) = +∞ 0 t j/2 e -t dt.

Now we are able to prove Theorem 2.6. Before starting the proof, let us rewrite the representation formula in (5.6) as follows: Proof of Theorem 2.6. Let χ : R → R be a cutoff function such that χ(λ) = 1 for |λ| ≤ ν/2 and χ(λ) = 0 for |λ| ≥ ν. For j = 0, 1, • • • , N , we define χ j (λ) = χ(λ -λ j ), where λ 0 = 0 and λ j ∈ σ + r (H), ∀j = 1, • • • , N . Set g (z, t) = e -ite -i z (H -z) -1 f, g .

First, we prove the part (b) of the theorem. We begin by introducing at the second member of (5.13) the resolvent expansions near zero energy and positive resonances which are obtained in theorems 2.3 and 2.5 respectively

Γ ν (η, ) g (z, t) dz = 1 s=-2 R (2)
s f, g

Γ ν (η, )
z s/2 e -ite -i z χ 0 (Re z) dz

+ N j=1 l-2 s=-1 R s (λ j )f, g Γ ν (η, )
(z -λ j ) j e -ite -i z χ j (Re z) dz

+ Γ ν (η, )
e -ite -i z R

(2) 1 (z)f, g χ 0 (Re z) dz

+ N j=1 Γ ν (η, )
e -ite -i z R (z -λ j )f, g χ j (Re z) dz

+ Γ ν (η, )
e -ite -i z R(z)f, g χ + (Re z) dz

:= I ,η 0 (t) + N j=1 -2
s=-1 I ,η s (t, λ j ) + J ,η 0 (t) (5.14)

+ N j=1
J ,η (t, λ j ) + J ± (t), (5.15) where χ + = 1 -N j=0 χ i . Let t > 0 be fixed. We see that

I ,η 0 (t) = R (2)
-2 f, g

-2 f, g + 2(-iπ) 1/2 R

(2)

-1 f, g t -1/2 + O(t -2 ), where the decay rate t -1/2 can be seen from Lemma 5.4. In addition, the integral I ,η 1 (t) can be easily estimated using Lemma 5.4 as follows lim η→0 + lim →0 + I ,η 1 (t) = 2 R

(2)

1 f, g +∞ 0 √ λe -itλ χ(λ) dλ = -(iπ) 1/2 R (2)
1 f, g t -3/2 + O(t -2 ).

(5.16)

Moreover, for j = 1, • • • , N , we can show that I ,η -1 (t, λ j ) in (5.14) converge as → 0 + to integrals

I η -1 (t, λ j ) = R -1 (λ j )f, g e -itλj
Lη e -itξ ξ χ(Re ξ) dξ, ξ = z -λ j , along the contour L η =] -∞, -η] ∪ {ξ = ηe i(π-θ) , 0 < θ < π} ∪ [η, +∞[ traveled from -∞ to +∞. Then, the limit integrals I η -1 (t, λ j ) converge as η → 0, in the sense of generalized function of t, to I -1 (t, λ j ) := R -1 (λ j )f, g e -itλj

+∞

-∞ e -itλ λ + i0 χ(λ)dλ = -2iπ R -1 (λ j )f, g e -itλj + O(t -2 ), ∀t > 0.

However, for r = 0, 1, • • • , l -2, we have

I η, r (t, λ j ) -→ η→0 + , →0 + R r (λ j )f, g e -itλj
-∞

+∞

λ r e -itλ χ(λ)dλ, ∀t > 0, where the right member decays rapidly at infinity as the r-th derivative of the Fourier transform of the cut off function χ such that the convergence holds in the sense of regularized function.

Let now estimate J ± (t) and J ,η j (t) as t → +∞ given in (5.15). We decompose J ± (t) as follows:

J ± (t) = Γ1(η, )∪Γ+( ) g (z, t)χ + (Re z) dz

+ Γ ν -( )
g (z, t)χ + (Re z) dz (5.17) := J + (t) + J -(t).

(5.18)

The integral J -(t) has an exponential time-decay O(e -tcν ) for some c ν > 0 independent on and t.

Regarding the integral J + (t), it follows from (5.12) that lim →0 + |J + (t)| = O(t -2 ) f 0,s g 0,s , as t → +∞.

Next, we have to estimate J 0 (t). By Theorem 2.3, if ρ > 7, z → R Finally, let j = 1, • • • , N . We have by Theorem 2.5 that if ρ > 2 -1 and s > -1/2, ∈ N with ≥ 2, the remainder term R -2 (z -λ j ) can be continuously extended to R -2 (λ -λ j + i0) ∈ C -2 ({λ > 0, |λ -λ j | < δ}, B(-1, s, 1, -s)) .

(5.20)

Then, as the above integrals I η, r (t, λ j ), J ,η (t, λ j ) converges as , η → 0 in the sense of regularized functions to the Fourier transform in t of the regular and compactly supported function λ → R -2 (λ + i0)χ(λ). This in view of (5.20) gives lim η→0 lim →0 J ,η (t, λ j ) = o(t -+2 ) f 0,s g 0,s , t → +∞.

(5.21)

See [START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF]Lemma 10.1].

We have proved the part (b). In this theorem, the stronger condition ρ > 7 and s > 7/2 is required to obtain o(t -2 ) in (5.21) ( = 3) and to get o(t -3/2 ) in (5.19). But this condition can be relaxed to ρ > 4 to obtain (2.16) with remainder estimate o(|z| -1/2 ) and then to get the remainder o(t -1/2 ) in (5.19). Now, we return to the proof of the part (a) of Theorem 2.6. We have only to compute the integral I ,η 0 (t) which does not have the same behavior as in the previous proof. Indeed, when zero is a resonance and not an eigenvalue, I ,η 0 (t) is replaced by

I ,η 0 (t) = 1 j=-1 R (1)
j f, g

C0(η, )
e -ite -i z z j/2 dz + R

(1)

-1 f, g +∞ η( ) e -ite -i (λ+i ) √ λ + i - e -ite -i (λ-i ) √ λ -i χ(λ) dλ + I ,η 1 (t),
where R

-1 is the one rank operator defined in Theorem 2.2. It is easily to show that the first integral at the right hand side vanishes as η → 0. However, the second integral tends as → 0 + , in the sense of generalized functions, to e -itλ √ λ (χ(λ) -1) dλ = 2(-iπ) 1/2 t -1 2 + O(t -2 ).

Also, see (5.16) for the estimate of I ,η 1 (t).

Regarding the estimate of J 0 (t) in (5.19), to obtain the decay o(t -1/2 ) it is required that ρ > 3 and s > 3/2. However, in presence of positive resonances our assumption ρ > 5 and s > 5/2 is needed to obtain o(t -1 ) in (5.21), according to Theorem 2.5.

We end the paper by the following remark: Remark 5.5. If we assume that the condition (2.14) is satisfied instead of (2.13), then using the expansion (4.17) of R(z) near λ j ∈ σ + r (H), the oscillating term N j=1 e -itλj R -1 (λ j ) in (2.26) will be replaced by N j=1 e -itλj R j (t, λ j ),

where R j (t, λ j ) is a polynomial of t of degree at most µ j -N j with values in B(0, s, 0, -s) and with leading term -(-it) µj -Nj (µ j -N j )! R(λ j ). See Theorem 2.5. Indeed, for = 0, 1, • • • , µ j -N j , (λ + i0) --1 is defined as the -th derivative, in the sense of generalized functions, of (-1) ! (λ + i0) -1 . Then, by integrating by part we have e -itλ (λ + i0) d j χ dλ j (λ)dλ = a t + a -1 t -1 + • • • + a 1 t + a 0 , with a = (-i) +1 2π ! .

- 4 1 )- 4

 414 (z) has the same properties as R ((z) in Theorem 2.3.

  .42) Finally, (3.35) with the previous equation implies that for η ∈ {η ∈ C + , |η| < δ}

( 1 ) 1 }

 11 and Ker L 2 (Id+K 0 ) = Span{v

  .69) such that µ 11 = α 11 = a -1 and µ 1j = 0 for j = 2, • • • , k. Moreover, the remainder term E (-1) -+,l-4 (z) satisfies the estimates in (3.45).

Proposition 5 . 1 .

 51 Assume that ρ > 2 and the hypothesis (H3) on positive resonances holds. If zero is an eigenvalue of H we assume in addition that ρ > 3 and (H1) or (H2) holds. Then σ + d (H) is finite. Proof. Since the eigenvalues located in the closed upper half-plane can only accumulate at points of {0} ∪ σ +

. 11 )

 11 However, the integrand of the second integral tends to e -itλ (H -(λ + i0)) -1 f, g as → 0 + which by Proposition 5.2 belongs to C 2 ([R 1 , +∞[, C), with the following estimate:

Figure 2 .

 2 Figure 2. The curve Γ ν (η, ).

  e -ite -i (λ+i ) √ λ + i -e -ite -i (λ-i ) √ λ -i χ(λ)dλ + I ,η 1 (t),whereω 0 := [ν -i , η cos η -i ] ∪ C 0 (η, ) ∪ [η cos η + i , ν + i ] ∪ [ν + i , ν -i ]is a closed curve enclosing zero traveled in the clockwise sense. It follows that I ,η 0 (t) -I ,η 1 (t) converges as → 0 + , η → 0 + respectively, in the sense of generalized functions of t, to

1 2

 1 can be continuously extended to C 2 ({|z| < δ, ±Im z ≥ 0}, B(-1, s, 1, -s)) for s > 7/2, such that d r dλ r R (2) 1 (λ ± i0) B(-1,s,1,-s) = o(λ -r ), 0 < λ < δ, r = 0, 1, 2.Then, it follows from Lemma 10.2 in[START_REF] Jensen | Spectral properties of schrödinger operators and time-decay of the wave functions[END_REF] thatlim η→0 lim →0 J 0 (t) = o(t -32 ) as t → +∞.(5.19)

Acknowledgment. I would like to express my gratitude to my supervisor Xue Ping Wang for helpful discussions and his useful comments on this research work.

where (F

-2

-2 ) (ij) if 2 ≤ i, j ≤ k and ( F

-2 ) (ij) = 0 elsewhere. Also, F

-2 -F

(3)

The rest of the proof follows directly from (3.27) and (3.73). Moreover, the same argument used to prove that the matrix F

-2 in (3.43) is of rank k can be applied to the matrix F

-2 to show that it has rank k -1, unless in the present case we show that (α ij ) 2≤i,j≤k is a (k -1) × (k -1) invertible matrix. Indeed, we can check that

which denotes the (i -1, j -1)-th entry of the invertible matrix Φ k-1 (see (3.61)). Thus

We end this section by proving Theorem 2.4.

Proof of Theorem 2.4. Since the same proof of Theorem 2.3 can be done here, we will omit details.

Using the asymptotic expansion of

where, by the help of the matrices defined in (3.61) and (3.74), we have

1 , and

where α ij and µ ij are respectively entries of the matrices F

-2 and F

-1 given in (3.68) and (3.69). Here we used G 1 V χ

1 . Thus the resolvent expansion in (2.22) is proved. Moreover, we refer to the proof of Theorem 2.3 for the properties of the remainder term R

(3) l-4 (z).

Resolvent expansions near positive resonances

In this section, we prove Theorem 2.5. First, we use the condition (2.13) of the hypothesis (H3) to establish the asymptotic expansion of the resolvent R(z) = (H -z) -1 near an outgoing positive resonance. Note that the study of incoming positive resonance can be done in a similar way.

Let us begin with the known results on the behavior of the free resolvent R 0 (z) on the boundary of the right half-plane (the real half axis ]0, +∞[). Taking the analytic continuation of the integral kernel R 0 (z)(x, y) to C+ \ {0}, the expansion of R 0 (z) at order r for every r ∈ N and for z ∈ Ω + δ , δ > 0, is written

where

are integral operators with corresponding kernels r + j (x, y, λ 0 ) := lim z→λ0,z∈C+

For simplicity, we will use in the following the variable ξ = z -λ 0 . We denote by R 0 (λ + i0) the boundary value of R 0 (z).

Let λ 0 be an outgoing positive resonance of the operator H = -∆ + V . Note that the two subspaces Ker(Id+K + (λ 0 )) and {ψ ∈ Ker(H-λ 0 ); ψ satisfies the radiation condition (1.3) with sign +} coincide in H 1,-s , 1/2 < s < ρ -1/2 (see Section 2.1). Denote dim Ker(Id + K + (λ 0 )) = N 0 . Let Π λ0 1 be the Riesz projection associated with the eigenvalue -1 of the operator K + (λ 0 )

we also denote E + λ0 = Ran Π λ0 1 and m = rank Π λ0 1 .

The same strategy used in Section 3 to prove Theorem 2.3 and Theorem 2.4 will be followed in this section. First, note that the decomposition made in Lemma 3.1 can be done in the present case for E + λ0 with just a change of notation } its dual with respect to the non degenerate bilinear form Θ. In particular, Ker(Id + K + (λ 0 )) is the subspace of L 2,-s generalized by {u }. Then, we will study the Grushin problem for the operator Id + K(z), that we have constructed in Section 3.3.