Philippe Briet
email: briet@univ-tln.fr

André Georges Martinez

Andr É Martinez
email: andre.martinez@unibo.it

Estimates On The Molecular Dynamics For The Predissociation Process

Keywords: Resonances, Born-Oppenheimer approximation, eigenvalue crossing, quantum evolution, survival probability. Subject classifications: 35P15, 35C20, 35S99, 47A75

We study the survival probability associated with a semiclassical matrix Schrödinger operator that models the predissociation of a general molecule in the Born-Oppenheimer approximation. We show that it is given by its usual time-dependent exponential contribution, up to a reminder term that is small in the semiclassical parameter and for which we find the main contribution. The result applies in any dimension, and in presence of a number of resonances that may tend to infinity as the semiclassical parameter tends to 0.

Introduction

The molecular predissociation is one of most well known quantum phenomena giving rise to metastable states and resonances. This corresponds when a bound state molecule dissociates to the continuum through tunneling see e.g. [Kr, La, St, Ze]. The rigorous description of this phenomena goes back to [Kl] with further developments in [DuMe] and, more recently, in [GrMa].

In the context of the Born-Oppenheimer approximation, the transition can occur when a confining electronic curve near a given energy E (e.g. E is a local minimum) crosses a dissociative electronic level (that is, a curve having a limit smaller than E at infinity). Such a situation occurs for instance in the SH molecule : see [LeSu].

After reduction to an effective Hamiltonian, this phenomena can be described by a 2 × 2 matrix H of semiclassical pseudodifferential operators (see, e.g., [KMSW, MaSo]), with small parameter h corresponding to the square root of the inverse of the mass of the nuclei, and with principal part that is diagonal and consists of two Schrödinger operators.

In this paper we consider predissociation resonances from a dynamical point of view, i.e. in terms of exponential behavior in time of the quantum evolution e -itH associated with that system.

Our main motivation is the recent series of works around the case where H = H 0 + κV is the perturbation of an operator with an embedded eigenvalue: See, e.g., [START_REF] Cattaneo | A general resonance theory based on Mourre's inequality[END_REF][START_REF] Costin | Resonance theory for Shrödinger operators[END_REF][START_REF] Jensen | The Fermi golden rule and its form at thresholds in odd dimensions[END_REF][START_REF] Herbst | Exponential decay in the Stark effect[END_REF][START_REF] Hunziker | Resonances, metastable states and exponential decay laws in perturbation theory[END_REF] and references therein. In all of these papers, denoting by ϕ the corresponding eigenfunction of H 0 , the survival probability e -itH ϕ, ϕ is studied. Roughly speaking, they show that the embedded eigenvalue gives rise to a resonance ρ, and the previous quantity behaves like e -itρ ϕ 2 with an error-term typically O(κ 2 ). Moreover, inserting a cutoff in energy, the error-term has a polynomial decay in time at infinity. The starting point of our work is the following observation: in the case of the molecular predissociation, H can be seen as a perturbation of a matrix Schrödinger operator admitting embedded eigenvalues. Therefore, a similar procedure can be done in order to study the quantum evolution. However, in contrast with the case H = H 0 + κV , the small parameter is involved in the unperturbed operator, too, making very delicate the extension of the methods used for it. In order to overcome this difficulty, we use the definition of resonances based of complex distortion (see, e.g., [START_REF] Hunziker | Distortion analyticity and molecular resonance curves[END_REF]), and we replace the arguments of regular perturbation theory (used, e.g., in [CGH]) by those of semiclassical microlocal analysis.

In this way, we can essentially generalize the previous results, and in the case of an isolated resonance ρ with a gap a(h) >> h 2 , our result takes the form,

e -itH g(H)ϕ, ϕ = e -itρ b(ϕ, h) + O h 2 a(h) min 0≤k≤ν { 1 [(1 + t)a(h)] k } ϕ 2 ,
where b(ϕ, h) is the residue at ρ of z → (z -H) -1 ϕ, ϕ , and ν ≥ 0 depends on the regularity of the energy cutoff g (see Theorem 4.1 for a more complete result with several resonances). In addition, we also have an expression for the main contribution of the remainder term (see Remark 4.3). In the case where ν can be taken positive, this also leads to the fact that the error term remains negligible up to times of order Ch -1 |Im ρ| -1 with C > 0, C ∼ ν as ν → ∞, that is much beyond the life-time of the resonant state (see Remark 4.4).

Our results must also be compared with that of [NSZ], where a polynomial bound is obtained for the rest in the quantum evolution, in the case of a scalar semiclassical Schrödinger operator.

Let us briefly describe the content of the paper. In the next section, we give a precise description of the model and assumptions. Section 3 is devoted to the definition of resonances by means of complex distortion theory. Our main result is given in Section 4, whose proof is spread over Sections 5 to 9.

Section 10 contains the proof of a corollary where the energy cutoff has been removed and we discuss in Section 11 the non-trapping case. Finally, some examples of application are given in Section 12, and the Appendix contains the proof of some technical results.

Assumptions

We consider the semiclassical 2 × 2 matrix Schrödinger operator, (2.1)

H = H 0 + hW(x, hD x ) = P 1 0 0 P 2 + hW(x, hD x )
on the Hilbert space

H := L 2 (R n ) ⊕ L 2 (R n ), with, P j := -h 2 ∆ + V j (x) (j = 1, 2), where x = (x 1 , . . . , x n ) is the current variable in R n (n ≥ 1), h > 0 denotes
the semiclassical parameter, and

W(x, hD x ) = 0 W W * 0 with W = w(x, hD x ) is a first-order semiclassical pseudodifferential opera- tors, in the sense that, for all α ∈ N 2n , ∂ α w(x, ξ) = O(1 + |ξ|) uniformly on R 2n .
This is typically the kind of operator one obtains in the Born-Oppenheimer approximation, after reduction to an effective Hamiltonian (see [KMSW, MaSo]).

We assume, Assumption 1. The potentials V 1 and V 2 are smooth and bounded on R n , and satisfy,

The set

U := {V 1 ≤ 0} is bounded ; (2.2) lim inf |x|→∞ V 1 > 0; (2.3)
V 2 has a strictly negative limit -Γ as |x| → ∞;

(2.4)

V 2 > 0 on U. (2.5) In particular, H with domain D H := H 2 (R n ) ⊕ H 2 (R n ) is selfadjoint.
Since we have to consider the resonances of H near the energy level E = 0, we also assume, Assumption 2. The potentials V 1 and V 2 extend to bounded holomorphic functions near a complex sector of the form, S R 0 ,δ := {x ∈ C n ; |Re x| ≥ R 0 , |Im x| ≤ δ|Re x|}, with R 0 , δ > 0. Moreover V 2 tends to its limit at ∞ in this sector and Re V 1 stays away from 0 in this sector.

Assumption 3. The symbol w(x, ξ) of W extends to a holomorphic func-

tions in (x, ξ) near, S R 0 ,δ := S R 0 ,δ × {ξ ∈ C n ; |Im ξ| ≤ δ Re x },
and, for real x, w is a smooth function of x with values in the set of holomorphic functions of ξ near {|Im ξ| ≤ δ}. Moreover, we assume that, for any α ∈ N 2n , it satisfies

(2.6) ∂ α w(x, ξ) = O( Re ξ ) uniformly on S R 0 ,δ ∪ (R n × {|Im ξ| ≤ δ}) .
Under the previous assumption we plan to study the quantum evolution of the operator P given in (2.1), where W is defined as

W := 0 Op L h (w) Op R h (w) 0
where for any symbol a(x, ξ) we use the following quantizations,

Op L h (a)u(x) = 1 (2πh) n e i(x-y)ξ/h a(x, ξ)u(y)dydξ; Op R h (a)u(x) = 1 (2πh) n e i(x-y)ξ/h a(y, ξ)u(y)dydξ.
Finally, we assume,

Assumption [V] (Virial condition) 2V 2 (x) + x∇V 2 (x) < 0 on {V 2 ≤ 0},
or, more generally,

Assumption [NT] E = 0 is a non-trapping energy for V 2 .
The fact that 0 is a non-trapping energy for V 2 means that, for any (x, ξ) ∈

p -1 2 (0), one has | exp tH p 2 (x, ξ)| → +∞ as t → ∞, where p 2 (x, ξ) := ξ 2 + V 2 (x)

Resonances

In the previous situation, the essential spectrum of H 0 is [-Γ, +∞). The resonances of H can be defined by using a complex distortion in the following

way: Let f (x) ∈ C ∞ (R n , R n ) such that f (x) = 0 for |x| ≤ R 0 , f (x) = x for
|x| large enough. For θ = 0 small enough, we define the distorted operator H θ as the value at ν = iθ of the extension to the complex of the operator

U ν HU -1 ν
which is defined for ν real, and analytic in ν for ν small enough, where we have set,

(3.1) U ν φ(x) := det(1 + νdf (x)) 1/2 φ(x + νf (x)).
Since we have a pseudodifferential operator w(x, hD x ), the fact that U ν HU -1 ν is analytic in ν is not completely standard but can be done without problem (thanks to Assumption 3). By using the Weyl Perturbation Theorem, one can also see that there exists ε 0 > 0 such that for any ±θ > 0 small enough, the spectrum of Let us observe that the resonances of H can also be viewed as the poles of the meromorphic extension, from {Im z > 0}, of some matrix elements of the resolvent R(z) := (Hz) -1 (see, e.g., [ReSi, HeMa]).

H θ is discrete in {z ∈ C ; Re z ∈ [-ε 0 , ε 0 ], ±Im z ≥ ∓ε 0 θ},
By adapting techniques of [START_REF] Helffer | Multiple Wells in the Semiclassical Limit I[END_REF][START_REF] Helffer | Résonances en limite semi-classique[END_REF] (see also [Kl, GrMa]), one can prove that, in our situation, the resonances of H near 0 are close to the eigenvalues of the operator

(3.2) H := -h 2 ∆ + V 1 0 0 -h 2 ∆ + V 2 + hW(x, hD x ), where V 2 ∈ C ∞ (R n ; R) coincides with V 2 in {V 2 ≥ δ} (δ > 0 is fixed arbitrar- ily small)
, and is such that inf V 2 > 0. The precise statement is the following one : Let I(h) be a closed interval included in (-ε 0 , ε 0 ), and a(h) > 0 such that a(h) → 0 as h → 0 + , and, for all ε > 0 there exists C ε > 0 satisfying,

(3.3) a(h) ≥ 1 C ε e -ε/h ; (3.4) σ( H) ∩ ((I(h) + [-3a(h), 3a(h)])\I(h)) = ∅,
for all h > 0 small enough. Then, there exist two constants ε 1 , C 0 > 0 and a bijection,

β : σ( H) ∩ I(h) → Res(H) ∩ Ω(h),
where we have set,

Ω(h) := (I(h) + [-a(h), a(h)) + i[-ε 1 , 0],
such that,

β(λ) -λ = O(e -C 0 /h ),
uniformly as h → 0 + .

In particular, since the eigenvalues of P are real, one obtains that, for any ε > 0, the resonances ρ in Ω(h) satisfy,

Im ρ = O(e -C 0 /h ).
In what follows, we will show that, under an additional assumption, these resonances are also closed to the eigenvalues of P 1 .

Remark 3.1. Actually, under an assumption of analyticity on W slightly stronger that Assumption 3 (see [GrMa]), or if W has a simpler form (see [Kl]), C 0 can be taken arbitrarily close to 2d(U, {V 2 ≤ 0}), where d stands for the Agmon distance associated with the potential min(V 2 , V 1 ), that is, the pseudo-distance associated with the pseudo-metric max(0, min(V 2 , V 1 ))dx 2 .

Main Result

For our purpose, we need to have a stronger gap between I(h) and the rest of the spectrum of P 1 . Namely, we assume the existence of a(h) > 0, such that, (4.1)

h 2 a(h) → 0 as h → 0 + ; σ(P 1 ) ∩ ((I(h) + [-3a(h), 3a(h)])\I(h)) = ∅,
Then, we denote by u 1 , . . . , u m an orthonormal basis of eigenfunctions of

P 1 corresponding to its eigenvalues λ 1 , . . . , λ m in I(h) (we recall that m = m(h) = O(h -n ))
. For j = 1, . . . , m, we also set,

φ j := u j 0 ∈ L 2 (R n ) ⊕ L 2 (R n ),
so that φ j is an eigenvector of,

H 0 := -h 2 ∆ + V 1 0 0 -h 2 ∆ + V 2 ,
with eigenvalue λ j imbedded in its continuous spectrum [Γ, +∞).

Theorem 4.1. Suppose Assumptions 1-3, (4.1), and Assumption [V] or

[NT]. Let g ∈ L ∞ (R) supported in (I(h) + (-2a(h), 2a(h))) with g = 1 on I(h) + [-a(h), a(h)],
and such that, for some ν ≥ 0, one has,

(4.2) g, g ′ , . . . , g (ν) ∈ L ∞ (R); g (k) = O(a(h) -k ) (k = 1, . . . , ν).
Then, for all t ∈ R and ϕ ∈ Span{φ 1 , . . . , φ m }, one has,

(4.3) e -itH g(H)ϕ, ϕ = m j=1 e -itρ j b j (ϕ, h) + r(t, ϕ, h), where ρ 1 , . . . , ρ m are the resonances of H lying in Ω(h) := I(h)+[-a(h), a(h)]- i[0, ε 1 ],
and satisfy, (4.4)

ρ j = λ j + O(h 2 ), r(t, ϕ, h) is such that, (4.5) r(t, ϕ, h) = O h 2 a(h) min 0≤k≤ν { 1 [(1 + t)a(h)] k } ϕ 2 ,
uniformly with respect to h > 0 small enough, t ≥ 0, and ϕ ∈ Span(φ 1 , . . . φ m ).

Here b j (ϕ, h) is the residue at ρ j of the meromorphic extension from {Im z > 0} of the function,

z → (z -H) -1 ϕ, ϕ .
and satisfies: There exists a m × m matrix M (z) depending analytically on

z ∈ Ω(h), with (4.6) M (z) = O(h 2 ), such that, (4.7) b j (ϕ, h) is the residue at ρ j of the meromorphic function z → (z -Λ + M (z)) -1 α ϕ , α ϕ C m ,
where α ϕ := ( ϕ, φ 1 , . . . , ϕ, φ m ) and Λ := diag(λ 1 , . . . , λ m ).

If in addition one assumes that λ 1 , . . . , λ m are all simple, and the gap a(h) :=

min j =k |λ j -λ k | is such that, (4.8) h 2 / a(h) → 0 as h → 0 + , then, b j (ϕ, h) satisfies, (4.9) b j (ϕ, h) = | ϕ, φ j | 2 + O (h 2 + h 4 (a a) -1 ) ϕ 2 ,
uniformly with respect to h > 0 small enough and ϕ ∈ Span(φ 1 , . . . φ m ).

Remark 4.2. Actually, our proof also gives a generalization of a result

given in [CGH] for the case m = 1 : see Propositions 7.1 and 7.3.

Remark 4.3. Concerning the remainder term, we will see in the proof that it is of the form

r(t, ϕ, h) = r 0 (t, ϕ, h) + O h 4 (a(h)) 2 min 0≤k≤ν { 1 [(1 + t)a(h)] k } ϕ 2 with (4.10) r 0 (t, ϕ, h) = -h 2 lim ε,ε ′ →0 + j,k ϕ, φ j ϕ, φ k × e -itP 2 g(P 2 )(P 2 -λ j -iε) -1 W * u j , (P 2 -λ k + iε ′ ) -1 W * u k . Remark 4.4. In particular, one has |r(t, ϕ, h)| << |e -itρ j | as long as 0 ≤ t << 1 |Im ρ j | ln(a(h)/h 2 )
that is much beyond the life time. In the case ν ≥ 1, since |Im ρ j | is exponentially small w.r.t. h , this can indeed be improved by allowing times up to C 0 ν h|Im ρ j | for some constant C 0 > 0 independent of ν.

Remark 4.5. Let us observe that, in the particular case where m = 1,

one obtains b 1 (ϕ, h) = | ϕ, φ 1 | 2 + O (h 2 + h 4 /a 2 ) ϕ 2 .
Therefore, in the situation of the Theorem with (4.8), a mere application of the previous result for each λ j would give b

j (ϕ, h) = | ϕ, φ j | 2 + O (h 2 + h 4 / a 2 ) ϕ 2 ,
and compared with (4.9) this is a weaker result if a(h) << a(h).

As a corollary, for the case without energy cutoff, we also obtain, Corollary 4.6. In the general situation of Theorem 4.1 (without the assumption on the simplicity of the λ j 's), one has,

e -itH ϕ, ϕ = m j=1 e -itρ j b j (ϕ, h) + O (h 2 + h 4 a(h) -2 ) ϕ 2 .
In the sequels, we will concentrate on the detailed proof of Theorem 4.1 in the case of Assumption [V]. The more general case of Assumption [NT] can be proved in a similar way by using the Helffer-Sjöstrand framework of resonances theory [START_REF] Helffer | Résonances en limite semi-classique[END_REF], and will be outlined in Section 11.

Preliminaries

In order to prove Theorem 4.1, we start from the Stone formula,

(5.1)

e -itH g(H)ϕ, ϕ = lim ε→0 + 1 2iπ R e -itλ g(λ) (R(λ + iε) -R(λ -iε))ϕ, ϕ dλ,
where R(z) := (Hz) -1 . In the sequels, we also denote by R θ (z) := (H θz) -1 the distorted resolvent, and by ϕ θ := U iθ ϕ the distortion of ϕ (observe that, thanks to the analyticity of V 1 and the ellipticity of P 1 , each function u j can be distorted without problem). In particular, by standard arguments (see, e.g., [ReSi, HeMa]), one has

R(z)ϕ, ϕ = R θ (z)ϕ θ , ϕ -θ .
From now on, we fix θ > 0 small enough and, thanks to the fact that g = 1 on I(h) + [-a(h), a(h)], we can slightly deform the contour of integration in this region, and rewrite (5.1) as,

(5.2)

e -itH g(H)ϕ, ϕ = 1 2iπ γ + e -itz g(Re z) R θ (z)ϕ θ , ϕ -θ dz - 1 2iπ γ - e -itz g(Re z) R -θ (z)ϕ -θ , ϕ θ dz,
where the complex contour γ ± can be parametrized by Re z, coincides with R away from I(h) + (-a(h), a(h)), and is included in {±Im z > 0} on I(h) + (-a(h), a(h)).

Here we anticipate by using (4.4) and, proceeding as in [CGH], we see that

(5.2) can be transformed into,

(5.3) e -itH g(H)ϕ, ϕ = m j=1 e -itρ j b j (ϕ, h) + r(t, ϕ, h),
where b j (ϕ, h) is the residue at ρ j of the meromorphic function

z → -R θ (z)ϕ θ , ϕ -θ ,
and r(t, ϕ, h) is given by,

(5.4)

r(t, ϕ, h) := 1 2iπ γ - e -itz g(Re z) ( R θ (z)ϕ θ , ϕ -θ -R -θ (z)ϕ -θ , ϕ θ ) dz,
where γ -is chosen in such a way that it stays below the ρ j 's. Thus, the proof will consist in estimating both b j (ϕ, h) and r(t, ϕ, h).

The Grushin problems

From now on (up to Section 11), we suppose Assumption [V].

In order to have good enough estimates on the resolvent, and in particular to compare it with that of P 1 , for z in Ω(h) :

= (I(h)+[-a(h), a(h)])+i[-ε 1 , ε 1 ],
we specify our choice of distorsion. In (3.1), we take F such that, (6.1)

f (x) = x in a neighborhood of the sea {V 2 ≤ 0}; F = 0 in a neighborhood of the well U = {V 1 ≤ 0}.
With such a distorsion, it is well known (see, e.g., [BCD]) that, under Assumption [V], the distorted operator P θ 2 satifies, (6.2)

(P θ 2 -z) -1 L(L 2 (R n )) = O(1),
uniformly with respect to h > 0 small enough and z ∈ Ω(h).

We introduce the two following Grushin problems,

G(z) := H θ -z L - L + 0 : D H × C m → H × C m , G 0 (z) := H θ 0 -z L - L + 0 : D H × C m → H × C m ,
where H θ 0 stands for the distorted Hamiltonian obtained from H 0 , and L ± are defined as, (6.3)

L -(α 1 , . . . , α m ) := m j=1 α j φ θ j ;

(6.4)

L + u := L * -u = ( u, φ -θ 1 , . . . , u, φ -θ m ).
with φ ±θ j := U ±iθ φ j .

It is elementary to check that G 0 (z) is invertible, with inverse given by,

G 0 (z) -1 = Π θ R θ 0 (z) Π θ L - L + z -Λ ,
where Λ = diag(λ 1 , . . . , λ m ), Π θ := 1 -Π θ with Π θ the spectral projection of H θ 0 associated with the eigenvalues (λ 1 , . . . , λ m ), that is,

Π θ u := m j=1 u, φ -θ j φ θ j ,
and R θ 0 (z) is the reduced resolvent of H θ 0 i.e. the inverse of the restriction of H θ 0z to the range of Π θ .

In addition to (6.2), we have, Lemma 6.1.

( P ±θ 1 -z) -1 L(L 2 (R n )) = O(a(h) -1 ),
uniformly with respect to h > 0 small enough and z ∈ Ω(h).

Proof. See Appendix 1.

In order to prove that G(z) is invertible, too, and to compare its inverse with G 0 (z) -1 , we compute the product,

G(z)G 0 (z) -1 =: A 11 A 12 A 21 A 22 .
Using that H θ = H θ 0 + hW θ (where W θ stands for the distorted operator obtained from W), we find,

A 11 = 1 + hW θ R θ 0 (z); A 12 = hW θ L -; A 21 = 0; A 22 = I C m .
Then, we observe,

Π θ = Π θ 1 0 0 1 , and, R θ 0 (z) = Π θ 1 R θ 1 (z) Π θ 1 0 0 R θ 2 (z)
, where R θ 2 (z) is the resolvent of P θ 2 (the distorted operator obtained from P 2 ), and R θ 1 (z) is the reduced resolvent of P θ 1 . Thus, denoting by W θ the distorted operator obtained from W = w(x, hD x ), and W * θ that obtained from W * , we find,

hW θ R θ 0 (z) = 0 hW θ R θ 2 (z) hW * θ Π θ 1 R θ 1 (z) Π θ 1 0 .
Here we must be aware that this operator is not

O(h), since Π θ 1 R θ 1 (z) Π θ 1 is O(a(h) -1 ) only. However, the other off-diagonal operator hW θ R θ 2 (z) is O(h)
, and this is enough, for instance, to invert 1 + hW θ R θ 0 (z) without problem. From now on, we set,

(6.5) Q 1 (z) := W * θ Π θ 1 R θ 1 (z) Π θ 1 = O(a(h) -1 ) ; Q 2 (z) := W θ R θ 2 (z) = O(1).
In particular, (6.6)

K(z) := h 2 Q 1 (z)Q 2 (z) = O(h 2 /a(h)),
and thus, by assumption (4.1), the operator 1 -K is invertible for h > 0 small enough. Then, a straightforward computation shows that G(z)G 0 (z) -1 is invertible, with inverse given by,

F(z) := B 1 (z) B 2 (z) 0 I C m ,
where,

B 1 (z) := 1 + h 2 Q 2 (1 -K) -1 Q 1 -hQ 2 (1 -K) -1 -h(1 -K) -1 Q 1 (1 -K) -1 , and
(6.7) B 2 (z) := h 2 0 Q 2 (1 -K) -1 0 (1 -K) -1 W θ L -.
(Here, we have also used the fact that the first component of W θ φ j is identically 0.)

A similar computation shows that G 0 (z) -1 G(z) is invertible, too, and, as a consequence, so is G(z), with inverse, (6.8)

G(z) -1 = G 0 (z) -1 F(z) = E(z) E + (z) E -(z) E -+ (z)
,

where, (6.9)

E(z) := Π θ R θ 0 (z) Π θ B 1 (z) E + (z) := L -+ Π θ R θ 0 (z)B 2 (z) E -(z) := L + B 1 E -+ (z) := z -Λ + L + B 2 (z).
We set,

M (z) := L + B 2 (z) = M 0 (z) + M 1 (z), with M 0 (z) := h 2 L + 0 Q 2 (z) 0 1 W θ L -, and
(6.10) M 1 (z) := L + B 2 (z) -M 0 (z).
One can prove, Lemma 6.2. One has,

M 0 (z) L(C m ) = O(h 2 ); M 1 (z) L(C m ) = O(h 4 /a(h)) = o(h 2 ),
uniformly with respect to h > 0 small enough and z ∈ Ω(h).

Proof. See Appendix 2.

In particular, (6.11)

M (z) L(C m ) = O(h 2 ),
uniformly with respect to z ∈ Ω(h) and h > 0 small enough. Since h 2 /a(h) → 0, by standard perturbation theory we deduce,

Sp(Λ + M (z)) = {λ 1 (z), . . . , λ m (z)},
with,

λ j (z) = λ j + O(h 2 ).
As a consequence the solutions z ∈ Ω(h) of the problem,

0 ∈ σ(E -+ (z)),
are all of the form,

z = λ j + O(h 2 ),
for some j. Deforming continuously (E -+ (z)) into z -Λ (e.g., by setting Λ t (z) := z -Λ + tM (z), 0 ≤ t ≤ 1), and following continuously the roots of the determinant of Λ t (z) as t varies from 0 to 1, we also see that all the values of j are reached by such solutions. Since we also know that these solutions are precisely the resonances of H in Ω(h) (see (7.4)), we have proved (4.4).

The reduced resolvent

In this section, we still consider the Grushin problem given by G(z), but we will solve it in a different way, in order to obtain the inverse in terms of the reduced resolvent R θ (z) of H θ (instead of that of H θ 0 ), as in the usual Feshbach method. Indeed, denoting by H θ the restriction of Π θ H θ to the range of Π θ , for all z such that Im z > 0 we can define the reduced resolvent R θ (z) as the inverse of H θz, and it is straightforward to verify that, for such z, the inverse of

G(z) is given by, G(z) -1 = E(z) E + (z) E -(z) E -+ (z) ,
with, (7.1)

E(z) := Π θ R θ (z) Π θ E + (z) := (1 -h Π θ R θ (z) Π θ W θ )L - E -(z) := L + (1 -hW θ Π θ R θ (z) Π θ ) E -+ (z) := z -Λ + h 2 ( W θ Π θ R θ (z) Π θ W θ φ θ k , φ -θ j ) 1≤j,k≤m .
Comparing with (6.9), we obtain in particular (still for Im z > 0, for which the computations of the previous section remain valid),

(7.2) Π θ R θ (z) Π θ = Π θ R θ 0 (z) Π θ B 1 (z).
Now, since both expressions are holomorphic in {Im z > 0}, and the righthand side extends analytically in Ω(h), we conclude that so does Π θ R θ (z) Π θ , and the identity remains valid in Ω(h).

In addition, the expression

W θ Π θ R θ (z) Π θ W θ φ θ k , φ -θ j
is actually independent of θ, and is nothing but the meromorphic extension to Ω(h) of the function (holomorphic in {Im z > 0}),

(7.3) F j,k (z) := W Π R(z) ΠWφ k , φ j
Finally, in order to estimates the residues appearing in (5.3), let us recall the well known formula for the whole resolvent of H θ . For z ∈ Ω(h)\{ρ 1 , . . . , ρ m }, one has,

(7.4) R θ (z) = E(z) -E + (z) (E -+ (z)) -1 E -(z).
In view of (7.1)-( 7.2), we know that the operators E(z), E ± (z) and E -+ (z) depend analytically on z in Ω(h). Therefore, in formula (7.4), the only possible poles come from (E -+ (z)) -1 .

Therefore, we have proved, Proposition 7.1. The distorted resolvent R θ (z) of H is given by (7.4),

where the operators E(z), E ± (z) and E -+ (z) are given in (7.1). Moreover, the resonances of H in Ω(h) are exactly the roots of the equation,

det(z -Λ + h 2 F (z)) = 0, where F (z) is the m × m matrix with coefficients F j,k (z) (1 ≤ j, k ≤ m)
given by (7.3).

In the particular case where m = 1, let us observe that, at first glance, F 1,1 (z) can be estimated by O(a(h) -1 ), and its holomorphic derivative F ′ 1,1 (z) by O(a(h) -2 ) (this is because of the presence of the reduced resolvent in F 1,1 (z)).

For the resonance, this leads to,

ρ 1 = λ 1 -h 2 F 1,1 (ρ 1 ) = λ 1 + O(h 2 /a(h)) = λ 1 -h 2 F 1,1 (λ 1 ) + O(h 4 /a(h) 3 ),
which, compared to the result given in [CGH] seems much less interesting.

But actually, looking more precisely to the expression of F (z), one can prove, Lemma 7.2. In the case m = 1, one has,

|F 1,1 (z)| + |F ′ 1,1 (z)| = O(1),
uniformly with respect to h > 0 small enough and z ∈ Ω(h).

Proof. Using (7.2), we have, 1), and since also

F 1,1 (z) = W θ Π θ R θ 0 (z) Π θ B 1 (z)W θ φ θ 1 , φ -θ 1 = B 1 (z)W θ φ θ 1 , Π -θ R -θ 0 (z) Π -θ (W θ ) * φ -θ 1 , and since, (W θ ) * φ -θ 1 = 0 W -θ u -θ j , we have, (7.5) Π -θ R -θ 0 (z) Π -θ (W θ ) * φ -θ 1 = 0 R -θ 2 (z)W -θ u -θ j , Hence, Π -θ R -θ 0 (z) Π -θ (W θ ) * φ -θ 1 L 2 = O(
B 1 (z) L(L 2 ) = O(1), we deduce, F 1,1 (z) = O(1).
(Here, we have used the fact that u ±θ j L 2 = O(1).)

On the other hand, taking the derivate with respect to z, we obtain,

F ′ 1,1 (z) = W θ Π θ R θ (z) 2 Π θ W θ φ θ 1 , φ -θ 1
Then, applying (7.2) with θ replaced by -θ, and z replaced by z, and then taking the adjoint, we obtain,

(7.6) Π θ R θ (z) Π θ = B * 1 (z) Π θ R θ 0 (z) Π θ , with B * (z) = I + O(h 2 /a) in L(L 2
). Using both (7.2) and (7.6), we are led to,

F ′ 1,1 (z) = B * 1 (z) Π θ R θ 0 (z) Π θ W θ φ θ 1 , B 1 (z) * Π -θ R -θ 0 (z) Π -θ (W θ ) * φ -θ 1 .
Thus, we can conclude as before (see (7.5)) that

F ′ 1,1 (z) = O(1).
As a consequence, we obtained the following generalization of the result of [CGH]:

Theorem 7.3. Suppose Assumptions 1-3, (4.1), and m = 1. Then, the resonance ρ 1 (h) of H that is the closest one to λ 1 (h) satisfies,

ρ 1 (h) = λ 1 (h) -h 2 F 1,1 (λ 1 (h)) + O(h 4 ),
uniformly for h > 0 small enough. Here, F 1,1 (z) is defined in (7.3).

Estimates on the residues

Going back to (5.3), and using (7.4), we deduce,

(8.1) b j (ϕ, h) = Residue z=ρ j E + (z) (E -+ (z)) -1 E -(z)ϕ θ , ϕ -θ .
Since ϕ ∈ Span(φ 1 , . . . , φ m ), it can be written as,

(8.2) ϕ = m j=1 α j φ j , (α j ∈ C)
, and thus we see on (7.1) that we actually have,

E -(z)ϕ θ = L + ϕ θ = (α 1 , . . . , α m ).
In a similar way, since Π * θ = Π -θ , we also find,

E + (z) * ϕ -θ = (α 1 , . . . , α m ).
Inserting into (8.1), and setting,

α ϕ := (α 1 , . . . , α m ) ∈ C m ,
we obtain,

(8.3) b j (ϕ, h) = Residue z=ρ j E -+ (z) -1 α ϕ , α ϕ C m .
Therefore, using (6.9)-(6.11), we deduce (4.6)-(4.7). Now, assuming that the λ j 's are simple and that (4.8) is satisfied, we write, (8.4)

E -+ (z) = (z -Λ + M 0 (z)) 1 + (z -Λ + M 0 (z)) -1 M 1 (z) .
Moreover, using (8.3) and denoting by γ j the oriented boundary of the disc centered in λ j of radius a(h)/2, we have,

(8.5) b j (ϕ, h) = 1 2iπ γ j E -+ (z) -1 α ϕ , α ϕ dz.
When z ∈ γ j , we have (z -Λ) -1 = O( a -1 ) and thus, using (4.8),

(z -Λ + M 0 (z)) -1 = (1 + (z -Λ) -1 M 0 (z)) -1 (z -Λ) -1 = O( a -1 ).
Moreover, using (6.11), we have,

(z -Λ + M 0 (z)) -1 M 1 (z) = O(h 4 /(a a)) = o(1),
and thus, by (8.4), for z ∈ γ j ,

E -+ (z) -1 = 1 + O(h 4 /(a a) (z -Λ -M 0 (z)) -1 = (z -Λ + M 0 (z)) -1 + O(h 4 /(a a 2 )),
and thus, since the length of γ j is O( a), (8.6)

γ j E -+ (z) -1 α ϕ , α ϕ dz = γ j (z -Λ + M 0 (z)) -1 α ϕ , α ϕ dz + O h 4 a a ϕ 2 .
On the other hand, we see on its definition that we have,

M 0 (z) = h 2 W θ R θ 2 (z)W * θ u θ k , u -θ j 1≤j,k≤m
, and, introducing the operator P 2 := -h 2 ∆ + V 2 where V 2 is as in (3.2), the exponential decay of u ±θ j away from U and Agmon estimates (see [START_REF] Helffer | Résonances en limite semi-classique[END_REF]) show that,

W θ R θ 2 W * θ u θ k , u -θ j = W R 2 (z)W * u k , u j + O(e -δ/h ),
for some constant δ > 0, and with R 2 (z) :

= ( P 2 -z) -1 . Setting M 0 (z) := W θ R θ 2 W * θ u θ k , u -θ j = W R 2 (z)W * u k , u j 1≤j,k≤m
, we deduce as before, (8.7)

γ j E -+ (z) -1 α ϕ , α ϕ dz = γ j (z -Λ + M 0 (z)) -1 α ϕ , α ϕ dz + O h 4 a a ϕ 2 ,
where the matrix M 0 (z) is O(h 2 ), depends analytically on z ∈ Ω(h), and is selfadjoint when z is real. As a consequence, thanks to the gap condition on the λ j 's, we see that the matrix Λ -M 0 (z) can be diagonalized in a basis (e 1 (z), . . . , e m (z)) of C m , that depends analytically on z ∈ Ω(h), is orthonormal when z is real, and the corresponding change of basis is given by a matrix A(z) satisfying,

t A(z)A(z) = I C m ; A(z) = I C m + O(h 2 ); t A(z)(z -Λ + M 0 (z)) -1 A(z) = diag 1 z -µ 1 (z) , . . . , 1 z -µ m (z)
,

where the eigenvalues µ 1 (z), . . . , µ m (z) of Λ -M 0 (z) satisfy,

µ j (z) = λ j + f j (z)
with f j (z) = O(h 2 ). Note that f j are real on the real. Since

d dz M 0 (z) = O(h 2 ),
we see by a standard Hellmann-Feynman argument that, in this situation, we also have,

µ ′ j (z) = f ′ j (z) = O(h 2 ).
Moreover, the poles λ 1 , . . . , λ m of (z-Λ-M 0 (z)) -1 α ϕ , α ϕ are the solutions of an equation,

z = µ j (z)
for some j = 1, . . . , m. Thus, they are necessarily simple, and since µ j (z) = µ j (z), they must be real. Finally, we obtain,

(8.8) 1 2iπ γ j (z -Λ -M 0 (z)) -1 α ϕ , α ϕ dz = (1 -f ′ j ( λ j )) -1 |α j | 2 + O(h 2 ϕ 2 ) = |α j | 2 + O(h 2 ϕ 2 ),
and (4.9) follows from (8.5), (8.7) and (8.8).

Estimates on the rest

We have to estimate the quantity, (9.1)

S θ (z) := R θ (z)ϕ θ , ϕ -θ -R -θ (z)ϕ -θ , ϕ θ , for z ∈ γ -where, setting I = [α, β] := I(h) + [-a, a],
we choose the contour γ -as,

γ -:= (R\ I) ∪ (α -i[0, a]) ∪ ([α, β] -ia) ∪ (β -i[0, a]).
We first compute v = (v 1 , v 2 ) := R θ (z)ϕ θ . Denoting by u := j α j u j the first component of ϕ, we find,

v 1 = (P θ 1 -z) -1 (1 -T θ ) -1 u θ v 2 = -h(P θ 2 -z) -1 W * θ v 1 , with T θ := h 2 W θ (P θ 2 -z) -1 W * θ (P θ 1 -z) -1 = O(h 2 /a(h)).
Then, using that (P

1 -z) -1 u = k α k (λ k -z) -1 u k ,
that the u j 's are orthogonal to each other, and that z stays at a distance greater than a/2 from the λ j 's, we deduce,

R θ (z)ϕ θ , ϕ -θ = v 1 , u -θ = j |α j | 2 λ j -z + j,k α j α k λ k -z T θ u θ j , u -θ k +O(h 4 a -3 ).
Using again that the u j 's are eigenfunction of P 1 , this lead us to,

R θ (z)ϕ θ , ϕ -θ = j |α j | 2 λ j -z + h 2 j,k α j α k (λ k -z)(λ j -z) W θ R θ 2 (z)W * θ u θ j , u -θ k + O(h 4 a -3 ϕ 2 ).
Here we observe that the quantity

W θ R θ 2 (z)W * θ u θ j , u -θ k
is nothing but the holomorphic continuation from {Im z > 0} through the real axis of the function z → R 2 (z)W * u j , W * u k . From now on, we denote this continuation by R 2 (z)W * u j , W * u k .

Changing θ into -θ, we also find an analog expression for R -θ (z)ϕ -θ , ϕ θ , and making their difference, we obtain, (9.2)

S θ (z) = h 2 j,k α j α k (λ k -z)(λ j -z) ( R 2 (z)-R 2 (z))W * u j , W * u k +O(h 4 a -3 ϕ 2 ).
Multiplying by e -itz g(Re z) and integrating over γ -, we obtain the required estimate of r(t, ϕ, h) in the case ν = 0

For the case ν > 0, as in [CGH] we use the formula,

e -izt = (1 + t) -ν 1 + i d dz ν e -izt ,
and we make k integrations by parts with respect to z (0 ≤ k ≤ ν). This makes appear the composition of a finite number of resolvents and additional negative powers of λ jz, and the estimate follows in the same way.

Moreover, setting (9.3)

r 0 (t, ϕ, h) := h 2 2iπ j,k γ - e -itz g(Re z) (λ k -z)(λ j -z) ( R 2 (z) -R 2 (z))W * u j , W * u k
we see on (9.2) that we have r(t, ϕ, h) = r 0 (t, ϕ, h) + O(h 4 a -2 ϕ 2 ). In addition, in (9.3), we can change (λ j , λ k ) into (λ j + iε, λ k + iε ′ ) and take the limit as ε, ε ′ → 0 + . Before taking this limit, we can also deform γ -into R, and this transforms We first prove, Lemma 10.1.

R 2 (z) -R 2 (z) into R 2 (λ + i0) -R 2 (λ -i0).
m j=1 b j (ϕ, h) = 1 + O(h 2 + h 4 /a 2 ) ϕ 2 .
Proof. We write, (10.1) and, using (8.3) and denoting by γ the oriented boundary of the rectangle

E -+ (z) = (z -Λ + M 0 (z)) 1 + (z -Λ + M 0 (z)) -1 M 1 (z) ,
{z ∈ C ; Re z ∈ I(h) + [-a(h), a(h)], |Im z| ≤ ε 1 }, we have, (10.2) m j=1 b j (ϕ, h) = 1 2iπ γ E -+ (z) -1 α ϕ , α ϕ dz.
We divide γ into its vertical part γ v and its horizontal one γ h .

When z ∈ γ h , since z remains at a distance ε 1 of R, we have (z -Λ) -1 = O(1) and thus,

(z -Λ + M 0 (z)) -1 = (1 + (z -Λ) -1 M 0 (z)) -1 (z -Λ) -1 = O(1).
Moreover, still for z ∈ γ h , we see on (6.6) that K(z) = O(h 2 ), and thus, by (6.7) and (6.10), M 1 (z) = O(h 4 ). As a consequence

(z -Λ + M 0 (z)) -1 M 1 (z) = O(h 4 ), (z ∈ γ h ).
Therefore, by (10.1), for such z we can write,

E -+ (z) -1 = 1 + O(h 4 ) (z -Λ -M 0 (z)) -1 = (z -Λ -M 0 (z)) -1 + O(h 4 ),
and thus, (10.3)

γ h E -+ (z) -1 α ϕ , α ϕ dz = γ h (z -Λ -M 0 (z)) -1 α ϕ , α ϕ dz + O(h 4 ) ϕ 2 .
On the other hand, when z ∈ γ v , we can write

z = z 1 + iz 2 with z 1 , z 2 ∈ R, dist(z 1 , I(h)) = a(h), |z 2 | ≤ ε 1 . Therefore, for such z we have, (z -Λ + M 0 (z))) -1 = O((a + |z 2 |) -1 ), K(z) = O(h 2 (a + |z 2 |) -1 ), and M 1 (z) = O(h 4 (a + |z 2 |) -1
). Proceeding as before, we deduce,

E -+ (z) -1 = (z -Λ -M 0 (z)) -1 + O(h 4 /(a + |z 2 |) 3 ) ϕ 2 ,
and thus, integrating in z 2 on [-ε 1 , ε 1 ],

(10.4)

γ v E -+ (z) -1 α ϕ , α ϕ dz = γ v (z-Λ-M 0 (z)) -1 α ϕ , α ϕ dz+O(h 4 /a(h) 2 ) ϕ 2 .
We deduce from (10.3)-(10.4), (10.5)

γ E -+ (z) -1 α ϕ , α ϕ dz = γ (z-Λ-M 0 (z)) -1 α ϕ , α ϕ dz+O(h 4 /a(h) 2 ) ϕ 2 .
At this point, we make the key observation that, by definition, M 0 (z) extends analytically in some h-independent complex neighborhood of I(h), where it is O(h 2 ) in norm. As a consequence, modifying the complex contour γ into another one that stays at some fix positive distance from I(h), we deduce from (10.5), (10.6)

γ E -+ (z) -1 α ϕ , α ϕ dz = γ (z -Λ) -1 α ϕ , α ϕ dz + O(h 2 + h 4 /a(h) 2 ) ϕ 2 .
Going back to (10.2), this gives,

m j=1 b j (ϕ, h) = m j=1 |α j | 2 +O(h 2 +h 4 /a(h) 2 ) ϕ 2 = 1 + O(h 2 + h 4 /a 2 ) ϕ 2 ,
and (4.9) is proved. Now, applying Theorem 4.1 with t = 0, we obtain,

g(H)ϕ, ϕ = m j=1 b j (ϕ, h) + O(e -M/h )
and thus, by the previous lemma,

g(H)ϕ, ϕ = ϕ 2 + O(h 2 + h 4 /a 2 ) ϕ 2 .
Hence,

(10.7) (1 -g(H))ϕ, ϕ = O(h 2 + h 4 /a 2 ) ϕ 2 ,
and we can chose g in such a way that 0 ≤ g ≤ 1. In that case, (10.7) can be re-written as,

(1g(H))

1 2 ϕ 2 = O(h 2 + h 4 /a 2 ) ϕ 2 ,
and Corollary 4.6 follows by writing,

e -itH ϕ, ϕ = e -itH g(H)ϕ, ϕ + e -itH (1 -g(H))ϕ, ϕ = e -itH g(H)ϕ, ϕ + e -itH (1 -g(H)) 1 2 ϕ, (1 -g(H)) 1 2 ϕ = e -itH g(H)ϕ, ϕ + O( (1 -g(H)) 1 2 ϕ 2 ).

The non-trapping case

In the case when only Assumption [NT] is assumed (instead of Assumption [V]), the strategy of the proof is the same. However, an important ingredient for the estimates on the residues was the uniform boundedness of the resolvent of P θ 2 . Therefore, in order to generalize this proof one needs a framework where (P 2z) -1 becomes bounded uniformly with respect to h. This is provided by the theory of resonances developed by Helffer and Sjöstrand in [START_REF] Helffer | Résonances en limite semi-classique[END_REF]. Without entering too much into details, let us just recall that this theory consists in changing L 2 (R n ) into a space H θG , that contains C ∞ 0 (R n ), and that depends on a positive small enough parameter θ and a function G ∈ C ∞ (R 2n ; R) supported near {p 2 = 0} (where p 2 (x, ξ) := ξ 2 + V 2 (x)), and satisfying,

|p 2 (x, ξ) -iθH p 2 G(x, ξ)| ≥ θ C ξ 2 ,
for some constant C > 0. Then, one has, (11.1)

(P 2 -z) -1 L(H θG ) = O(1/θ),
uniformly with respect to h > 0 small enough and z close to 0. Let us also recall that pseudodifferential operators with analytic symbols on complex sectors can act on H θG , and their representation involves the restriction of their symbol to the complex Lagrangian manifold,

Λ θG := {(x + iθ∂ ξ G(x, ξ), ξ -iθ∂ x G(x, ξ)) ; (x, ξ) ∈ R 2n }.
Moreover, a whole symbolic calculus can be performed for such operators, where only the restrictions to Λ θG of the symbols are involved. Finally, as in the L 2 -case, an analog of Sobolev spaces can be introduced by inserting a weight, and we denote by H 2 θG the analog of H 2 (R n ) in this context. In particular, we have,

P 1 , P 2 : H 2 θG → H θG .
Then, setting D θG := H 2 θG × H 2 θG and H θG := H θG × H θG , we consider the two Grushin problems G(z) and G 0 (z) as in Section 6, but this time without distortion, as operators : D θG × C m → H θG × C m , and with the scalar product replaced (in the definition of L + ) by the duality-bracket between H θG and H -θG .

Then the proof of the estimates on the residues proceeds in the same way, in particular the fact that G is supported near {p 2 = 0} (thus, away from the well U ) makes valid an analog of Lemma 6.1 in this context. Indeed, the norm in H θG is equivalent to a weighted norm of the same type as in (??), but this time with a weight G that is no more compactly supported (but still supported in a neighborhood of {p 2 = 0}): see [START_REF] Helffer | Résonances en limite semi-classique[END_REF], Formula (9.48).

For the same reason, the estimates of Lemma 6.2 on M 0 (z) and M 1 (z) can be generalized, too, and all of Sections 8 and 10 remain valid.

The same procedure applies to estimate the remainder term r(t, ϕ, h).

12. Examples 12.1. The one dimensional case. When n = 1, if we assume,

V ′ 1 = 0 on {V 1 = 0},
then it is well known (see, e.g., HeRo) that the eigenvalues of P 1 are all simple and separated by a gap of order h. Then, we can take |I(h)| = O(h), a = a ∼ h , and we also have m = O(1). Moreover, in this case Assumption

[NT] on V 2 is equivalent to,

V ′ 2 = 0 on {V 2 = 0}, and {V 2 ≤ 0} has no bounded connected component.

For instance V 2 (x) = -Γ + α(1 + x 2 ) -1 , (with α > 0 sufficiently large, so that V 2 > 0 on {V 1 ≤ 0}) satisfies all the assumptions (including Assumption [V]).

In such a situation, (4.9) becomes, 12.2. The non-degenerate point-well. In addition to Assumption 1, let us suppose,

U = {0}, Hess V 1 (0) > 0.
Then, it is well known (see [START_REF] Helffer | Multiple Wells in the Semiclassical Limit I[END_REF][START_REF] Simon | Semiclassical analysis if low lying eigenvalues I. Non-degenerate minima: Asymptotic expansions[END_REF]) that the spectrum of P 1 near 0 consists of eigenvalues admitting asymptotic expansions as h → 0 + , of the form,

λ j (h) ∼ k≥0 λ j,k h 1+ k 2 ,
where λ j,0 is the j-th eigenvalue of the harmonic oscillator -∆+ 1 2 Hess V 1 (0)x, x .

As for V 2 , one can take V 2 (x) = -Γ + α(1 + x 2 ) -1 with α, Γ > 0 arbitrary.

Then Assumption [V] is satisfied, and choosing I(h) = [0, Ch] with C / ∈ {λ j,0 ; j ≥ 1}, we see that the general assumptions of Theorem 4.1 are satisfied with a(h) ∼ h. Thus, (12.1) remains valid in this case.

uniformly with respect to h > 0 small enough. In order to prove (13.5), we take α = (α 1 , . . . , α m ) ∈ C m , and we write, uniformly with respect to h and α. Hence, inserting in (13.6), we find,

M A α 2 = m j=1 | m k=1 α k Au θ k , u -θ j L 2 (R n ) | 2 = m j=1 | A α, u -θ j L 2 (R n ) | 2 ,
M A α 2 = O( A α 2 L 2 (D) + e -c/h A α 2 L 2 ),
and thus,

M A α 2 = O( A α 2 L 2 ) = O( A 2 • α 2 L 2 ),
and the result follows by observing that (using the decay properties of the u ±θ k 's and (13.7) again),

α L 2 = O( α L 2 (D) + e -c/h α C m ) = O( α C m ).

  and contained in {±Im z ≤ 0}. When θ is positive, the eigenvalues of H θ are called the resonances of H [Hu1, HeSj2, HeMa], they form a set denoted by Res(H) (on the contrary, when θ < 0, the eigenvalues of H θ are just the complex conjugates of the resonances of H, and are called anti-resonances).

  By the spectral theorem, this leads to the expression (4.10) of Remark 4.3. 10. Proof of Corollary 4.6

(

  12.1) b j (ϕ, h) = | ϕ, φ j | 2 + O(h 2 ) ϕ 2 , and, with Corollary 4.6, this gives, (12.2) e -itH ϕ, ϕ = m j=1 e -itρ j | ϕ, φ j | 2 + O(h 2 ) ϕ 2 .

  where α := m k=1 α k u θ k . Then, we denote by D ⊂ R n and open set such thatU ⊂ D ⊂ R n \Supp F,In particular, on D we have u ±θ k = u k , and, by Agmon estimates, we know that the norms u ±θ k L 2 (R n \D) are exponentially small, uniformly with respect to θ. Therefore, since m = O(h -n ), we can write, (13.6)M A α 2 = m j=1 | A α, u j L 2 (D) | 2 + O(e -c/h ) A α 2 L 2 ,where c > 0 is independent of α, θ, and h. Then, we use the fact that, for the same reason (and sinceu θ k , u -θ j L 2 = δ j,k ), we have, (13.7) u k , u j L 2 (D) = δ j,k + O(e -c/h ),where the positive constant c may be different from the previous one. This permits us to show (e.g., by diagonalizing the family (u k ) 1≤k≤m in L 2 (D) by means of a matrix B = I + O(e -δ/h )) that one has, m j=1 | A α, T u j L 2 (D) | 2 = O( A α 2 L 2 (D) ),

  is the symbol of P 2 , and H p 2 := (∇ ξ p 2 , -∇ x p 2 ) is the Hamilton field of

	p 2 . It is equivalent to the existence of a function G ∈ C ∞ (R 2n ; R) supported
	near {p 2 = 0} (where p 2 (x, ξ) := ξ 2 + V 2 (x)), and satisfying,
	(2.7)	H p 2 G(x, ξ) > 0 on {p 2 = 0}.
	Note that Assumption [V] is nothing but (2.7) with G(x, ξ) = x•ξ. Moreover,
	thanks to Assumption 2, we see that this condition is automatically satisfied
	for |x| large enough.

Acknowledgements A. Martinez was partially supported by the University of Toulon. He is also indebted to the Centre de Physique Théorique de Marseille, where most of this work was done, for his warm hospitality in February and March 2015.

Moreover, in the case n = 1, all the λ j,0 's are simple, and thus so are the λ j 's, with a gap a ∼ h, and (12.2) is valid, too.

When n ≥ 2, some λ j,0 may have some multiplicity. This is for instance the case if we take n = 2 and V 1 (x 1 , x 2 ) = x 2 1 + 4x 2 2 + x 2 1 x 2 + O(|x| 4 ) uniformly near 0. Then (see [START_REF] Helffer | Multiple Wells in the Semiclassical Limit I[END_REF], end of Section 3), the asymptotic of the first eigenvalues of P 1 can be computed, and one finds,

with α := y 2 1 y 2 v 1 (y 1 )w 2 (y 2 )v 3 (y 1 )w 1 (y 2 )dy 1 dy 2 > 0, where v j stands for the normalized j-th eigenfunction of -d 2 y 1 + y 2 1 , and w j for the normalized j-th eigenfunction of -d 2 y 2 + 4y 2 2 .

Thus, we can apply Theorem 4.1 with

13. Appendix 13.1. Appendix 1: Proof of Lemma 6.1. We do it for P θ 1 only, since the sign of θ is not involved in the proof.

We denote by,

1 where the well U has been filled with η (the so-called "filled-well" operator). By analogy with a technique used in [START_REF] Helffer | Résonances en limite semi-classique[END_REF], Section 9 (in particular Formula (9.22)), we consider the operator,

By a straightforward computation, we have, (13.1)

with,

Then, denoting by d 1 the Agmon distance associated with V 1 , one observes that both d 1 (Supp ∇χ, Supp ψ) and d 1 (Supp η, Supp (1ψ) are positive numbers. Therefore, one can apply e.g. the Propositions 9.3 and 9.4 in [START_REF] Helffer | Résonances en limite semi-classique[END_REF] (or, more directly, Agmon estimates on P θ 1 , uniformly with respect to θ) to deduce the existence of some δ 1 > 0, independent of θ, such that, (13.2)

and Agmon estimates on P θ 1 show the existence of δ 2 > 0, still independent of θ, such that, for all j = 1, . . . , m, one has,

and therefore, since

From (13.2)-( 13.3), we obtain,

for some constant δ 3 > 0. Going back to (13.1), we deduce, (13.4)

On the other hand, since the distortion coincides with the identity on the supports of χ and of ψ, we have,

and by construction ( P θ 1z) -1 = O(1) and ( P 1z) -1 = O(1/a). Hence, by (13.4), Lemma 6.1 follows.

13.2. Appendix 2: Proof of Lemma 6.2. In view of (6.5), (6.6), it is enough to prove that, if A is a bounded operator on L 2 (R n ), then the matrix