FRANC Xiaoyao Jia

¸ois Nicoleau

Xue Ping Wang

A NEW LEVINSON'S THEOREM FOR POTENTIALS WITH CRITICAL DECAY

Keywords: 2000 Mathematics Subject Classification. 35P25, 47A40, 81U10 Schrödinger operators, resolvent expansion, zero energy resonance, spectral shift function, Levinson's theorem

|x| 2 ). We prove a generalized Levinson's for this class of potentials in presence of zero eigenvalue and zero resonance. Contents 1. Introduction 1 2. A representation formula 5 3. Resolvent asymptotics near the threshold 9 4. Generalized residue of the trace function 18 5. Proof of Levinson's theorem 31 References 37

Introduction

Threshold spectral analysis of Schrödinger operators plays an important role in many problems arising from non-relativistic quantum mechanics, such as low-energy scattering, propagation of cold neutrons in a crystal or the Efimov effect in manyparticle systems (cf. [START_REF] Bollé | Schrödinger operators at threshold. (English summary) Ideas and methods in quantum and statistical physics[END_REF][START_REF] Newton | Scattering Theory of waves and particles[END_REF][START_REF] Wang | On the existence of N -body Efimov effect[END_REF]). Presence of zero resonance of Schrödinger operators is responsible for several striking physical phenomena. The Levinson's theorem which relates the phase shift to the number of eigenvalues and the zero resonance is one of the oldest topics in this domain (cf. [START_REF] Levinson | On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase[END_REF][START_REF] Ma | The Levinson theorem[END_REF][START_REF] Newton | Scattering Theory of waves and particles[END_REF]). Threshold spectral analysis is usually carried out for potentials decaying faster than 1 |x| 2 (cf. [START_REF] Jensen | Spectral properties of Schrödinger operators and timedecay of the wave functions[END_REF][START_REF] Murata | Asymptotic expansions in time for solutions of Schrödinger type equations[END_REF][START_REF] Wang | Asymptotic behavoir of resolvent for N-body Schrödinger operators near a threshold[END_REF]). Potentials with the critical decay 1 |x| 2 appear in many interesting situations such as spectral analysis on manifolds with conical end or ion-atom scattering for N -body Schrödinger operators. The threshold spectral properties for potentials with critical decay are quite different from those of more quickly decaying potentials and the contribution of zero resonance and zero eigenvalues to the asymptotics of the resolvent at low-energy and to the long-time expansion of wave functions are studied in [START_REF] Carron | Le saut en zero de la fonction de decalage spectral[END_REF][START_REF] Wang | Threshold energy resonance in geometric scattering[END_REF][START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF]. In this work, we shall study the low-energy asymptotics of some spectral shift function and prove a generalized Levinson's theorem for potentials with critical decay in presence of zero eigenvalue and zero resonance.

Recall that for a pair of selfadjoint operators (P, P 0 ) on some Hilbert space H, if (P + i) -k -(P 0 + i) -k is of trace class for some k ∈ N * , (1.1) the spectral shift function ξ(λ) is defined as distribution on R by

Tr (f (H) -f (H 0 )) = - R f ′ (λ)ξ(λ) dλ, ∀f ∈ S(R). (1.2)
In fact, this relation only defines ξ(•) up to an additive constant. It can be fixed by, for example, requiring ξ(λ) = 0 for λ sufficiently negative if P and P 0 are both bounded from below.

In this paper, we are interested in the threshold spectral analysis of the Schrödinger operator P = -∆ + v(x) on L 2 (R n ) with n ≥ 2 and v(x) a real function satisfying

v(x) = q(θ) r 2 + O( x -ρ 0 ), |x| >> 1, (1.3)
for some q ∈ C(S n-1 ) and ρ 0 > 2. Here x = rθ with r = |x| and θ = x |x| . S n-1 is the unit sphere . Let -∆ S n-1 denote the Laplace-Beltrami operator on S n-1 . We assume throughout this paper that the smallest eigenvalue λ 1 of -∆ S n-1 + q(θ) verifies

λ 1 > - 1 4 (n -2) 2 . (1.4)
This assumption ensures that the form associated to P 0 = -∆ + q(θ) r 2 on C ∞ 0 (R n \ {0}) is positive. We still denote by P 0 its Friedrich's realization as selfadjoint operator on L 2 (R n ). Notice that the condition (1.1) is not satisfied for the pair (P, -∆) by lack of decay on v, nor for the pair (P, P 0 ) because of the critical singularity at 0. This leads us to introduce a model operator P 0 in the following way. Let 0 ≤ χ j ≤ 1 (j = 1, 2) be smooth functions on R n such that supp χ 1 ⊂ B(0, R 1 ), χ 1 (x) = 1 when |x| < R 0 and χ 1 (x) 2 + χ 2 (x) 2 = 1. Set P 0 = χ 1 (-∆)χ 1 + χ 2 P 0 χ 2 , on L 2 (R n ). When q(θ) = 0, we can take P 0 = -∆ and the main results of this work still hold. Under the assumption (1.4), one has P 0 ≥ 0. The operator P will mainly be considered as a perturbation of P 0 . Under the condition that v(x) is bounded and satisfies (1.3) for some ρ 0 > n, (1.1) is satisfied for k > n 2 and the spectral shift function ξ(λ) for the pair (P, P 0 ) is well defined by (1.1).

High energy asymptotics of the spectral shift function of Schrödinger operators has been studied by many authors (see for example [START_REF] Albeverio | Low-energy parameters in nonrelativistic scattering theory[END_REF], [START_REF] Pushnitski | Spectral shift function of the Schrödinger operator in the large coupling constant limit. II. Positive perturbations[END_REF], [START_REF] Robert | Relative time-delay for perturbations of elliptic operators and semi-classical asymptotics[END_REF], [START_REF] Robert | Semiclassical asymptotics for the spectral shift function[END_REF], [START_REF] Yafaev | The Schrödinger operator: perturbation determinants, the spectral shift function, trace identities, and more[END_REF]). In particular, D. Robert proved in [START_REF] Robert | Relative time-delay for perturbations of elliptic operators and semi-classical asymptotics[END_REF] that for the pair (P, P 0 ) introduced as above, if the potential v is smooth and satisfies |∂ α

x (v(x) -q(θ) r 2 )| ≤ C α x -ρ 0 -|α| , for |x| large. (1.5) for some ρ 0 > n, then one has (i). ξ(λ) is C ∞ in (0, ∞); (ii). d k dλ k ξ(λ) has a complete asymptotic expansion for λ → ∞,

d k dλ k ξ(λ) ∼ λ n/2-k-1 j≥0 α (k) j λ -j .
We are mainly interested in the low-energy asymptotics of the spectral shift function ξ(•) for (P, P 0 ). Although ξ(•) depends on the choice of cut-offs χ 1 and χ 2 , we shall see that physically interesting quantities in low-energy limit are independent of such choices. The low-energy resolvent asymptotics for P is studied in [START_REF] Wang | Threshold energy resonance in geometric scattering[END_REF][START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF] for v of the form v(x) = q(θ) r 2 + w(x) with w(x) bounded and satisfying w(x) = O( x -ρ 0 ) for some ρ 0 > 2. In this work, in order to define the spectral shift function, we assume v(x) to be bounded and in the last section on Levinson's theorem we even need to assume it smooth. Therefore, we need to slightly modify the proofs of [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF] to suit with the present situation.

The eigenvalues of the operator -∆ S n-1 + q(θ) play an important role in the threshold spectral analysis of P (cf. [START_REF] Carron | Le saut en zero de la fonction de decalage spectral[END_REF][START_REF] Wang | Threshold energy resonance in geometric scattering[END_REF][START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF]). Let

σ ∞ = ν; ν = λ + (n -2) 2 4
, λ ∈ σ(-∆ S n-1 + q(θ)) .

(1.6)

Denote σ k = σ ∞ ∩]0, k], k ∈ N * . (1.7)
σ 1 is closely related to the properties of zero resonance. We say that 0 is a resonance of P if the equation P u = 0 admits a solution u such that x -1 u ∈ L 2 , but u ∈ L 2 . u is then called a resonant state of P . For ν ∈ σ ∞ , let n ν denote the multiplicity of

λ ν = ν 2 -(n-2) 2 4
as the eigenvalue of -∆ S n-1 + q(θ). For the class of potentials with critical decay, if zero is a resonance of P , its multiplicity is at most ν∈σ 1 n ν . Let u be a resonant state of P . Then one has the following asymptotics for r = |x| large

u(x) = ψ(θ) r n-2 2 +ν (1 + o(1)) (1.8)
for some ν ∈ σ 1 and ψ = 0 an eigenfunction of -∆ S n-1 + q(θ) with eigenvalue λ ν . We call u a ν-resonant state (or a ν-bound state). This terminology is consistent with the historical half-bound state for rapidly decreasing potentials in three dimensional case, since the set σ 1 is then reduced to { 1 2 }. The multiplicity m ν of ν-resonant states is defined as the dimension of the subspace spanned by all ψ such that the expansion (1.8) holds for some resonant state u. The main result of this paper is the following generalized Levinson's theorem in presence of zero eigenvalue and zero resonance.

Theorem 1.1. Let n ≥ 2. Assume that v is smooth and satisfies (1.5) for some ρ 0 > max{6, n + 2}. Let ξ(•) denote the spectral shift function for the pair (P, P 0 ). Then there exist some constants c j and

β n/2 such that ∞ 0 (ξ ′ (λ) - [ n 2 ] j=1 c j λ [ n 2 ]-j-1 )dλ = -(N -+ N 0 + ν∈σ 1 νm ν ) + β n/2 .
(1.9)

Here β n/2 = 0 if n is odd and c [ n 2 ] = 0 if n is even. β j for j = 1,
2 is computed at the end of paper. N -is the number of negative eigenvalues of P , N 0 the multiplicity of the zero eigenvalue and m ν that of ν-resonance of P . It is understood that if 0 is not an eigenvalue (resp., if there is no ν-resonant states), then N 0 = 0 (resp. m ν = 0).

For the class of potentials under consideration, zero resonance of P may appear in any space dimension with arbitrary multiplicity depending on q. See [START_REF] Carron | Le saut en zero de la fonction de decalage spectral[END_REF][START_REF] Wang | Threshold energy resonance in geometric scattering[END_REF] in the case of manifolds with conical end and [START_REF] Jia | Some threshold spectral problems of Schrödinger operators[END_REF] in the case of R n . In [START_REF] Jia | Some threshold spectral problems of Schrödinger operators[END_REF], the existence of zero resonance is studied. Levinson's theorem in presence of zero eigenvalue and zero resonance is mostly proved for spherically symmetric potentials by using Sturm-Liouville method or techniques of Jost function. See [START_REF] Ma | The Levinson theorem[END_REF][START_REF] Newton | Scattering Theory of waves and particles[END_REF] for overviews on the topic, where the reader can also find some discussions on radial potentials with critical decay. The proof of Levinson's theorem for non-radial potentials is much more evolved. See [START_REF] Newton | Noncentral potentials the generalized Levinson theorem and the structure of the spectrum[END_REF] and a series of works of D. Bollé et al. [START_REF] Albeverio | Low-energy parameters in nonrelativistic scattering theory[END_REF][START_REF] Bollé | Threshold scattering in two dimensions[END_REF][START_REF] Bollé | Schrödinger operators at threshold. (English summary) Ideas and methods in quantum and statistical physics[END_REF] for potentials satisfying x s v ∈ L 1 (R n ) for some suitably large s depending on n and for n = 1, 2, 3. For example, one needs s > 4 if n = 3, s > 8 if n = 2 according to [START_REF] Bollé | Schrödinger operators at threshold. (English summary) Ideas and methods in quantum and statistical physics[END_REF]. A formula as (1.9) in its general setting seems to be new.

To prove Theorem 1.1, we use the result of [START_REF] Robert | Relative time-delay for perturbations of elliptic operators and semi-classical asymptotics[END_REF] on the high energy asymptotics of ξ ′ (λ). The main task of our work is to analyze the spectral shift function in a neighbourhood of 0. Our approach is to calculate the trace of the operator-valued function z → (R(z) -R 0 (z))f (P ) and its generalized residue at 0, where R 0

(z) = (P 0 -z) -1 , R(z) = (P -z) -1 and f ∈ C ∞ 0 (R) is equal to 1 near 0.
As the first step, we prove the asymptotics expansions of the resolvents R(z) and R 0 (z) for z near 0 by adapting the methods of [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF] to our situation. The main difficulties arise from the fact that the set σ ∞ may be arbitrary (depending on q(θ)). The interplays between the zero energy resonant states and between the zero resonant states and the zero energy eigenfunctions can be produced in such a way that their contributions may be dominant over that of a single ν-resonant state with itself. This is overcome by carefully examining the terms obtained from asymptotic expansion of the resolvent and by making use of the properties of ν-resonant states. Remark that the decay condition can be improved if σ 1 contains only one point and our proofs (both for the resolvent expansions and for the Levinson's theorem) can be very much simplified if the potential v(x) is spherically symmetric.

This work is organized as follows. In Section 2, we establish a representation formula of the spectral shift function which will be used to prove Levinson's theorem. In Section 3, we use the asymptotic expansion of R 0 (z) = ( P 0z) -1 to get the asymptotic expansion for the resolvents R 0 (z) and R(z). The methods are the same as in [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF]. We indicate only the modifications to make and omit the details of calculation. The hard part of the proof of Theorem 1.1 is to calculate the generalized residue at 0 of the trace function z → T r(R(z) -R 0 (z))f (P ), which is carried out in Section 4. Here f is some smooth function with compact support equal to 1 near 0. This result is used to study the low-energy asymptotics of the derivative of the spectral shift function in Section 5 and to prove the Levinson's theorem.

Notation. The scalar product on L 2 (R + ; r n-1 dr) and L 2 (R n ) is denoted by •, • and that on L 2 (S n-1 ) by (•, •). H r,s , r, s ∈ R, denotes the weighted Sobolev space of order r with the weight x s . The duality between H 1,s and H -1,-s is identified with L 2product. Denote H 0,s = L 2,s . L(r, s; r ′ , s ′ ) stands for the space of continuous linear operators from H r,s to H r ′ ,s ′ . The complex plane C is slit along positive real axis so that z ν = e ν ln z and ln z = log |z| + i arg z with 0 < arg z < 2π are holomorphic for z near 0 in the slit complex plane.

A representation formula

Let (P, P 0 ) be a pair of self-adjoint operators, semi-bounded from below, in some separable Hilbert space H. We assume that for some k ∈ N * ,

||(P -i) -k -(P 0 -i) -k || tr < ∞, (2.1)
where ||.|| tr denotes the trace-class norm in H. Under the assumption (2.1), it is wellknown that for any f ∈ S(R), f (P )f (P 0 ) is of trace class. For z ∈ C in the resolvent set of P 0 and P , we denote by R 0 (z) = (P 0z) -1 , (resp. R(z) = (Pz) -1 ) the resolvent of P 0 , (resp. of P ). Writing

(R(z) -R 0 (z))f (P ) = [R(z)f (P ) -R 0 (z)f (P 0 )] -R 0 (z)(f (P ) -f (P 0 )), (2.2)
we see immediately that (R(z) -R 0 (z))f (P ) is of trace class.

The spectral shift function (SSF, in short) ξ(λ) ∈ L 1 loc (R) is defined up to a constant as a Schwartz distribution on R by the equation

Tr (f (P ) -f (P 0 )) = - R f ′ (λ)ξ(λ) dλ, ∀f ∈ S(R).
(2.

3)

The right hand side can be interpreted as ξ ′ , f , where ξ ′ is the derivative of ξ in the sense of the distributions, and •, • denotes the pairing between S ′ (R) and S(R).

In this section, we give a representation formula for the SSF under the following assumptions :

• The spectra of P and P 0 are purely absolutely continuous in ]0, +∞[ :

σ(P 0 ) = σ ac (P 0 ) = [0, +∞[, (2.4
)

σ ac (P ) = [0, +∞[. (2.5)
In particular, we assume there are no embedded eigenvalues of P and P 0 in ]0, +∞[.

• The total number, N -, of negative eigenvalues of P is finite.

• For any f ∈ C ∞ 0 (R), there exists some ǫ 0 > 0, δ 0 < 1 and C > 0 (depending on f ), such that

| Tr [(R(z) -R 0 (z))f (P )] | ≤ C |z| 1+ǫ 0 , (2.6)
for z ∈ C with |z| large and z / ∈ σ(P ) and

| Tr [R 0 (z)(f (P ) -f (P 0 ))] | ≤ C |z| δ 0 , (2.7)
for z ∈ C with |z| small and z / ∈ σ(P 0 ).

• Let f ∈ C ∞ 0 (R) with f (t) = 1 for t near 0. The generalized residue of the function z → Tr [(R(z) -R 0 (z))f (P )] at z = 0 is finite in the following sense: if we denote, for ǫ ≪ δ, γ(δ, ǫ) = {z ∈ C ; |z| = δ, dist(z, R + ) ≥ ǫ},
we assume that

J 0 = - 1 2πi lim δ→0 lim ǫ→0 γ(δ,ǫ) Tr [(R(z) -R 0 (z))f (P )] dz exists, (2.8)
where γ(δ, ǫ) is positively oriented.

• The derivative of ξ in the sense of distributions satisfies

ξ ′ (λ) ∈ L 1 loc (]0, ∞[).
(2.9)

Remark that conditions (2.1), (2.4)-(2.7) imply that if J 0 exists for some f as above, then it exists for all f ∈ C ∞ 0 (R) with f (t) = 1 for t near 0 and J 0 is independent of f . In fact, if f 1 and f 2 are two such functions, applying (2.2) to f = f 1f 2 , one sees that Tr [(R(z) -R 0 (z))f (P )] can be decomposed into two terms, one is holomorphic in z near 0 and the other is of the order O( 1 |z| δ 0 ), δ 0 < 1. Hence the generalized residue of z → Tr [(R(z) -R 0 (z))f (P )] at 0 is equal to zero. We have the following representation formula for the spectral shift function ξ(λ).

Theorem 2.1. Let f ∈ C ∞ 0 (R) such that f (λ) = 1
for λ in neighborhood of σ pp (P )∪{0}. Under the above assumptions, the limit

∞ 0 f (λ) ξ ′ (λ) dλ = lim δ→0 + lim R→∞ R δ f (λ) ξ ′ (λ) dλ exists and one has T r (f (P ) -f (P 0 )) - ∞ 0 ξ ′ (λ) f (λ) dλ = N -+ J 0 .
(2.10)

Proof. Let us consider the application F (z) = T r [(R(z) -R 0 (z))f (P )] which is holomorphic outside σ(P ). We want to deduce (2.10) from Cauchy's formula applied to F (z).

Let us begin by some notation. Let E j , 1 ≤ j ≤ k, be the distinct eigenvalues of P with multiplicity m j . Denote for z 0 ∈ C and δ > 0,

C(z 0 ; δ) = { z ∈ C; |z -z 0 | = δ } ; D(z 0 ; δ) = { z ∈ C; |z -z 0 | ≤ δ }.
(2.11)

For δ > 0 small enough, σ(P

) ∩ D(E i ; δ) = {E i }. For R ≫ 1 and 0 < ǫ ≪ δ, let us denote γ(R, ǫ) = {z ∈ C; |z| = R, dist(z, R + ) ≥ ǫ} (2.12) d ± (R, δ, ǫ) = [ δ 2 -ǫ 2 ± iǫ, R 2 -ǫ 2 ± iǫ]. (2.13)
We denote by Γ δ,ǫ,R the curve defined by

Γ δ,ǫ,R =   k j=1 C(E j ; δ)   ∪ γ(δ, ǫ) ∪ d + (R, δ, ǫ) ∪ γ(R, ǫ) ∪ d -(R, δ, ǫ).
Γ δ,ǫ,R is positively oriented according to the anti-clockwise orientation of the big circle γ(R, ǫ). Since F (z) is holomorphic in the domain limited by Γ δ,ǫ,R , the Cauchy integral formula gives

1 2πi Γ δ,ǫ,R F (z) dz = 0.
We split the integral into four terms

1 2iπ Γ δ,ǫ,R F (z)dz = 4 j=1
I j with (2.14)

I 1 = 1 2iπ γ(R,ǫ) F (z) dz , I 2 = k j=1 1 2iπ C(E j ;δ)
F (z) dz.

I 3 = 1 2iπ γ(δ,ǫ) F (z) dz , I 4 = 1 2iπ d + (R,δ,ǫ) ∪ d -(R,δ,ǫ) F (z) dz.
By condition (2.6), one has I 1 = O(R -ǫ 0 ) when R → ∞, uniformly in δ and ǫ > 0. For j = 1, ..., k, E j is an eigenvalue of P with finite multiplicity m j , and the spectral projection associated to E j is given by

Π j = 1 2iπ C(E j ;δ) R(z) dz, (2.15)
since C(E j ; δ) is oriented according the clockwise orientation. Using that R 0 (z) is holomorphic in D(E j , δ), we obtain :

I 2 = k j=1 T r (Π j f (P )) = k j=1 m j f (E j ) = k j=1 m j = N -. (2.16)
Using assumption (2.8), we have by definition lim δ→0 lim ǫ→0

I 3 = J 0 , (2.17)
and as a conclusion, we have obtained :

lim R→+∞ lim δ→0 lim ǫ→0 I 4 = -N --J 0 . (2.18)
Now, let us establish a new expression for I 4 ; we split F (z) = F 1 (z) + F 2 (z) where

F 1 (z) = Tr [R(z)f (P ) -R 0 (z)f (P 0 )], (2.19)
F 2 (z) = Tr [R 0 (z)(f (P 0 ) -f (P ))]. (2.20)
First, let us investigate the contribution coming from F 1 (z). By the definition of the SSF,

F 1 (z) = - R ξ(λ) ∂f ∂λ (λ, z) dλ (2.21) where f (λ, z) = 1 λ -z f (λ). Thus, F 1 (z) = G 1 (z) -G 2 (z) with G 1 (z) = R ξ(λ) 1 (λ -z) 2 f (λ) dλ, (2.22) G 2 (z) = R ξ(λ) 1 λ -z f ′ (λ) dλ. (2.23)
For simplicity, we set Γ = d + (R, δ, ǫ) ∪ d -(R, δ, ǫ). We have

1 2iπ Γ F 1 (z) dz = 1 2iπ Γ G 1 (z) dz - 1 2iπ Γ G 2 (z) dz := (a) -(b). (2.24)
Let us study (a). Using Fubini's theorem, we can write

(a) = 1 2iπ R ξ(λ) f (λ) Γ ∂ ∂z 1 λ -z dz dλ. (2.25) Set k(λ, s, ǫ) = 1 λ -( √ s 2 -ǫ 2 + iǫ) - 1 λ -( √ s 2 -ǫ 2 -iǫ) .
One has

(a) = 1 2iπ R ξ(λ) f (λ)(k(λ, R, ǫ) -k(λ, δ, ǫ)) dλ. (2.26)
Let us look at the term I ǫ related to k(λ, R, ǫ) and let us show that lim ǫ→0

I ǫ = ξ(R)f (R).
Fix 0 < R 0 < R and split I ǫ = J ǫ + K ǫ , where

J ǫ = 1 2iπ R 0 -∞ ξ(λ) f (λ) k(λ, R, ǫ) dλ, K ǫ = 1 2iπ +∞ R 0 ξ(λ) f (λ)k(λ, R, ǫ) dλ. (2.27)
By the basic properties of the SSF, one has ξf ∈ L 1 (R) and

|J ǫ | ≤ 1 π R 0 -∞ |ξ(λ) f (λ)| ǫ (λ - √ R 2 -ǫ 2 ) 2 + ǫ 2 dλ ≤ Cǫ ||ξf || 1 .
(2.28)

On the other hand, we have

K ǫ = 1 π +∞ R 0 - √ R 2 -ǫ 2 ǫ (ξf )( R 2 -ǫ 2 + ǫs) 1 s 2 + 1 ds. (2.29)
Under the assumption (2.9), ξ ∈ C 0 (]0, +∞[), so by making use of the dominated convergence theorem, one has

lim ǫ→0 K ǫ = ξ(R)f (R).
(2.30)

As a conclusion, we have shown

lim ǫ→0 (a) = ξ(R)f (R) -ξ(δ)f (δ) = -ξ(δ), (2.31)
for δ > 0 sufficiently small and R ≫ 1. Now, let us study (b). Using Fubini's theorem,

(b) = R ξ(λ) f ′ (λ) 1 2iπ Γ 1 λ -z dz dλ. (2.32) It is easy to check that 1 2iπ Γ 1 λ -z
dz is uniformly bounded with respect to ǫ and

lim ǫ→0 1 2iπ Γ 1 λ -z dz = 1 [δ,R] (λ) a.e λ. (2.33)
So, making use of the dominated convergence theorem and then the assumption (2.9), a straightforward application of distribution theory gives

lim ǫ→0 (b) = R δ ξ(λ) f ′ (λ) dλ = -ξ(δ) - R δ ξ ′ (λ) f (λ) dλ, (2.34)
for R ≫ 1 and δ > 0 sufficiently small. As a conclusion, we have obtained,

lim ǫ→0 1 2iπ Γ F 1 (z) dz = R δ ξ ′ (λ) f (λ) dλ. (2.35)
To study the contribution coming from F 2 (z), we write

1 2iπ Γ F 2 (z) dz = T r 1 2iπ Γ R 0 (z) dz (f (P 0 ) -f (P )) . (2.36)
Using the spectral theorem for P 0 and the assumption (2.1), the same argument as (2.33) gives s-lim

ǫ→0 1 2iπ Γ R 0 (z) dz = 1 [δ,R] (P 0 ). (2.37)
Since f (P )f (P 0 ) is of trace class, and s-lim

δ→0,R→∞ 1 [δ,R] (P 0 ) = Id, (2.38) one can deduce (see Lemma 2.2 below) that lim R→∞ lim δ→0 lim ǫ→0 1 2iπ Γ F 2 (z) dz = T r (f (P 0 ) -f (P )).
(2.39) So, using (2.18), (2.35) and (2.39), we obtain the result.

To complete the proof of Theorem 2.1, we prove the following elementary lemma.

Lemma 2.2. Let A be an operator of trace class. Assume that f (λ) is an operator valued function, uniformly bounded in λ, such that s-lim

λ→λ 0 f (λ) = B exists.
Then, f (λ)A converges to BA in the trace norm class, as λ → λ 0 . In particular,

lim λ→λ 0 T r (f (λ)A) = T r(BA). (2.40)
Proof. For any ǫ > 0, let F be a finite rank operator such that ||A -F || tr < ǫ. Then,

||f (λ)A -BA|| tr ≤ ||(f (λ) -B)F || tr + ||(f (λ) -B)(A -F )|| tr .
Since F is a finite rank operator and s-lim

λ→λ 0 f (λ) = B, we have ||(f (λ) -B)F || tr ≤ Cǫ for |λ -λ 0 | ≤ δ with some δ > 0 small enough. Since f (λ) -B is a uniformly bounded in λ, we have also ||(f (λ) -B)(A -F )|| tr ≤ Cǫ. This ends the proof. 2
The remaining part of this work is to apply Theorem 2.1 to Schrödinger operator, using the known results on the asymptotic expansion of ξ ′ (λ) as λ → ∞. The main task is to study ξ(λ) for λ near 0. If one can calculate the generalized residue J 0 and can show that ξ ′ (λ) is integrable in ]0, 1], then one can take a family of functions f R (λ) = χ( λ R ), where χ is smooth and 0 ≤ χ(s) ≤ 1, χ(s) = 1 for s near 0, χ(s) = 0 for s > 1 and expand both the terms

∞ 0 ξ ′ (λ)f R (λ) dλ, and Tr(f R (P ) -f R (P 0 ))
in R large. Theorem 1.1 can be derived from Theorem 2.1 by comparing the two asymptotic expansions in R.

Resolvent asymptotics near the threshold

Consider the Schrödinger operator P = -∆ + v(x) where the potential v(x) is bounded and has the asympotic behavior

v(x) = q(θ) r 2 + O( x -ρ 0 ), |x| → ∞ (3.1)
where ρ 0 > 2, (r, θ) is the polar coordinates on R n and q(•) is continuous on the sphere. Let P 0 = -∆ + q(θ) r 2 be the homogeneous part of P . Assume n ≥ 2 and

-∆ S n-1 + q(θ) > - 1 4 (n -2) 2 , on L 2 (S n-1 ). (3.2)
In particular, (3.2) implies that the quadratic form defined by P 0 on C ∞ 0 (R n \ {0}) is positive. We still denote by P 0 its Friedrich's realization as selfadjoint operator in L 2 (R n ). Due to troubles related to the critical local singularity in the definition of spectral shift function, the model operator P 0 we will use in the next Section is a modification of P 0 inside some compact set. Let 0 ≤ χ j ≤ 1 (j = 1, 2) be smooth functions on R n such that suppχ 1 ⊂ B(0, R 1 ), χ 1 (x) = 1 when |x| < R 0 and

χ 1 (x) 2 + χ 2 (x) 2 = 1. Define P 0 = χ 1 (-∆)χ 1 + χ 2 P 0 χ 2 , on L 2 (R n ) (if q = 0
, we take P 0 = -∆). In the next Section, P will be regarded as a perturbation of P 0 . But to study the low energy asymptotics of R 0 (z) and R(z), we regard both P and P 0 as perturbations of P 0 . Denote

P 0 = P 0 -W, (3.3)
P = P 0 + V = P 0 -W (3.4)
where

W = χ 2 1 q(θ) r 2 - 2 j=1 |∇χ j | 2 V = - 2 i=1 |∇χ i | 2 + v -χ 2 2 q(θ) r 2 W = W -V = -v + q(θ) r 2 .
Note that W and W contain a critical singularity at zero. For n ≥ 3, the Hardy's inequality implies that they map continuously H 1 to H -1 . When n = 2, the same remains true under the condition (1.4) if one defines H 1 as the form domain of P 0 . V is bounded and satisfies

|V (x)| ≤ C x -ρ 0 , ρ 0 > 2. ( 3.5)
P 0 is still a Schrödinger operator with a potential of critical decay at the infinity. But it has nice threshold spectral property at zero. Let

σ ∞ = ν; ν = λ + (n -2) 2 4 , λ ∈ σ(-∆ S n-1 + q(θ)) . (3.6) Denote σ k = σ ∞ ∩]0, k], k ∈ N. (3.7) For ν ∈ σ ∞ , let n ν denote the multiplicity of λ ν = ν 2 -(n-2) 2 4
as the eigenvalue of

-∆ S n-1 + q(θ). Let ϕ (j) ν , ν ∈ σ ∞ , 1 ≤ j ≤ n ν denote an orthonormal basis of L 2 (S n-1 ) consisting of eigenfunctions of -∆ S n-1 + q(θ): (-∆ S n-1 + q(θ))ϕ (j) ν = λ ν ϕ (j) ν , (ϕ (i) ν , ϕ (j) ν ) = δ ij .
Let π ν denote the orthogonal projection in L 2 (R + × S n-1 ; r n-1 drdθ) onto the subspace spanned by the eigenfunctions of -∆ S n-1 + q(θ) associated with the eigenvalue λ ν :

π ν f = nν j=1 (f, ϕ (j) ν ) ⊗ ϕ (j) ν , f ∈ L 2 (R + × S n-1 ; r n-1 drdθ). Let Q ν = - d 2 dr 2 - n -1 r d dr + ν 2 -(n-2) 2 4 r 2 , in L 2 (R + ; r n-1 dr).
The asymptotic expansion of the resolvent R 0 (z) = (P 0z) -1 near zero can be obtained as in [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF], by considering P 0 as a perturbation of P 0 . For the later purpose, we also need the asymptotic expansion of d dz R 0 (z) for z near 0, where R 0 (z) = ( P 0z) -1 for z / ∈ σ(P 0 ). Decomposition the resolvent R 0 (z) as

R 0 (z) = ν∈σ∞ (Q ν -z) -1 π ν , z / ∈ R The Schwartz kernel K ν (r, τ ; z) of (Q ν -z) -1
, ℑz > 0, can be calculated explicitly. In fact, the Schwartz kernel of e -itQν is given by (see [START_REF] Taylor | Partial differential equations II[END_REF])

1 2it (rτ ) -n-2 2 e -r 2 +τ 2 4it -i πν 2 J ν ( rτ 2t ), t ∈ R, (3.8)
where J ν (•) is the Bessel function of the first kind of order ν. Since

(Q ν -z) -1 = i ∞ 0 e -it(Qν -z) dt for ℑz > 0, the Schwartz kernel of (Q ν -z) -1 is K ν (r, τ ; z) = (rτ ) -n-2 2 ∞ 0 e -r 2 +τ 2 4it +izt-i πν 2 J ν ( rτ 2t ) dt 2t (3.9) = (rτ ) -n-2 2 ∞ 0 e iρ t +izrτ t-i πν 2 J ν ( 1 2t ) dt 2t for ℑz > 0, where ρ = ρ(r, τ ) ≡ r 2 + τ 2 4rτ .
Note that the formula of K ν (r, τ ; z) used in [START_REF] Carron | Le saut en zero de la fonction de decalage spectral[END_REF][START_REF] Wang | Threshold energy resonance in geometric scattering[END_REF] contains a sign error. The asymptotic expansion for (Q νz) -1 as z → 0 and ℑz > 0 is deduced from (3.9) by splitting the last integral into two parts according to t ∈]0, 1 or t ∈ [1, ∞[. The integral for t ∈]0, 1] gives rise to a formal power series expansion in z near 0. The expansion corresponding to the integral of t ∈ [1, ∞[ needs a lengthy calculation. See [START_REF] Wang | Threshold energy resonance in geometric scattering[END_REF]. Set

f (s; r, τ, ν) = D ν (r, τ ) 1 -1 e i(ρ+θ/2)s (1 -θ 2 ) ν-1/2 dθ, ν ≥ 0, with D ν (r, τ ) = a ν (rτ ) -(n-2) 2 , a ν = e -iπν/2 2 2ν+1 π 1/2 Γ(ν + 1/2) . (3.10)
Then f can be expanded in a convergent power series in s

f (s; r, τ, ν) = ∞ j=0 s j f j (r, τ, ν), s ∈ R, (3.11)
with f j (r, τ, ν) = (rτ ) -1 2 (n-2) P j,ν (ρ), (3.12) with P j,ν (ρ) a polynomial in ρ of degree j:

P j,ν (ρ) = i j a ν j! 1 -1 (ρ + 1 2 θ) j (1 -θ 2 ) ν-1 2 dθ.
In particular,

f 0 (r, τ, ν) = d ν (rτ ) -n-2 2 , d ν = e -1 2 iπν 2 2ν+1 Γ(ν + 1) ; f 1 (r, τ, ν) = id ν (rτ ) -n-2 2 ρ.
In [START_REF] Wang | Threshold energy resonance in geometric scattering[END_REF][START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF], the expansion of ( P 0z) -1 is given for z near 0. For the later use, we need the derivation of this expansion. By an abuse of notation, we denote by the same letter F an operator on L 2 (R + , r n-1 dr) and its distributional kernel.

For

ν ∈ σ ∞ , denote [ν] the integral part of ν and ν ′ = ν -[ν]. Set z ν = z ν ′ , if ν / ∈ N, z ln z, if ν ∈ N.
For ν > 0, let [ν] -be the largest integer strictly less than ν.

When ν = 0, set [ν] -= 0. Define δ ν by δ ν = 1, if ν ∈ σ ∞ ∩ N, δ ν = 0, otherwise. One has [ν] = [ν] -+ δ ν .
Proposition 3.1. Assume the condition (3.2). One has (a). The following asymptotic expansion holds for z near 0 with ℑz > 0.

R 0 (z) = N j=0 z j F j + ν∈σ N N -1 j=[ν] - z ν z j G ν,j+δν π ν + R (N ) 0 (z), (3.13
)

in L(-1, s; 1, -s), s > 2N + 1. The remainder term R (N ) 0 (z) can be estimated by R (N ) 0 (z) = O(|z| N +ǫ ) ∈ L(-1, s; 1, -s), s > 2N + 1,
for some ǫ > 0 and F j is of the form

F j = ν∈σ∞ F ν,j π ν ∈ L(-1, s; 1, -s), s > 2j + 1. (3.14)
F ν,j and G ν,j can be explicitly calculated from the asymptotic expansion of K ν (r, τ, z) in z. In particular,

G ν,j (r, τ ) =      b ν ′ ,j (rτ ) j+ν ′ f j-[ν] (r, τ ; ν), ν / ∈ N - (irτ ) j j! f j-[ν] (r, τ ; ν), ν ∈ N with f k defined by (3.12) and b ν ′ ,j = - i j e -iν ′ π/2 Γ(1 -ν ′ ) ν ′ (ν ′ + 1) • • • (ν ′ + j) , (3.15)
for 0 < ν ′ < 1.

(b). The expansion (3.13) can be differentiated with respect to z and one has

d dz R (N ) 0 (z) = O(|z| N -1+ǫ ) ∈ L(-1, s; 1, -s), s > 2N + 1, for some ǫ > 0 small enough.
See [START_REF] Wang | Threshold energy resonance in geometric scattering[END_REF] for the proof of (a). To show that it is possible to differentiate the asymptotic expansion in z, one first shows that it can be done for each (Q νz) -1 , then one utilizes the properties of Bessel function to control the dependence on ν, including the remainders, and finally takes the sum in ν. See [START_REF] Jia | Some threshold spectral problems of Schrödinger operators[END_REF] for details. Remark 3.2. For the later use, let us precise a few terms in R 0 (z). By an abuse of notation, we denote by same letter an operator on L 2 (R + ; r n-1 dr) and its distribution kernel. Then one has

F ν,0 = (rτ ) -1 2 (n-2) ∞ 0 e i ρ t -i πν 2 J ν ( 1 2t ) dt 2t , ν ∈ σ ∞ , (3.16
)

G ν,0 = - e -iπν Γ(1 -ν) ν2 2ν+1 Γ(1 + ν) (rτ ) -n-2 2 +ν , ν ∈]0, 1[, (3.17)
G 1,1 = - 1 8 (rτ ) -n-2 2 +1 . (3.18)
By (3.16), one can derive that

|F ν,0 (r, τ )| ≤ C(rτ ) -n-2 2 ( rτ r 2 + τ 2 ) min{1,ν} , (3.19
)

for some C > 0 independent of ν ∈ σ ∞ .
The uniformity in ν is obtained by examining the dependence of J ν (r) on ν.

The asymptotic expansion for R 0 (z) = (P 0z) -1 can be deduced by P 0 as perturbations of P 0 . One has

R 0 (z) = (1 -F (z)) -1 R 0 (z), R(z) = (1 -F (z)) -1 R 0 (z) (3.20)
where

F (z) = R 0 (z)W, F (z) = R 0 (z) W . (3.21)
For n ≥ 3, the multiplication by 1 |x| 2 belongs to L(1, s; -1, s) for any s, by the Hardy inequality. By (3.26), the same is true for n = 2 if we define H 1,s as x -s Q( P 0 ), where Q( P 0 ) is the form-domain of P 0 . Therefore although W and W have a critical singularity -s) for s > 1 (here and in the following, H 1,s is replaced by x -s Q( P 0 ) when n = 2). Similar result holds for F (z). Note that F 0 W and F 0 W are not compact operators.

1 |x| 2 at zero, Proposition 3.1 implies that F (z) = F 0 W + O(|z| ǫ ) in L(1, -s; 1,
Definition 3.3. Set N (P ) = { u; F 0 W u = u, u ∈ H 1,-s , ∀s > 1}. A function u ∈ N (P )\L 2 is
called a resonant state of P at zero. If N (P ) = {0}, we say that 0 is the regular point of P . The multiplicity of the zero resonance of P is defined as µ r = dim N /(ker L 2 P ). Zero resonance and resonant states of P 0 are defined in the same way with W replaced by W .

For u ∈ H 1,-s for any s > 1 and u ∈ N , one can show that P u

= ( P 0 -W )u = 0 If W = O( x -ρ 0 ) with ρ 0 > 3, it is proved in [32] that u(rθ) = ν∈σ 1 nν j=1 1 2ν W u, |y| -n-2 2 +ν ϕ (j) ν ϕ (j) ν (θ) r n-2 2 +ν + u, (3.22)
where u ∈ L 2 (|x| > 1) , and (•, •) is the scalar product in L 2 (S n-1 ). In particular, (3.22) shows that the multiplicity of the zero resonance of P is bounded by the total multiplicity of eigenvalues λ ν of -∆ S n-1 + q(θ) with ν ∈ σ 1 .

For ν ∈ σ 1 , we shall say u is a ν-resonant state, or ν-bound state, of P if u ∈ N (P ) and if u has an asymptotic behavior like

u(x) = ψ(θ) r n-2 2 +ν + o( 1 r n-2 2 +ν
),

for some ψ = 0, as r → ∞. In the case n = 3 and q(θ) = 0, one has σ 1 = { 1 2 }. The only possible zero energy resonant states of P are half-bound states, which is in agreement with the usual terminology on this topic. In the general case, (3.22) shows that ψ is an eigenfunction of -∆ S n-1 + q(θ) associated with the eigenvalue λ ν . We shall say that m ν-resonant states of P , denoted as u 1 , • • • , u m , are linearly independent if

u l (x) = ψ l (θ) r n-2 2 +ν (1 + o(1)), r → ∞, with {ψ 1 , • • • , ψ m } linearly independent in L 2 (S n-1
). Let m ν denote the maximal number of linearly independent ν-resonant states of P . Then, m ν does not exceed the multiplicity of the eigenvalue λ ν of -∆ S n-1 + q(θ) and

ν∈σ 1 m ν = µ r . (3.23)
Note in particular that if u is a ν-resonant state, then one has

W u, |y| -n-2 2 +µ ϕ (j) µ = 0 (3.24)
for all µ ∈ σ 1 with µ < ν and for all j with 1 ≤ j ≤ n µ and if u is an eigenfunction of P associated with the eigenvalue 0, the above equality remains true for all ν ∈ σ 1 and all j with 1 ≤ j ≤ n µ (see (3.22) ). These properties will be repeatedly used in the calculation of the generalized residue in Section 4. Finally, remark that if m ν = 0 for some ν ∈ σ 1 , we can choose m ν ν-resonant states u l = ψ l (θ)

r n-2 2 +ν (1 + o(1)), 1 ≤ l ≤ m ν , such that {ψ l } is orthonormal in L 2 (S n-1
). Modifying the basis {ϕ (j) ν } used in the definition of the spectral projection π ν , we can assume without loss that ϕ

(l) ν = ψ l , 1 ≤ l ≤ m ν .
The model operator P 0 to be used in the next Section has the following nice threshold spectral property. Lemma 3.4. Assume n ≥ 2 and (3.2). Zero is a regular point of P 0 .

Proof. Recall that

P 0 = P 0 -W with W of compact support. Let u ∈ N (P 0 ). Let ν 0 = min σ 1 > 0, by (3.2). Since W is of compact support, W u ∈ H -1,t for any t > 0. It follows from (3.22) that u ∈ H 1,-s ′ for any s ′ > 1 -ν 0 . On the other hand, P 0 u = 0 implies that -∆u = (W -q(θ) |x| 2 )u ∈ L 2,2-s ′ .
By the assumption (3.2), one can deduce that there exists c > 0 such that

|x| -2 χ 2 u, χ 2 u ≤ c P 0 χ 2 u, χ 2 u . (3.25) In fact, let ρ ∈ C ∞ 0 (R n ) with ρ(x) = 1 for |x| ≤ 1. Set u m = ρ(x/m)χ 2 u, m ∈ N * . Then u m ∈ H 1 ,
and by the ellipticity of -∆ one has in fact u m ∈ H 2 . The assumption (3.2) implies that there exists ǫ 0 > 0 such that for f ∈ H 1 with compact support in R n \ {0}, one has

P f, f ≥ (| ∂f ∂r | 2 + (ǫ 0 - (n -2) 2 4 ) |f | 2 r 2 )r n-1 drdθ.
Making use of the Hardy inequality, one obtains for n ≥ 2

|x| -2 f, f ≤ ǫ -1 0 P 0 f, f . (3.26)
Since u ∈ H 1,-s and -∆u ∈ L 2,2-s for any s > 1ν 0 , ν 0 > 0, we can take s ∈]1ν 0 , 1[. Applying (3.26) to f = u m and taking the limit m → ∞, we derive (3.25) by noticing that the term related to [-∆, ρ(x/m)] tends to 0, due to the decay of u. (3.25) implies in particular that P 0 χ 2 u, χ 2 u ≥ 0. The equation

-∆χ 1 u, χ 1 u + P 0 χ 2 u, χ 2 u = P 0 u, u = 0
shows that each term of the above sum vanishes. The estimate (3.25) gives in turn χ 2 u = 0. Now the unique continuation theorem shows that u = 0. This proves that zero is a regular point of P 0 .

The existence of the asymptotic expansion of the resolvent R 0 (z) can easily be obtained by a method of perturbation. Concretely, let K(z) be defined by

K(z) = (χ 1 (-∆ + 1 -z) -1 χ 1 + χ 2 ( P 0 -z) -1 χ 2 )(P 0 -z) -1, z ∈ R + .
(3.27) K(z) is compact operator on H 1,-s , s > 1. By Proposition 3.1, the limit K(0) = lim z→0 K(z) exists and is compact. The kernel of 1 + K(0) in H 1,-s coincides with N (P 0 ) (both are equal to solutions to the equation P 0 u = 0, u ∈ H 1,-s )) which by Lemma 3.4 is {0}. So (1 + K(0)) -1 is invertible in H 1,-s . From Proposition 3.1, we can derive the asymptotic expansion of R 0 (z) in suitable weighted spaces from the formula

R 0 (z) = (1 + K(z)) -1 (χ 1 (-∆ + 1 -z) -1 χ 1 + χ 2 ( P 0 -z) -1 χ 2 ). (3.28)
In particular, R 0 (0) = lim z→0,z ∈R + R 0 (z) exists in L(-1, s; 1, -s), s > 1 and

R 0 (0) = (1 + K(0)) -1 (χ 1 (-∆ + 1) -1 χ 1 + χ 2 F 0 χ 2 ). (3.29)
However since K(z) contains several terms induced by cut-offs, the expansion obtained in this way is too complicated to be useful in the proof of Levinson's theorem which requires detailed information on higher order terms. For this purpose, we use the resolvent equation R 0 (z) = (1 -R 0 (z)W ) -1 R 0 (z) to obtain a more concise expansion.

Proposition 3.5. Assume (3.2), n ≥ 2 and ρ 0 > 2. (a). 1 -F 0 W is invertible on H 1,-s , s > 1. (b). Let s ∈]1, ρ 0 /2[. 1-F 0 W is a Fredholm operator on H 1,-s with indices (m, m), m = dim N (P ) < ∞. One has H 1,-s = ker(1 -F 0 W ) ⊕ ran(1 -F 0 W ). Proof. (a). Lemma 3.4 shows that 1 -F 0 W is injective. Since (1 -F 0 W ) * = 1 -W F 0 , F 0 is injective from coker (1 -F 0 W ) to ker(1 -F 0 W ). Therefore coker (1 -F 0 W ) = {0} and ran(1 -W F 0 ) is dense in H 1,-s . For any u ∈ H 1,-s , set f = u + R 0 (0)(W u)
with R 0 (0) given by (3.29). Then f ∈ H 1,-s . Since F 0 and R 0 (0) are limits of R0 (z) and R 0 (z) in L(-1, s; 1, -s), we can check that

(1 -F 0 W )f = u + (R 0 (0) -F 0 -F 0 W R 0 (0))W u = u. (3.30)
This proves that u ∈ ran(1 -F 0 W ) and 1 -F 0 W is bijective on H 1,-s . The open mapping theorem shows that 1 -F 0 W is invertible on H 1,-s . To show (b), recall that 1+R 0 (0)V is a Fredholm operator with equal indices (m, m), m = dim ker(1 + R 0 (0)V ), and that H 1,-s = ker(1 + R 0 (0)V ) ⊕ ran (1 + R 0 (0)V ) (see [START_REF] Jensen | Nenciu: A unified approach to resolvent expansions at thresholds[END_REF][START_REF] Wang | Asymptotic behavoir of resolvent for N-body Schrödinger operators near a threshold[END_REF]). By the relation P = P 0 + V = P0 -W , one can show that N (P ) = ker

P = ker(1 + R 0 (0)V ) in H 1,-s , s ∈]1, ρ 0 /2[ and that coker (1 -F 0 W ) is also of dimension m. To show that the range of 1 -F 0 W is closed, let u ∈ H 1,-s and {u n } ⊂ ran(1 -F 0 W ) such that u n → u in H 1,-s . Let v n ∈ H 1,-s such that u n = (1 -F 0 W )v n . Then P 0 u n = P v n = (P 0 + V )v n . It follows that (1 + R 0 (0)W )u n = (1 + R 0 (0)V )v n .
(3.31)

Since 1 + R 0 (0)W is continuous on H 1,-s , the left-hand side of (3.31) converges to (1+R 0 (0)W )u as n → ∞, while the right-hand side is clearly in the range of 1+R 0 (0)V . Since R 0 (0)V is compact, the range of 1 + R 0 (0)V is closed. It follows from (3.31) that there exists v ∈ H 1,-s such that

(1 + R 0 (0)W )u = (1 + R 0 (0)V )v. One can check that u = (1 -F 0 W )v ∈ ran(1 -F 0 W ), which proves that the range of (1 -F 0 W ) is closed. It follows that (1 -F 0 W
) is a Fredholm operator with equal indices. The other affirmation of (b) can be proved in the same way as in [START_REF] Wang | Asymptotic behavoir of resolvent for N-body Schrödinger operators near a threshold[END_REF].

Denote

ν = (ν 1 , • • • , ν k ) ∈ (σ N ) k , z ν = z ν 1 • • • z ν k , { ν} = k j=1 ν ′ j , [ ν] -= k j=1 [ν j ] -, [ ν] = k j=1 [ν j ].
Here ν ′ j = ν j -[ν j ] -for ν j > 0. From Propositions 3.1 and 3.5 (a) and the resolvent equation R 0 (z) = (1 -R 0 (z)W ) -1 R 0 (z), we obtain the following Proposition 3.6. The following asymptotic expansion holds for z near 0 with ℑz > 0.

(a). Let N ∈ N and s > 2N + 1. Then there exists N 0 ∈ N depending on N and min σ ∞ such that

R 0 (z) = N j=0 z j R j + (1) { ν}+j≤N z ν z j R ν,j + R (N ) 0 (z), (3.32)
in L(-1, s; 1, -s). Here the notation (l)

{ ν}+j≤N means the finite sum taken over all

{ ν} ∈ σ k N , k ≥ l, and j ≥ [ ν] -such that { ν} + j ≤ N , R 0 = AF 0 ; R 1 = AF 1 A * ; (3.33) R ν,0 = AG ν 1 ,δν 1 π ν 1 W AG ν 2 ,δν 2 π ν 2 W • • • AG ν k ,δν k π ν k A * (3.34) for ν = (ν 1 , ν 2 , • • • , ν k ) with A = (1 -F 0 W ) -1 . In particular, if k = 1 and ν = ν 1 , one has R ν,0 = AG ν 1 ,δν 1 π ν 1 A * . (3.35)
R j (resp. R ν,j ) are in L(-1, s; 1, -s) for s > 2j + 1 (resp. for s > 2j + { ν} + 1), and R

(N ) 0 (z) = O(|z| N +ǫ ) in L(-1, s; 1, -s), s > 2N + 1. (

b). The above expansion (3.32) can be differentiated in z and one has the estimate

d dz R (N ) 0 (z) = O(|z| N -1+ǫ ), (3.36)
in L(-1, s; 1, -s), s > 2N + 1, with some ǫ > 0.

For the operator P , zero may be an eigenvalue or a resonance of P . The multiplicity of zero resonance may be large, but does not exceed the sum of multiplicities of the eigenvalues λ of -∆ S n-1 + q(θ) such that ν = λ + (n-2) 2 4 ∈]0, 1]. The existence of an asymptotic expansion of R(z) can be obtained as in [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF] by regarding P as perturbation of P 0 . In order to obtain a more "concise" expansion, we regard P as perturbation of

P 0 . Let 0 < ς 1 < • • • < ς κ 0 ≤ 1 (3.37)
be the points in σ 1 such that P has m ς j linearly independent ς j -resonant states with κ 0 j=1 m ς j = µ r , µ r being the multiplicity of zero resonance of P . Modifying the basis of the eigenfunctions {ϕ (j) ν } of -∆ S n-1 + q(θ) if necessary, one can show (see (3.22) and the remarks following it) that there exists a basis of ς j -resonant states, u

(i) j , i = 1, • • • , m ς j verifying |c ς j | 1/2 W u (l) j , |x| -n-2 2 +ς j ϕ (l ′ ) ς j = δ ll ′ , 1 ≤ l ≤ m ς j , 1 ≤ l ′ ≤ n ς j , 1 ≤ j ≤ κ 0 , (3.38)
where c ν is the coefficient of G ν,δν given by

c ν = - e -iπν Γ(1 -ν) ν2 2ν+1 Γ(1 + ν) , ν ∈]0, 1[; c 1 = - 1 8 . (3.39)
(see (3.17) and (3.18)) and δ ll ′ = 1 if l = l ′ ; 0 otherwise. As seen in [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF], we can choose a basis

{φ j ; j = 1, • • • , µ} of N , µ = dim N , such that φ i , W φ j = δ ij .
Without loss, we can assume that for 1 ≤ j ≤ µ r , φ j is resonant state of P , while for µ r + 1 ≤ j ≤ µ, φ j is an eigenfunction of P . Define

Q r = µr j=1 W φ j , • φ j (3.40) Q e = µ j=µr+1 W φ j , • φ j . (3.41)
The following result can be proved by studying an appropriate Grushin problem for (1 -R 0 (z) W ) as in [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF]. See [START_REF] Jia | Some threshold spectral problems of Schrödinger operators[END_REF] for the details. Theorem 3.7. Let µ = dim N = 0 and N ∈ N. Assume ρ 0 > max{4N -2, 2N +4}. One has the following asymptotic expansion for R(z) in L(-1, s; 1, -s), s > max{2N + 1, 2}:

R(z) = N -1 j=0 z j T j + (1) { ν}+j≤N -1 z ν z j T ν;j + T e (z) + T r (z) + T er (z) + O(|z| N -1+ǫ ) (3.42) Here T j (resp., T ν,j ) is in L(1, -s; -1, s) for s > 2j + 1 (resp., for s > 2j + 1 + { ν}), T 0 = AF 0 , T 1 = AF 1 (1 + W AF 0 ) with A = (Π ′ (1 -F 0 W )Π ′ ) -1 Π ′ , Π ′ is the projection from H 1,-s onto ran (1 -F 0 W ) corresponding to the decomposition H 1,-s = ker(1 -F 0 W ) ⊕ ran(1 -F 0 W ). The sum (1)
{ ν}+j≤N has the same meaning as in (3.32) and the first term in this sum is z ν 0 with coefficient T ν 0 ,0 given by

T ν 0 ,0 = AG ν 0 ,δν 0 π ν 0 (1 + W AF 0 ),
where ν 0 is the smallest value of ν ∈ σ ∞ . T e (z), T r (z) describe the contributions up to the order O(|z| N -1+ǫ ) from eigenfunctions and resonant states, respectively, and T er (z) the interaction between eigenfunctions and resonant states. One has

T e (z) = -z -1 Π 0 + (-) j, { ν}+j≤N -1 z ν z j T e; ν;j T r (z) = κ 0 j=1 z -1 ς j (Π r,j + +,N α,β, ν,l z ν z |β| (z ς ) -α-β z l T r; ν,α,β,l,j ), with Π r,j = e iπς ′ j mς j l=1 •, u (l) j u (l) j , j = 1, • • • , κ 0 , T er (z) = κ 0 j=1 z -1 ς j (Π 0 W Q e F 1 W Π r,j + Π r,j W Q r F 1 W Π 0 + +,N α,β, ν,l z ν z |β| (z ς ) -α-β z l T er; ν,α,β,l,j ).
Here ς ′ j = ς j -[ς j ], Π 0 is the spectral projection of P at 0, and T e (z) is of rank not exceeding Rank Π 0 with leading singular parts given by ν j ∈ σ 2 :

T e; ν;-1 = (-1) k ′ +1 (Π 0 W G ν 1 ,1+δν 1 π ν 1 W ) • • • (Π 0 W G ν k ′ ,1+δν k ′ π ν k ′ W )Π 0 , (3.43) for ν = (ν 1 , • • • , ν k ′ ) ∈ σ k ′ 2 with { ν} ≤ 1, (z ς ) -α = (z ς 1 ) -α 1 • • • (z ςκ 0 ) -ακ 0 . The sum- mation (-) j, { ν}+j≤N -1 is taken over all indices ν ∈ (σ N ) k , k ∈ N *
and all j ∈ Z with j ≥ [ ν] --1 such that { ν} + j ≤ N -1, and the summation +,N α,β, ν,l is taken over all possible α, β

∈ N κ 0 with 1 ≤ |α| ≤ N 0 , |β| ≥ 1, ν = (ν 1 , • • • , ν k ′ ) ∈ σ k ′ N , k ′ ≥ 2|α|
, for which there are at least α k values of ν j 's belonging to σ 1 with ν j ≥ ς k , for 1 ≤ k ≤ κ 0 , l ∈ N, satisfying

|β| + { ν} + l - κ 0 k=1 (α k + β k )ς k ≤ N.
This theorem is proved [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF] for the Schrödinger operator of the form P = P 0 + V with V bounded and satisfying |V | ≤ C x -ρ 0 . In the present work, V = -χ 2 1 r 2 q(θ) + W has a critical singularity at 0. So we can not directly use the result of [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF], but with the help of Proposition 3.5, one can follow the same line of proof to obtain Theorem 3.7 (see [START_REF] Jia | Some threshold spectral problems of Schrödinger operators[END_REF]). With the sign correction on c 1 and the formula (4.32) in [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF], one sees that a 1-resonant state u (l) j gives rise to a singularity of the leading term

1 z ln z •, u (l) j u (l) j instead of -1 z ln z •, u (l) j u (l)
j as stated in Theorem 4.6 of [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF].

Generalized residue of the trace function

Let f be a function satisfying the condition of Theorem 2.1. As we have seen before, (R(z) -R 0 (z))f (P ) is of trace class for z / ∈ σ(P ). Moreover, the application z → T (z) = Tr [(R(z) -R 0 (z))f (P )] is meromorphic on C\R + . The goal of this section is to calculate the generalized residue of T (z) at z = 0, in the sense of Section 2.

First, let us recall some well-known results for the trace ideals S p ⊂ L(L 2 (R n )), (see [START_REF] Reed | Methods of modern mathematical physics, II. Fourier Analysis, Self-adjontness[END_REF]IX.4] for details). By Proposition 3.6, for z near 0 with ℑz > 0, one has for s > 1 :

R 0 (z) = R 0 + R (0) 0 (z) in L(-1, s; 1, -s), (4.1)
with

R (0) 0 (z) = O(| z | ǫ ).
We deduce the following result :

Lemma 4.3. For m > n/2, s > 3 and z near 0, ℑz > 0, x -s R 0 (z) x -s ∈ S m and there exists a constant C of z, such that

|| x -s R 0 (z) x -s || m ≤ C. Moreover, x -s (R(z) -R 0 ) x -s = O(|z| ǫ ) in S m .
Proof. Let χ ∈ C ∞ 0 (R) such that χ(r) = 1 for |r| < 1. Then x -s R 0 (z) x -s can be written as

x -s R 0 (z) x -s = F 1 (z) + F 2 (z), with F 1 (z) = x -s R 0 (z)χ(P 0 ) x -s ; F 2 (z) = x -s R 0 (z)(1 -χ(P 0 )) x -s .
First, let us study F 2 (z). Using the resolvent identity, we can decompose F 2 (z) = F 21 + F 22 (z), where

F 21 = x -s R 0 (-1)(1 -χ(P 0 )) x -s , F 22 (z) = (1 + z) ( x -s R 0 (z) x -s ′ )( x s ′ R 0 (-1)(1 -χ(P 0 )) x -s ).
It is easy to check that F 21 and x s ′ R 0 (-1)(1χ(P 0 )) x -s are in S m , if s ′ > 1 is chosen close to 1. By (4.1), it is clear that x -s R 0 (z) x -s ′ is uniformly bounded, for z near 0, ℑz > 0. Then we deduce that F 2 (z) ∈ S m , and ||F 2 (z)|| m ≤ C with some constant C independent of z.

Now, let us study F 1 (z). We write F 1 (z) = x -s R 0 (z) x -s ′ ( x s ′ χ(P 0 ) x -s ) with s ′ > 1 close to 1. Using a similar argument as above, we can get that ||F 1 (z)|| m ≤ C with some constant C independent of z. Therefore x -s R 0 (z) x -s ∈ S m , and for some C independent of z,

|| x -s R 0 (z) x -s || m ≤ ||F 1 (z)|| m + ||F 2 (z)|| m ≤ C.
Repeating the same arguments with R 0 (z) replaced by R 0 (z) -R 0 , we obtain that

x -s (R(z) -R 0 ) x -s = O(|z| ǫ ) in S m .
We can now establish the main result of this section : Theorem 4.4. Assume that ρ > max{6, n + 2} and f satisfies the condition of Theorem 2.1. Then the generalized residue of T (z) = Tr [(R(z) -R 0 (z))f (P )] at z = 0 is given by

J 0 = N 0 + κ 0 j=1 ς j m j , (4.2)
where N 0 is the multiplicity of zero as the eigenvalue of P and m j the multiplicity of ς j -resonance of zero.

Proof. The proof is rather long and is divided in two steps. We write

f ≈ g if f (z) - g(z) = O(|z| -1+ǫ
) for some ǫ > 0, with z in a neighborhood of 0 and ℑz > 0.

We fix k ∈ N with k > n 2 + 1 and we use the following resolvent identities :

R(z) -R 0 (z) = -R 0 (z)V R(z) = k-1 j=1 (-1) j (R 0 (z)V ) j R 0 (z) + (-1) k R(z)(V R 0 (z)) k . Decompose T (z) = -Tr [R 0 (z)V f (P )R(z)] as T (z) = T 1 (z) + T 2 (z) where T 1 (z) = -Tr [R 0 (z)V f (P )(R 0 (z) + k-1 j=1 (-1) j (R 0 (z)V ) j R 0 (z))]. (4.3) T 2 (z) = (-1) k+1 Tr [R 0 (z)V f (P )R(z)(V R 0 (z)) k )]. (4.4)
First, let us study T 1 (z). Using the cyclicity of the trace, one has, for s > 1, close to 1,

Tr [R 0 (z)V f (P )R 0 (z)] = Tr [R 2 0 (z)V f (P )] = Tr[( x -s d dz R 0 (z) x -s ) ( x s V f (P ) x s )].
Since ρ > n + 2, x s V f (P ) x s ∈ S 1 . So, by Proposition 3.6, we deduce that

Tr [R 0 (z)V f (P )R 0 (z)] ≈ 0. (4.5)
Similarly, we can get that the other terms of T 1 (z) give the same contribution. Therefore, we have obtained :

T 1 (z) ≈ 0. (4.6) Now, let us study T 2 (z). Since k > n 2 + 1 > n 2 -2, we can define T 3 (z) for z / ∈ σ(P ) by : T 3 (z) = (-1) k+1 Tr [R 0 (z)V R(z)(V R 0 (z)) k )]. (4.7)
Then, for s > 1 close to 1, we have as previously,

T 2 (z) -T 3 (z) = (-1) k Tr [R 0 (z)V R(z)(1 -f (P ))(V R 0 (z)) k )]. = (-1) k Tr [( x -s d dz R 0 (z) x -s ) x s V (R(z)(1 -f (P ))) (V R 0 (z)) k-1 V x s ].
Since 1f (t) is equal to 0 for t near 0, R(z)(1f (P )) is uniformly bounded for z in a neighborhood of 0. Moreover, since ρ 0 > 6 and k > n 2 + 1, Proposition 4.2 (a) and Lemma 4.3 imply that, for z near 0 and z / ∈ R + ,

|| (V R 0 (z)) k-1 V x s || 1 = O(1). (4.8)
By Proposition 3.6 (b), we conclude that, for some ǫ > 0 and z near 0 with ℑz > 0,

T 2 (z) -T 3 (z) ≈ 0. (4.9)
The main task of the proof is to estimate T 3 (z) at 0.

• Step 1: Assume that 0 is not an eigenvalue of P .

Using the cyclicity of the trace, we remark that

T 3 (z) = (-1) k+1 Tr [( d dz R 0 (z))V R(z)V (R 0 (z)V ) k-1 ]. (4.10)
Let us introduce the following notation :

U 1 = sgn(V ) |V | 1 2 , U 2 = |V | 1 2 , S 1 (z) = U 1 d dz R 0 (z)U 2 , S 2 (z) = U 1 R(z)U 2 , S 3 (z) = U 1 R 0 (z)U 2 .
Then,

T 3 (z) = (-1) k+1 Tr [S 1 (z)S 2 (z)S k-1 3 (z)]. (4.11)
As previously, by Lemma 4.3, we have for k > n 2 + 1, ρ 0 > 6 and z near 0 with ℑ(z) > 0,

||S k-1 3 (z)|| 1 = O(1). (4.12)
By Proposition 3.6 (b), with N = 1,

d dz R 0 (z) = R 1 + (1)
{ ν}≤1 d dz z ν R ν,0 + O(|z| ǫ ), (4.13)
in L(-1, s; 1, -s), for s > 3 and some ǫ > 0. Since ρ 0 > 6, we deduce that

S 1 (z) = S 11 (z) + S 12 (z), (4.14)
with

S 11 (z) = U 1 R 1 U 2 + (1)
{ ν}≤1 d dz z ν U 1 R ν,0 U 2 (4.15)
and

S 12 (z) = O(|z| ǫ ) in L(L 2 (R n )). (4.16)
In the same way, we can use Theorem 3.7, with N = 1 (and ρ 0 > 6) to decompose S 2 (z). Note that T e (z) = 0, T er (z) = 0 since 0 is not an eigenvalue of P . So we have,

S 2 (z) = S 21 (z) + S 22 (z) + S 23 (z) (4.17)
with

S 21 (z) = κ 0 j=1 z -1 ς j U 1 Π r,j U 2 , (4.18)
S 22 (z) = κ 0 j=1 +,1 α,β, ν,l z -1 ς j z ν z |β| (z ς ) -α-β z l U 1 T r; ν,α,β,l,j U 2 , (4.19)
S 23 (z) = O(1) in L(L 2 (R n )). (4.20)
It follows from the previous discussion that

T 3 (z) = T 31 (z) + T 32 (z) + T 33 (z) (4.21)
where

T 31 (z) = (-1) k+1 Tr [S 11 (z)S 21 (z)S k-1 3 (z)], (4.22)
T 32 (z) = (-1) k+1 Tr [S 11 (z)S 22 (z)S k-1 3 (z)], (4.23)
T 33 (z) = (-1) k+1 Tr [(S 11 (z)S 23 (z) + S 12 (z)S 21 (z) +S 12 (z)S 22 (z) + S 12 (z)S 23 (z))S k-1 3 (z)]. (4.24)
First, let us study T 33 (z). We have the following result :

Lemma 4.5. T 33 (z) ≈ 0. (4.25)
Proof. Clearly, we have for some ǫ > 0,

S 11 (z) = O(|z| -1+ǫ ) , S 21 (z) = O(|z ln z| -1 ) in L(L 2 (R n )). (4.26)
In the same way, we want to estimate S 22 (z). We note that the summation +,1 α,β, ν,l is taken over all possible α, β

∈ N κ 0 with 1 ≤ |α| ≤ N 0 , |β| ≥ 1, ν = (ν 1 , • • • , ν k ′ ) ∈ σ k ′
2 , k ′ ≥ 2|α|, for which there are at least α k values of ν j 's belonging to σ 1 with ν j ≥ ς k , for 1 ≤ k ≤ κ 0 and l ∈ N, satisfying the condition

|β| + { ν} + l - κ 0 k=1 (α k + β k )ς k ≤ 1. It follows that z |β| (z ς ) -β = O(1) , z ν (z ς ) -α = O(|z| ǫ ). (4.27)
We emphasize that in the second estimate of (4.27), we have used k ′ ≥ 2|α|. Then, we have obtained,

S 22 (z) ≈ 0 in L(L 2 (R n )). (4.28)
Then, the lemma follows from (4.12), (4.16), (4.20), (4.26) and (4.28).

The following elementary lemma will be useful to estimate T 31 (z) and T 32 (z).

Lemma 4.6. Let u ∈ N (P ), A = (1 -F 0 W ) -1 . Then A * V u = -W u , R 0 V u = -u. Proof. By definition, if u ∈ N (P ), F 0 W u = u. Thus, V u = (W -W )u = -(1-W F 0 ) W u
which proves the first equality. Using the same argument, if P u = (P 0 + V )u = 0 then Proposition 3.6 implies that R 0 V u = -u.

Now, let us study T 31 (z); we shall see that T 31 (z) gives the leading term of the generalized residue. We denote δ ij the Kronecker symbol :

δ ij = 1 if i = j, δ ij = 0 otherwise. Lemma 4.7. T 31 (z) ≈ - 1 z κ 0 j=1 ς j m ς j - δ 1,ςκ 0 z ln z m 1 - m 1 l=1 < R 1 V u (l) κ 0 , V u (l) κ 0 > . (4.29)
Proof. It follows from (4.22) that

(-1) k+1 T 31 (z) = Tr [U 1 R 1 V κ 0 j=1 z -1 ς j Π r,j U 2 S k-1 3 (z)] + Tr [ (1)
{ ν}≤1 d dz z ν U 1 R ν,0 V κ 0 j=1 z -1 ς j Π r,j U 2 S k-1 3 (z)] = (a) + (b).
First, let us study (b). As we will see, the cases ς κ 0 < 1 and ς κ 0 = 1 are slightly different.

• Case 1 : Assume that ς κ 0 < 1. It follows that ς ′ j = ς j , and for j = 1, ..., κ 0 ,

Π r,j = e iπς j mς j l=1 < • , u (l) j > u (l) j .
We deduce that

(b) = κ 0 j=1 mς j l=1 (1)
{ ν}≤1 e iπς j z -1

ς j d dz z ν Tr [U 1 R ν,0 V U 2 S k-1 3 (z)• , u (l) j u (l) j ]
=

κ 0 j=1 mς j l=1 (1)
{ ν}≤1 e iπς j z -1

ς j d dz z ν Tr [U 1 R ν,0 V • , (S k-1 3 (z)) * U 2 u (l) j u (l) j ] = κ 0 j=1 mς j l=1 (1)
{ ν}≤1 e iπς j z -1

ς j d dz z ν U 1 R ν,0 V u (l) j , (S k-1 3 (z)) * U 2 u (l) j = κ 0 j=1 mς j l=1 (1)
{ ν}≤1 e iπς j z -1

ς j d dz z ν R ν,0 V u (l) j , U 1 (S k-1 3 (z)) * U 2 u (l) j .
Using (3.34) with ν = (ν 1 , . . . , ν p ), we have

R ν,0 V u (l) j = AG ν 1 ,δν 1 π ν 1 W AG ν 2 ,δν 2 π ν 2 W • • • AG νp,δν p π νp A * V u (l) j . (4.30)
Thus, Lemma 4.6 implies

R ν,0 V u (l) j = -AG ν 1 ,δν 1 π ν 1 W AG ν 2 ,δν 2 π ν 2 W • • • AG νp,δν p π νp W u (l) j . (4.31)
It is easy to see that for ν = (ν 1 , . . . , ν p ) with

p i=1 ν i > ς j , one has z -1 ς j d dz z ν = O(|z| -1+ǫ
) for some ǫ > 0. It follows from (4.12) that it suffices to evaluate (b) for ν = (ν 1 , . . . , ν p )

with p i=1 ν i ≤ ς j , ⋄ Assume first ∃i ∈ {1, • • • , p} such that ν i = ς j .
One deduces that p = 1, i.e ν = ν 1 . Moreover, since ς κ 0 < 1, one has ν 1 = ς j < 1. Using (3.35), we obtain R ν,0 V u

(l) j = -AG ς j ,0 π ς j W u (l) j . (4.32)
Recalling that

π ς j = nς j l ′ =1 (• , ϕ (l ′ ) ς j ) ⊗ ϕ (l ′ ) ς j ,
and using (3.17) and (3.39), we deduce

G ς j ,0 π ς j W u (l) j = nς j l ′ =1 +∞ 0 G ς j ,0 (r, τ ) ( W u (l) j , ϕ (l ′ ) ς j )ϕ (l ′ ) ς j τ n-1 dτ = c ς j r -n-2 2 +ς j nς j l ′ =1 W u (l) j , | x | -n-2 2 +ς j ϕ (l ′ ) ς j ϕ (l ′ ) ς j . (4.33)
It follows from the normalization condition of resonant states given in (3.38) that

G ς j ,0 π ς j W u (l) j = -e -iπς j | c ς j | 1 2 r -n-2 2 +ς j ϕ (l) ς j . (4.34)
Thus, in this case, we have

R ν,0 V u (l) j = e -iπς j | c ς j | 1 2 A | x | -n-2 2 +ς j ϕ (l) ς j . (4.35)
⋄ Assume now that ∀i ∈ {1, • • • , p}, ν i < ς j .

In particular ν p < 1 and using (3.24), we obtain

G νp,0 π νp W u (l) j = G νp,0 nν p l ′ =1 ( W u (l) j , ϕ (l ′ ) νp ) ⊗ ϕ (l ′ ) νp (4.36) = c νp r -n-2 2 +νp nν p l ′ =1 W u (l) j , | x | -n-2 2 +νp ϕ (l ′ ) νp ϕ (l ′ ) νp (4.37) = 0. (4.38)
Thus, in this case, using (4.31), we have

R ν,0 V u (l) j = 0. (4.39)
As a conclusion of (4.35) and (4.39), we obtain

(b) ≈ κ 0 j=1 mς j l=1 | c ς j | 1 2 z -1 ς j d dz (z ς j ) A(| x | -n-2 2 +ς j ϕ (l) ς j ), U 1 (S k-1 3 (z)) * U 2 u (l) j . (4.40)
By Proposition 3.6, (S k-1

3 (z)) * = (U 2 R 0 U 1 ) k-1 + O(|z| ǫ ) in L 2 (R n ). Moreover, since ς j < 1, z -1 ς j d dz (z ς j ) = ς j z . Thus, (b) ≈ 1 z κ 0 j=1 mς j l=1 | c ς j | 1 2 ς j A(| x | -n-2 2 +ς j ϕ (l) ς j ), U 1 (U 2 R 0 U 1 ) k-1 U 2 u (l) j ≈ 1 z κ 0 j=1 mς j l=1 | c ς j | 1 2 ς j A(| x | -n-2 2 +ς j ϕ (l) ς j ), V (R 0 V ) k-1 u (l) j ≈ (-1) k-1 z κ 0 j=1 mς j l=1 | c ς j | 1 2 ς j A(| x | -n-2 2 +ς j ϕ (l) ς j ), V u (l) j ≈ (-1) k z κ 0 j=1 mς j l=1 | c ς j | 1 2 ς j | x | -n-2 2 +ς j ϕ (l) ς j , W u (l) j ,
where we have used Lemma 4.6 in the two last equations. Using again the normalization condition of resonant states given in (3.38), we obtain :

(b) ≈ (-1) k z κ 0 j=1 mς j l=1 ς j = (-1) k z κ 0 j=1 m ς j ς j . (4.41)
• Case 2 : assume that ς κ 0 = 1.

In this case, ς ′ j = ς j for j = 1, ..., κ 0 -1 and ς ′ κ 0 = 0. So we can write,

(b) = κ 0 -1 j=1 mς j l=1 (1)
{ ν}≤1

e iπς j z -1

ς j d dz z ν R ν,0 V u (l) j , U 1 (S k-1 3 (z)) * U 2 u (l) j + (b ′ ), (4.42)
where we set

(b ′ ) = m 1 l=1 (1)
{ ν}≤1

1 z ln z d dz z ν R ν,0 V u (l) κ 0 , U 1 (S k-1 3 (z)) * U 2 u (l) κ 0 . (4.43)
The first part of (b) in (4.42) can be calculated exactly as in the case 1. Thus, one has

(b) ≈ (-1) k z κ 0 -1 j=1 m ς j ς j + (b ′ ). (4.44)
Now, let us study (b ′ ). Using the same approach, an easy calculus gives

(b ′ ) ≈ (-1) k-1 m 1 l=1 1 z ln z d dz (z ln z) R 1,0 V u (l) κ 0 , V u (l) κ 0 . (4.45)
Using (3.35), we have

R 1,0 V u (l) κ 0 = AG 1,1 π 1 A * V u (l) κ 0 = -AG 1,1 π 1 W u (l) κ 0 . (4.46)
Recalling that

π 1 = n 1 l ′ =1 (• , ϕ (l ′ ) 1 ) ⊗ ϕ (l ′ ) 1 ,
we deduce

G 1,1 π 1 W u (l) κ 0 = n 1 l ′ =1 +∞ 0 G 1,1 (r, τ )( W u (l) κ 0 , ϕ (l ′ ) 1 )ϕ (l ′ ) 1 τ n-1 dτ = c 1 r -n-2 2 +1 n 1 l ′ =1 W u (l) κ 0 , | x | -n-2 2 +1 ϕ (l ′ ) 1 ϕ (l ′ ) 1 ,
where we have used (3.18) and (3.39) in the last equation. As previously, it follows from the normalization condition of resonant states that

G 1,1 π 1 W u (l) κ 0 = -| c 1 | 1 2 r -n-2 2 +1 ϕ (l) 1 . (4.47) Thus, R 1,0 V u (l) κ 0 =| c 1 | 1 2 A | x | -n-2 2 +1 ϕ (l) 1 . (4.48) It follows that (b ′ ) ≈ (-1) k-1 m 1 l=1 1 z ln z d dz (z ln z) | c 1 | 1 2 A | x | -n-2 2 +1 ϕ (l) 1 , V u (l) κ 0 ≈ (-1) k m 1 l=1 1 z ln z d dz (z ln z) | c 1 | 1 2 | x | -n-2 2 +1 ϕ (l) 1 , W u (l) κ 0 ≈ (-1) k m 1 1 z + 1 z ln z ,
where we have used again the normalization condition of resonant states. Thus, in this case, we have obtained

(b) ≈ (-1) k z κ 0 -1 j=1 m ς j ς j + (-1) k m 1 1 z + 1 z ln z . ( 4
k 0 j=1 mς j l=1 e iπς ′ j z -1 ς j R 1 V u (l) j , V u (l) j (4.51) If ς κ 0 < 1, it is clear that (a) is negligible. If ς κ 0 = 1, we obtain (a) ≈ (-1) k-1 z ln z m 1 l=1 R 1 V u (l) κ 0 , V u (l) κ 0 , (4.52)
and the lemma is proved.

For T 32 (z), we have the following Lemma 4.8.

T 32 (z) ≈ 0. (4.53)

Proof. Using (4.26) and (4.28), we see that it suffices to prove that the following term is negligible :

(c) = κ 0 j=1 (1)
{ ν}≤1

+,1 α,β, ν 1 ,l z -1 ς j z ν 1 z |β| (z ς ) -α-β z l d dz z ν Tr [U 1 R ν,0 V T r; ν 1 ,α,β,l,j U 2 S k-1 3 (z)].
First, let us remark that if p i=1 ν i ≥ ς j , the same argument as (4.27) implies that

z -1 ς j z ν 1 z |β| (z ς ) -α-β z l d dz z ν ≈ 0. (4.54)
So, it remains to estimate (c) when ν = (ν 1 , ..., ν p ) satisfies p i=1 ν i < ς j . From the proof of Theorem 3.7, we know that T r; ν 1 ,α,β,l,j is a linear combination of Π r,j B r; ν 1 ,α,β,l,j and A r; ν 1 ,α,β,l,j Π r,j W G µ,δµ π µ B r; ν 1 ,α,β,l,j , (

where A r; ν 1 ,α,β,l,j is a bounded operator in L(1, -s; 1, -s), s > 3 and B r; ν 1 ,α,β,l,j is a bounded operator in L(-1, s; -1, s) for s > 3.

• Let us study the contribution coming from Π r,j B r; ν 1 ,α,β,l,j .

To simplify the notation, we write B = B r; ν 1 ,α,β,l,j . In this case, as previously, we have

Tr [U 1 R ν,0 V Π r,j BU 2 S k-1 3 (z)] = mς j l=1 e iπς ′ j R ν,0 V u (l) j , U 1 (S k-1 3 (z)) * U 2 B * u (l) j . (4.56)
Since ν p < ς j , as in (4.39), we obtain R ν,0 V u (l) j = 0.

• Now, let us study the contribution coming from A r; ν 1 ,α,β,l,j Π r,j W G µ,δµ π µ B r; ν 1 ,α,β,l,j .

We remark that, if µ < ς j , the same argument as before and (3.24) imply Π r,j W G µ,δµ π µ = 0. Thus, it suffices to study the case µ ≥ ς j . To do this, we refer to the proof of Theorem 4.6 in [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF]. The term G µ,δµ π µ B r; ν 1 ,α,β,l,j comes from the expansion of L 1 (z) W (D 0 + D 1 (z)) R 0 (z) with D 0 ∈ L(1, -s; 1, -s) for any s > 1 and D 0 | N = 0 and

L 1 (z) = µ∈σ 1 z µ G µ,δµ π µ D 1 (z) = zD 1 + µ∈σ 1 z µ D µ,0
for some D 1 and D µ,0 in L(1, -s; 1, -s) with s > 3, so the coefficient in front of G µ,δµ π µ B r; ν 1 ,α,β,l,j is O(z µ ). In the same way, the term A r; ν 1 ,α,β,l,j Π r,j W comes from the the expansion of (1

-(D 0 + D 1 (z))L 1 (z) W )I r (z)Q r -Π r (z) W Q r ,
where I r (z) is of the form

I r (z) = (1 + O(|z| ǫ ))Π r (z) W , Π r (z) = κ 0 j=1 1 z ς j Π r,j
with Π r,j defined in Theorem 3.7 (see p. 1931 of [START_REF] Wang | Asymptotic expansion in time of the Schrödinger group on conical manifolds[END_REF]), so the coefficient in front of A r; ν 1 ,α,β,l,j Π r,j W is O(|z| -ς j +ǫ ). It follows that the coefficient of T r; ν 1 ,α,β,l,j in Theorem 3.7 is O(|z| ǫ+µ-ς j ) = O(|z| ǫ ). As a conclusion, (c) = O(|z| -1+ǫ ) and the lemma is proved.

• Step 2 : Assume that 0 is an eigenvalue of P .

We follow the same strategy. S 1 (z), S 2 (z), S 3 (z), S 11 (z), S 12 (z) are the same as before. Using Theorem 3.7 with N = 1, S 2 (z) can be decomposed as

S 2 (z) = U 1 R(z)U 2 = 5 j=1 S 2j (z), (4.57)
with

S 21 (z) = U 1 T r (z)U 2 S 22 (z) = - 1 z U 1 Π 0 U 2 S 23 (z) = (-) j, { ν}+j≤0 z ν z j U 1 T e; ν;j U 2 S 24 (z) = U 1 T er (z)U 2 S 25 (z) = O(1).
Thus, it follows that It follows from (4.12), (4.16) and (4.26) that T 35 (z) ≈ 0. Now, let us establish the following lemma which will be useful to estimate the other terms.

Lemma 4.9. For ν = (ν 1 , ..., ν p ) ∈ (σ 1 ) p , one has :

R ν,0 V Π 0 = 0 , Π 0 V R ν,0 = 0. ( 4

.61)

Proof. We only prove the first assertion since the other one is similar. Let Ψ j , j = 1, ..., N 0 , be an orthonormal basis of the eigenspace of P with eigenvalue 0. As previously, one has

R ν,0 V Φ j = -AG ν 1 ,δν 1 π ν 1 W AG ν 2 ,δν 2 π ν 2 W • • • AG νp,δν p π νp W Φ j , (4.62)
Now, let assume that ς k 0 = 1. We follow the same strategy as in the case ς κ 0 < 1. It is easy to see that, in this case,

T 34 (z) ≈ (-1) k+1 z ln z Tr [U 1 R 1 V (Π 0 W Q e F 1 W Π r,κ 0 + Π r,κ 0 W Q r F 1 W Π 0 )U 2 S k-1 3 (z)] ≈ (-1) k+1 z ln z Tr [R 1 V (Π 0 W Q e F 1 W Π r,κ 0 + Π r,κ 0 W Q r F 1 W Π 0 )(V R 0 ) k-1 V ].
Then, the lemma comes from (4.69).

Finally, T 31 (z) has been computed in the case where 0 is not an eigenvalue of P .

End of the proof of Theorem 4.4.

As a conclusion, it follows from the above discussion that in all the cases, one has

T (z) ≈ - 1 z   κ 0 j=1 ς j m ς j + N 0   + C z ln z , (4.70)
for some constant C and for all z near 0 with ℑz > 0. Using the relation T (z) = T (z) for ℑz < 0, we deduce that C is real and (4.70) holds for z near 0 and ℑz = 0. Recall that the generalized residue is defined as

J 0 = - 1 2πi lim δ→0 lim ǫ→0 γ(δ,ǫ)
T (z) dz if the limit exists, where γ(δ, ǫ) is positively oriented. We obtain from (4.70) that

- 1 2πi γ(δ,ǫ) T (z) dz = κ 0 j=1 ς j m ς j + N 0 + O( ǫ δ ) + O( 1 | log δ| ),
uniformly in 0 < ǫ << δ. Taking first the limit ǫ → 0, then the limit δ → 0, one derives J 0 = κ 0 j=1 ς j m ς j + N 0 . This proves Theorem 4.4.

Proof of Levinson's theorem

In this section, we shall use Theorem 2.1 to prove a Levinson's theorem for potentials with critical decay. First, we have to verify that the conditions (2.1), and (2.4) -(2.9) hold. In the last section, we have seen that (2.8) is satisfied if ρ 0 > max(6, n + 2).

Under the assumption (1.4), the Hamiltonian P 0 is positive, and it is well known that P and P 0 have no embedded positive eigenvalues, and the spectrum of P and P 0 is purely absolutely continuous on ]0, +∞[. Using Theorem 3.7 with N = 0, one has for s > 2, z small with ℑz > 0,

|| x -s R(z) x -s ||≤ C | z | (5.1)
We deduce that the negative eigenvalues of P can not accumulate at 0. Indeed, let ϕ be an eigenfunction of P associated with an eigenvalue λ ∈ [-δ, 0[, δ small. It is well known that that ϕ decays exponentially. So, using (5.1) with z = λ + iǫ, we obtain

1 ǫ || x -s ϕ ||≤ C | λ + iǫ | || x s ϕ ||, (5.2)
which gives the contradiction when ǫ → 0.

At least, for ρ 0 > n and k > n 2 , it is well known that (2.1) holds, (see for example, [START_REF] Robert | Relative time-delay for perturbations of elliptic operators and semi-classical asymptotics[END_REF], Theorems 1.1 -1.2). Now, let us prove the following elementary lemma in order to verify (2.6) : Lemma 5.1. Assume ρ 0 > n. Then, for every f ∈ C ∞ 0 (R), there exists C > 0, such that :

| Tr [(R(z) -R 0 (z))f (P )] | ≤ C |z| 2 ,
uniformly in z ∈ C with |z| large and z / ∈ σ(P ).

Proof. For z / ∈ σ(P ), using the resolvent identity, we have :

(R(z) -R 0 (z))f (P ) = -R(z)V R 0 (z)f (P ) = -R(z)V R(z)f (P ) -R(z)V R 0 (z)V R(z)f (P ). Let f 1 ∈ C ∞ 0 such that f 1 f = f .
Using the cyclicity of the trace, we obtain :

Tr [(R(z) -R 0 (z))f (P )] = (1) + (2), with (1)
= -Tr[(R(z)f 1 (P )) (f 1 (P )V ) (R(z)f (P ))], (5.3)
(2) = -Tr[(R(z)f 1 (P )) (f 1 (P ) x -ρ 0 2 ) ( x ρ 0 2 V R 0 (z)V x ρ 0 2 ) ( x -ρ 0 2 f 1 (P )) (R(z)f (P ))]. (5.4)
For |z| large, ℑz = 0, we have ( [START_REF] Robert | Asymptotique de la phase de diffusion a haute energie pour des perturbations du second ordre du laplacien[END_REF]),

|| x ρ 0 2 V R 0 (z)V x ρ 0 2 ||= O(|z| -1/2 ).
Moreover, by the spectral theorem,

|| R(z)f (P ) ||= 1 dist(z, Supp f ) = O( 1 | z |
).

The lemma follows now from the fact that x -ρ 0 2 f 1 (P ) is a Hilbert-Schmidt operator and f 1 (P )V is of trace class.

In the same way, using Theorem 3.6 with N = 0, one has for z / ∈ σ(P 0 ), | z | small, and

s > 1, ρ 0 > n + 2, Tr[R 0 (z)(f (P ) -f (P 0 ))] = Tr[ x -s (R 0 + O(| z | ǫ )) x -s x s (f (P ) -f (P 0 )) x s ] = O(1).
(5.5)

Then, the assumption (2.7) is satisfied.

In order to obtain a Levinson theorem, we have also to prove that ξ ′ (λ) is integrable on ]0, 1]. We have the following result :

Theorem 5.2. Under the conditions of Theorem 4.4, for some ǫ 0 > 0, ξ ′ (λ) = O λ -1+ǫ 0 , λ ↓ 0.

(5.6)

Proof. We shall prove that This implies in particular that the condition (2.9) is satisfied. We also need the following asymptotics which comes from the functional calculus on pseudodifferential operators (see [START_REF] Robert | Relative time-delay for perturbations of elliptic operators and semi-classical asymptotics[END_REF], Theorem 1.1): Let χ ∈ S(R), with χ ≡ 1 in a neighborhood of 0. We have :

λ 1-ǫ 0 ξ ′ (λ) ∈ L ∞ (]0, 1[) = L 1 (]0, 1[) ′ , (5.7
Tr [χ( P R ) -χ( P 0 R )] ∼ j≥1 β j R n 2 -j , R → +∞, (5.25)
where the coefficients β j are distributions on χ and is calculable, in principle, in terms of the symbols of P and P 0 .

The main result of this section is the following :

Theorem 5.3. Assume ρ 0 > max(6, n + 2). Then, we have : Proof. Let χ ∈ C ∞ 0 (R) such that χ(λ) ≡ 1 in a neighborhood of 0. We use Theorem 2.1 with f (λ) = χ( λ R ). The proof is divided in two steps.

Case 1. The dimension n is odd.

We first compute the second term on the left hand side of (2.10) as follows : Now, let us study the left hand side of (5.27). We have :

∞ 0 χ( λ R ) ξ ′ (λ) dλ = ∞ 0 χ( λ R ) [ξ ′ (λ) - [ n 2 ] j=1 c j λ n 2 -1-j ] dλ + [ n 2 ] j=1 c j ∞ 0 χ( λ R ) λ n 2 -1-j dλ = ∞ 0 χ( λ R ) [ξ ′ (λ) -
∞ 0 χ( λ R ) [ξ ′ (λ) - [ n 2 ] j=1 c j λ n 2 -1-j ] dλ = 1 0 χ( λ R ) [ξ ′ (λ) - [ n 2 ] j=1 c j λ n 2 -1-j ] dλ + ∞ 1 χ( λ R ) [ξ ′ (λ) -

Definition 4 . 1 .

 41 For 1 ≤ p < ∞, we say that a compact operator A ∈ S p if |A| p is a trace class operator, where |A| = √ A * A, and we set ||A|| p = (Tr |A| p ) 1/p . For p = ∞, S ∞ is the set of the compact operators with ||A|| ∞ = ||A||. We have the following properties : Proposition 4.2. ([28, IX.4]) Let 1 ≤ p ≤ ∞ and p -1 + q -1 = 1. (a) If A ∈ S p and B ∈ S q , then AB ∈ S 1 and ||AB|| 1 ≤ ||A|| p • ||B|| q . (b) S p is a Banach space with norm || • || p . (c) S 1 ⊂ S p . (d) If A ∈ S p , then A * ∈ S p and ||A * || p = ||A|| p .

  .49) As a conclusion, we have proved in all cases (b) ≈ (-1) k z κ 0 j=1 m ς j ς j + δ ςκ 0 ,1 (-1) k m 1 z ln z . (4.50) It remains to study (a). Using the same strategy as for (b), we obtain easily (a) ≈ (-1) k-1

T 3 ( 3 ( 3 (T 3 (

 3333 z) = (-1) k+1 Tr [S 1 (z)S 2 (z)S k-1 for j = 1, ..., 4, T 3j (z) = (-1) k+1 Tr [S 11 (z)S 2j (z)S k-1 35 (z) = (-1) k+1 Tr [(S 11 (z)S 25 (z) + S 12 (z)S 2 (z))S k-1

1 s

 1 )or equivalently,∃ C > 0, ∀ϕ ∈ C ∞ 0 (]0, 1[), | R λ 1-ǫ 0 ξ ′ (λ) ϕ(λ) dλ | ≤ C || ϕ || 1 .(5.8)Let us consider g ∈ C ∞ 0 (]0, 1[; R) and f ∈ C ∞ 0 (R; R) such that f ≡ 1 on [0, 2], so we have f g = g. For λ ∈]0, 1[ and ǫ > 0, we have :Tr [R(λ + iǫ)f (P ) -R 0 (λ + iǫ)f (P 0 )] (5.9) = Tr [(R(λ + iǫ) -R 0 (λ + iǫ))f (P )] + Tr [R 0 (λ + iǫ) (f (P )f (P 0 )].By the definition of the SSF,Tr [R(λ + iǫ)f (P ) -R 0 (λ + iǫ)f (P 0 )] = -R ξ(s) ∂f ∂s (s, λ + iǫ) ds,(5.10)where we have setf (s, λ + iǫ) = 1 sλiǫ f (s). (5.11)Thus, the left hand side of (5.9) is given by(LHS) = R ξ(s) 1 (sλiǫ) 2 f (s) ds -R ξ(s) 1 (sλiǫ) f ′ (s) ds.(5.12)Using (4.70) and (5.5), the right hand side of (5.9) satisfies for a suitable real constant C,(RHS) = -J 0 λ + iǫ + C (λ + iǫ) log(λ + iǫ) + r(λ, ǫ),(5.13)wherer(λ, ǫ) = O |λ + iǫ| -1+ǫ 0 + Tr [R 0 (λ + iǫ) (f (P )f (P 0 )](5.14)= O λ -1+ǫ 0 , λ ↓ 0.(5.15) Thus, we have obtained thatr(λ, ǫ) = J 0 λ + iǫ -C (λ + iǫ) log(λ + iǫ) + a(λ, ǫ)b(λ, ǫ), λiǫ) 2 f (s) ds, λiǫ f ′ (s) ds. (5.18)We shall multiply (5.16) by g(λ) and we shall integrate over λ. First, we remark thatR a(λ, ǫ) g(λ) dλ = -R ξ(s)f (s) R ∂ ∂λ 1 λs + iǫ g(λ) dλ ds = R ξ(s)f (s) R 1 λs + iǫ g ′ (λ) dλ ds. ǫ) g(λ) dλ = π R ξ(s) (f g) ′ (s) ds.(5.22)Now, we remark that :R ξ(s) (f g) ′ (s) ds = R ξ(s) g ′ (s) ds = -R ξ ′ (s) g(s) ds,(5.23)since ξ ∈ C ∞ (]0, +∞[). Then, we choose g(s) = s 1-ǫ 0 ϕ(s) with ϕ ∈ C ∞ 0 (]0, 1[), and using (5.14), we have proved(5.8). Now, we are able to prove a Levinson's theorem for critical potentials. To do it, we recall that if v satisfies (1.5), one has the high energy asymptotics, ([START_REF] Robert | Relative time-delay for perturbations of elliptic operators and semi-classical asymptotics[END_REF], Theorem 1.2) : ξ ′ (λ) ∼ j≥1 c j λ n 2 -j-1 , λ → +∞.(5.24)

c j λ [ n 2 ]

 2 -1-j ) dλ = -(N -+ J 0 ) + β n/2 , (5.26)where β n/2 depends only on n, v and V . If n is odd, β n/2 = 0. If n is even, c n/2 = 0.

  j ] dλ = -(N -+J 0 )+

  j ] dλ.By Theorem 5.2, ξ ′ (λ) is integrable on ]0, 1j ] dλ.(5.28) In the same way, (5.24) implies |ξ ′ (λ)j ] dλ.(5.30) Taking R → +∞ in both sides of (5.27), we deduce that β j = d j . Thus, we get the Levinson's Theorem for odd dimension :∞ 0 (ξ ′ (λ) -[ n 2 ] j=1 c j λ [ n 2 ]-1-j ) dλ = -(N -+ J 0 ). (5.31) (Xiaoyao Jia) Department of Mathematics, Henan University of Sciences and Technology, 471003 Luoyang, China (François Nicoleau) Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, Université de Nantes, F-44322 Nantes Cedex 3, FRANCE E-mail address: nicoleau@math.univ-nantes.fr (Xue Ping Wang) Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, Université de Nantes, 44322 Nantes Cedex, France E-mail address: xue-ping.wang@univ-nantes.fr

(z)]

Research supported in part by the French National Research Project NONAa, No. ANR-08-BLAN-0228-01.

and G νp,δν p π νp W Φ j = G νp,δν p nν p l=1 ( W Φ j , ϕ (l) νp ) ⊗ ϕ (l) νp ,

where we have used (3.22) in the last equation, with u = Φ j ∈ L 2 (R n ).

First, let us study T 32 (z). We have the following result : Lemma 4.10.

T 32 (z) ≈ -N 0 z . (4.63)

Proof. We can decompose T 32 (z) as

with

By Lemma 4.9, I 1 (z) = 0. Now, let us study I 2 (z). As previously, we have

The lemma comes from the following result (see [START_REF] Jensen | Spectral properties of Schrödinger operators and timedecay of the wave functions[END_REF], [START_REF] Wang | Asymptotic behavoir of resolvent for N-body Schrödinger operators near a threshold[END_REF]) :

Now, we prove : Lemma 4.11.

T 33 (z) ≈ 0.

Proof. Since S 23 (z) ≈ 0, it is easy to see that

{ ν}≤1

(-)

We note that T e; ν;j = Π 0 A e; ν;j with A e; ν;j be a bounded operator in L(-1, s; -1, s), s > 3. So the result comes from Lemma 4.9.

We have the following estimate for T 32 (z) :

Lemma 4.12.

Proof. We have

First, we assume that ς κ 0 < 1. In this case, U 1 T er (z)U 2 ≈ 0 and it follows that

where

By Lemma 4.9, J 1 (z) = 0. Now, let us study J 2 (z). As previously, the leading contribution is obtained when ν = ς j . So we have :

where we have used the cyclicity of the trace. Moreover, it is easy to see that Lemma 4.6 implies

Using the first assertion of (4.69) and Lemma 4.9, we deduce

Finally, using the same arguments as in Lemma 4.8, we can show that J 3 (z) ≈ 0.

Case 2: The dimension n is even.

First, let us study the case n = 2. Using Theorem 2.1, we have as previously :

(5.32)

We take R → +∞ in (5.32), and remarking that

we deduce c 1 = 0 and we have obtained the Levinson theorem in dimension n = 2 :

(5.34)

Now, assume n ≥ 4. We write n = 2p with p ≥ 2. We define for 1 ≤ j ≤ p -1,

As for the case n = 2, we obtain easily :

As previously, we take R → +∞ in (5.36) and we deduce that β j = d j and c p = 0. Therefore, we get the Levinson's theorem for even dimension:

(5.37)

Remark 5.4. Of course, the values of β p appearing in the Levison theorem are independent of the cutoff function χ. We can use the functional calculus on pseudodifferential operators ( [START_REF] Helffer | Calcul fonctionnel par la transformée de Mellin[END_REF]) with the Hamiltonians

)

(5.39)

The symbols a f,j are defined as

where the d j,k are universal polynomial functions on ξ which do not depend on f . For example, one has where Γ is the well-known Gamma function, we can show :

• In dimension n = 2 :

(5.48)

• In dimension n = 4 :

(2v(x)V (x) -V 2 (x)) dx.

(5.49)