Thomas Normand

METASTABILITY RESULTS FOR A CLASS OF LINEAR BOLTZMANN EQUATIONS

We consider a semiclassical linear Boltzmann model with a non local collision operator. We provide sharp spectral asymptotics for the small spectrum in the low temperature regime from which we deduce the rate of return to equilibrium as well as a metastability result. The main ingredients are resolvent estimates obtained via hypocoercive techniques and the construction of sharp Gaussian quasimodes through an adaptation of the WKB method. Contents 0 23 4.2. Solving for ℓ s j j≥1 26 5. Computation of the small eigenvalues 27 6. Return to equilibrium and metastability 33 Appendix A. Proof of Lemma 1.4 34 Appendix B. Linear algebra Lemma 36 Appendix C. Asymptotic expansions 36 Appendix D. Proof of Lemma 4.3 40 References 40

h∂ t u + v • h∂ x u -∂ x V • h∂ v u + Q H (h, u) = 0 u |t=0 = u 0
in a semiclassical framework (i.e in the limit h → 0), where h is a semiclassical parameter and corresponds to the temperature of the system. Here we denoted for shortness ∂ x and ∂ v the partial gradients with respect to x and v. This equation is used to model the evolution of a system of charged particles in a gas on which acts an electrical force associated to the real valued potential V that only depends on the space variable x. The interactions between the particles are modelled by the linear operator Q H which is called collision operator. Here the unknown is the function u : R + → L 1 (R 2d ) giving the probability density of the system of particles at time t ∈ R + , position x ∈ R d and velocity v ∈ R d . For our purpose, we introduce the square roots of the usual Maxwellian distributions

µ h (v) = e -v 2 4h (2πh) d/4 and M h = e -V 2h µ h . (1.2)
In many models, we have

Q H (h, M 2 h ) = 0 and Q * H (h, 1) = 0 (1.3) so in particular M 2
h is a stable state of (1.1). In order to do a perturbative study of the time independent operator near M 2 h , we introduce the natural Hilbert space

H = u ∈ D ′ ; M -1 h u ∈ L 2 (R 2d
) . It is clear from the Cauchy Schwarz inequality that H is indeed a subset of L 1 (R 2d ) provided that e -V 2h ∈ L 2 (R d x ). In view of (1.3) and the definition of H, it is more convenient to work with the new unknown f = M -1 h u : R + → L 2 (R 2d ) for which the new equation becomes (1.4)

h∂ t f + v • h∂ x f -∂ x V • h∂ v f + Q h (f ) = 0 f |t=0 = f 0 where Q h = M -1 h • Q H (h, •) • M h .
Our study will be focused on the new time independent operator

P h = v • h∂ x -∂ x V • h∂ v + Q h = X h
0 + Q h for some specific choices of the collision operator Q h , where the notation X h 0 will stand for the operator v • h∂ x -∂ x V • h∂ v , but also for the vector field (x, v) → h(v, -∂ x V (x)). There are plenty of different collision operators studied in the literature, their main properties being that these are symmetric integral operators acting as multiplicators in the position variable x and canceling the Maxwellian distribution. Our work is in particular motivated by the study of the mild relaxation operator introduced in [START_REF] Robbe | Étude semi-classique de quelques équations cinétiques à basse température[END_REF] and given by H 0 (1 + H 0 ) -1 with H 0 the harmonic oscillator in velocity defined by

H 0 = -h 2 ∆ v + v 2 4 - hd 2 . (1.5)
In this spirit, the collision operators we will be working with will always be bounded and self-adjoint so, (X h 0 , C ∞ c (R 2d )) being essentially skew-adjoint, the operator P h (endowed with the appropriate domain) is maximal accretive and (1.4) is well-posed. More generally, some interesting cases of collision operators are given by functions of H 0 (see for instance [START_REF] Hérau | Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation boltzmann equation[END_REF][START_REF] Lerner | Hermite basis diagonalization for the non-cutoff radially symmetric linearized boltzmann operator[END_REF][START_REF]Phase space analysis and functional calculus for the linearized landau and boltzmann operators[END_REF][START_REF]Spectral and phase space analysis of the linearized non-cutoff kac collision operator[END_REF][START_REF] Robbe | Étude semi-classique de quelques équations cinétiques à basse température[END_REF]) which is the setting that we will adopt.

This paper is concerned with the spectral study of the operator P h . This type of questions has recently known some major progress on the impulse of microlocal methods. In the case of the linear Boltzmann equation (1.4), the use of hypocoercive techniques in 2015 in [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF] enabled to get some resolvent estimates and establish a rough localization of the small spectrum of P h which consists of exponentially small eigenvalues in correspondance with the minima of the potential V . This type of result is similar to the one obtained for example for the Witten Laplacian by Helffer and Sjöstrand in [START_REF] Helffer | Puits multiples en mécanique semi-classique. iv. étude du complexe de witten[END_REF] in the 1980's. Such a localization already leads to return to equilibrium and metastability results which can be improved as the description of the small spectrum becomes more precise. For example, sharp asymptotics of the small eigenvalues of the Witten Laplacian were obtained later in the 2000's in [START_REF] Bovier | Metastability in reversible diffusion processes ii. precise asymptotics for small eigenvalues[END_REF] and [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a witten complex approach[END_REF] and later again for Kramers-Fokker-Planck type operators by Hérau et al. in [START_REF] Hérau | Tunnel effect for kramers-fokker-planck type operators[END_REF]. In these papers, the idea was to exhibit a supersymmetric structure for the operator and then study both the derivative acting from 0-forms into 1-forms and its adjoint with the help of basic quasimodes. In [START_REF] Robbe | Étude semi-classique de quelques équations cinétiques à basse température[END_REF], Robbe managed to show that the Boltzmann equation (1.4) with mild relaxation enjoys such a supersymmetric structure. However, in that case, the matrix appearing in the modification of the inner product does not obey good estimates with respect to the semiclassical parameter h. This is why our goal here will be to give precise spectral asymptotics for the operator P h through a more recent approach which consists in directly constructing a family of accurate quasimodes for our operator in the spirit of [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible metastable diffusion processes[END_REF] and [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF].

The aim of this paper is twofold. In a first time, we want to prove a result similar to the one obtained by Robbe in [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF] but for a large class of collision operators. The second goal is to provide complete asymptotics of the small eigenvalues of P h as it was done in [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a witten complex approach[END_REF] for the Witten Laplacian or in [START_REF] Hérau | Tunnel effect for kramers-fokker-planck type operators[END_REF][START_REF]Tunnel effect and symmetries for kramers fokker-planck type operators[END_REF] with recent improvements by Bony et al. in [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF] in the case of Fokker-Planck type differential operators. We manage to establish such results for the equation (1.4) for a class of pseudo-differential collision operators presenting nice symbol properties as well as a factorized structure.

1.2. Setting and main results. For d ′ ∈ N * and Z ∈ C d ′ , we use the standard notation ⟨Z⟩ = (1 + |Z| 2 ) 1/2 . In this paper, we will treat the case of collision operators of the form

Q h = ϱ(H 0 )
with ϱ satisfying the following: Hypothesis 1.1. The function ϱ : R + → R + vanishes at the origin and for all t ≥ 0,

ϱ(t) ≥ 1 C t ⟨t⟩ .
Moreover, it admits an analytic extension to {Re z > -1 C } for which there exist ϱ ∞ ∈ R + and α > 0 such that ϱ(z) = ϱ ∞ + O(⟨z⟩ -α ).

In particular, Q h will be bounded uniformly in h and self-adjoint. An example of such collision operator is the mild relaxation operator introduced in [START_REF] Robbe | Étude semi-classique de quelques équations cinétiques à basse température[END_REF] and given by H 0 (1 + H 0 ) -1 . In order to state the consequences of Hypothesis 1.1, let us introduce a few notations of semiclassical microlocal analysis which will be used in all this paper. These are mainly extracted from [START_REF] Zworski | Semiclassical analysis[END_REF], chapter 4. We will denote Ξ ∈ R d ′ the dual variable of X and use the semiclassical Fourier transform

F h (f )(Ξ) = R d ′ e -i h X•Ξ f (X) dX.
We consider the space of semiclassical symbols

S κ ⟨(X, Ξ)⟩ k = a h ∈ C ∞ (R 2d ′ ) ; ∀α ∈ N 2d ′ , ∃ C α > 0 such that |∂ α a h (X, Ξ)| ≤ C α h -κ|α| ⟨(X, Ξ)⟩ k
where k ∈ R and κ ∈ [0, 1/2]. Note that those symbols are allowed to depend on h; however, in order to shorten the notations, we will drop the index h in the rest of the paper when dealing with semiclassical symbols. Given a symbol a ∈ S κ (⟨(X, Ξ)⟩ k ), we define the associated semiclassical pseudo-differential operator for the Weyl quantization acting on functions u ∈ S(R d ′ ) by

Op h (a)u(X) = (2πh) -d ′ R d ′ R d ′ e i h (X-X ′ )•Ξ a X + X ′ 2 , Ξ u(X ′ ) dX ′ dΞ
where the integrals may have to be interpreted as oscillating integrals. We will denote Ψ κ (⟨(X, Ξ)⟩ k ) the set of such operators. In our setting, we will denote ξ (resp. η) the dual variable of x (resp. v). We also need to introduce the notion of analytic symbols. For our purpose, we almost always consider symbols that do not depend on the variable ξ.

Definition 1.2. For τ > 0, let us introduce the set

Σ τ = {z ∈ C ; |Im z| < τ } d ⊂ C d .
For k ∈ R, we denote S 0 τ (⟨(x, v, η)⟩ k ) the space of symbols a h ∈ S 0 (⟨(x, v, η)⟩ k ) independent of ξ such that:

(i) For all (x, v) ∈ R 2d , a h (x, v, •) is analytic on Σ τ (ii) For all β ∈ N 2d , there exists

C β > 0 such that |∂ β (x,v) a h | ≤ C β ⟨(x, v, η)⟩ k on R 2d × Σ τ .
We will also use the notation

a h = O S 0 τ (⟨(x,v,η)⟩ k ) (h N ) to say that for all α ∈ N 3d , there exists C α,N such that |∂ α a h | ≤ C α,N h N ⟨(x, v, η)⟩ k on R 2d × Σ τ .
Here again, we will drop the index h in the notations of analytic symbols. Using the Cauchy-Riemann equations, we see that item (i) from Definition 1.2 implies that for all β ∈ N 2d and (x, v) ∈ R 2d , the functions ∂ β (x,v) a(x, v, •) are also analytic on Σ τ . Besides, the Cauchy formula implies that for any τ < τ , α ∈ N d and β ∈ N 2d , there exists C α,β such that

|∂ α η ∂ β (x,v) a| ≤ C α,β ⟨(x, v, η)⟩ k on R 2d × Σ τ
i.e up to taking τ smaller, item (ii) from Definition 1.2 can be extended to β ∈ N 3d . Let us introduce a slightly unusual notion of "expansion" where the coefficients are allowed to depend on h: we will say that

a ∼ h j≥0 h j a j (1.6) in S 0 (⟨(x, v, η)⟩ k ) (resp. in S 0 τ (⟨(x, v, η)⟩ k )) if (a j ) j≥0 ⊂ S 0 (⟨(x, v, η)⟩ k ) (resp. (a j ) j≥0 ⊂ S 0 τ (⟨(x, v, η)⟩ k )
) is a family of symbols which may depend on h and are such that for all N ∈ N,

a - N -1 j=0 h j a j = O S 0 (⟨(x,v,η)⟩ k ) (h N ) resp. O S 0 τ (⟨(x,v,η)⟩ k ) (h N )
Finally, we also have the usual notion of classical expansion for a symbol:

a ∼ j≥0 h j a j in S 0 (⟨(x, v, η)⟩ k ) (resp. in S 0 τ (⟨(x, v, η)⟩ k )) means that a ∼ h j≥0 h j a j in S 0 (⟨(x, v, η)⟩ k ) (resp. in S 0 τ (⟨(x, v, η)⟩ k )
) and the (a j ) j≥0 are independent of h. We now extend these notions to matrix valued symbols: if

M = (m p,q ) 1≤p≤n1 1≤q≤n2
is a matrix of functions such that each m p,q ∈ S κ (⟨(x, v, η)⟩ k ) (resp. m p,q ∈ S 0 τ (⟨(x, v, η)⟩ k )), we say that

M ∈ M n1,n2 S κ (⟨(x, v, η)⟩ k ) resp. M ∈ M n1,n2 S 0 τ (⟨(x, v, η)⟩ k ) and we denote Op h (M ) = Op h (m p,q ) 1≤p≤n1 1≤q≤n2
.

The notation

M = O Mn 1 ,n 2 S 0 (⟨(x,v,η)⟩ k ) (h N ) resp. M = O Mn 1 ,n 2 S 0 τ (⟨(x,v,η)⟩ k ) (h N )
means that for all (p, q) ∈ 1,

n 1 × 1, n 2 , the symbol m p,q is O S 0 (⟨(x,v,η)⟩ k ) (h N ) (resp. O S 0 τ (⟨(x,v,η)⟩ k ) (h N )). Furthermore, the notions of expansions M ∼ h n≥0 h n M n and M ∼ n≥0 h n M n in M n1,n2 S 0 (⟨(x, v, η)⟩ k ) resp. M n1,n2 S 0 τ (⟨(x, v, η)⟩ k
) are straightforward adaptations of the ones for scalar symbols.

These notions enable us to introduce a new class of collision operators which appears to be more general that the one given by Hypothesis 1.1. Let us denote b h the twisted derivative

b h = h∂ v + v/2 (1.7)
so that in particular with the notation (1.5) we have H 0 = b * h b h . Hypothesis 1.3. There exists τ > 0 and a symmetric matrix of analytic symbols

M h (x, v, η) = m p,q (x, v, η) 1≤p,q≤d ∈ M d S 0 τ (⟨(v, η)⟩ -2 ) sending R 3d into M d (R)
and such that, with the notation (1.7), the collision operator

Q h satisfies a) Q h = b * h • Op h (M h ) • b h b) M h ∼ n≥0 h n M n in M d S 0 τ (⟨(v, η)⟩ -2 ) c) For all (x, v, η) ∈ R 3d , M h (x, v, η) = M h (x, v, -η) d) For all (x, v, η) ∈ R 3d , M 0 (x, v, η) ≥ 1 C ⟨(v, η)⟩ -2 Id.
Since the (M n ) n do not depend on h, we easily get that these matrices of symbols are also even in η, symmetric, independent of ξ and with values in M d (R); so in particular item d) makes sense. This will enable us to establish Lemma 2.1 which is sometimes reffered to as microscopic coercivity (see for instance [START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF]). As announced, we have the following Lemma which is proven in Appendix A:

Lemma 1.4. Hypothesis 1.1 implies Hypothesis 1.3.

We will also make a few confining assumptions on the function V , assuring for instance that the bottom spectrum of the associated Witten Laplacian is discrete. In particular, our potential will satisfy Assumption 2 from [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible metastable diffusion processes[END_REF] and Hypothesis 1.1 from [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF].

Hypothesis 1.5. The potential V is a smooth Morse function depending only on the space variable x ∈ R d with values in R which is bounded from below and such that

|∂ x V (x)| ≥ 1 C for |x| > C.
Moreover, for all α ∈ N d with |α| ≥ 2, there exists C α such that

|∂ α x V | ≤ C α .
In particular, for every 0 ≤ k ≤ d, the set of critical points of index k of V that we denote U (k) is finite and we set

n 0 = #U (0) . (1.8)
Finally, we will suppose that n 0 ≥ 2.

The last assumption comes from the fact that when n 0 = 1, the so-called small spectrum of the operator P h (i.e its eigenvalues with exponentially small modulus) is trivial, so there is nothing to study. It is shown in [START_REF] Menz | Poincaré and logarithmic sobolev inequalities by decomposition of the energy landscape[END_REF], Lemma 3.14 that for a function V satisfying Hypothesis 1.5, we have V (x) ≥ |x|/C outside of a compact. In particular, under Hypothesis 1.5, it holds e -V /2h ∈ L 2 (R d x ). Moreover, in our setting, X h 0 is a smooth vector field whose differential is bounded on R 2d , so the operator X h 0 endowed with the domain

D = {u ∈ L 2 (R 2d ) ; X h 0 u ∈ L 2 (R 2d )} is skew-adjoint on L 2 (R 2d
) and the set S(R 2d ) is a core for this operator. Therefore, (P h , D) * = (-

X h 0 + Q h , D) and (P h , D) is m-accretive on L 2 (R 2d ).
We can now state our first result which consists in giving a rough localization of the small spectrum of P h that we prove in Section 2.

Theorem 1.6. Assume that Hypotheses 1.3 and 1.5 are satisfied and recall the notation (1.8). Then the operator (P h , D) admits 0 as a simple eigenvalue. Moreover, there exists c > 0 and h 0 > 0 such that for all 0 < h ≤ h 0 , Spec(P h ) ∩ {Re z ≤ ch 2 } consists of exactly n 0 eigenvalues (counted with algebraic multiplicity) that are exponentially small with respect to 1/h and for all 0 < c ≤ c, the resolvent estimate

(P h -z) -1 = O(h -2 )
holds uniformly in {Re z ≤ ch 2 }\B(0, ch 2 ). Finally, except for 0, the real parts of these small eigenvalues are positive.

This result can be seen as a generalization of Theorem 3.0.2 from [START_REF] Robbe | Étude semi-classique de quelques équations cinétiques à basse température[END_REF] (up to the h 2 instead of h) as we saw that the mild relaxation operator (which is the collision operator studied in this reference) satisfies our hypotheses. In our case we get a localization of order h 2 because we adopt a simpler proof based on hypocoercivity (inspired by [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF]) than the one presented in [START_REF] Robbe | Étude semi-classique de quelques équations cinétiques à basse température[END_REF].

In order to study the long time behavior of the solutions of (1.4), we need a precise description of the small spectrum of P h . To this aim, we construct in Sections 3 and 4 in the spirit of the WKB method a family of accurate quasimodes localized around the minima of V that enables us to establish sharp asymptotics of the small eigenvalues of P h . This leads in Section 5 to the following Theorem which is the main result of this paper. For the sake of simplicity, we make in the statement an additionnal assumption (Hypothesis 3.10) on the topology of the potential V that could actually be omitted (see [START_REF] Michel | About small eigenvalues of witten laplacian[END_REF] or [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF]).

Theorem 1.7. Suppose that Hypotheses 1.3, 1.5 and 3.10 are satisfied and denote m a global minimum of V . According to Theorem 1.6, we can associate to each m ∈ U (0) \{m} a non zero exponentially small eigenvalue of P h that we denote λ(m, h). These eigenvalues satisfy the following formula:

λ(m, h) = he -2 S(m) h det(Hess m V ) 1/2 2π B h (m)
with S defined in Definition 3.6 and B h (m) admitting a classical expansion whose first term is

s∈j(m) | det(Hess s V )| -1/2 M 0 (s, 0, 0)ν s 2 • ν s 2
where the map j is also defined in Definition 3.6, the matrix M 0 (s, 0, 0) is introduced in Hypothesis 1.3 and the vector ν s 2 ∈ R d is defined in Proposition 4.7. When Hypothesis 1.3 is replaced by Hypothesis 1.1, we can give a slightly more precise statement. In that case, denoting µ s the only negative eigenvalue of Hess s V , the first term of B h (m) is 1 2

s∈j(m) | det(Hess s V )| -1/2 -ϱ ′ (0) + ϱ ′ (0) 2 -4µ s .
Indeed, under Hypothesis 1.1, it is shown in Appendix A, more precisely in (A.13) that M 0 (s, 0, 0) = ρ(0) Id = ϱ ′ (0) Id. Thanks to Proposition 4.7, we then have

Hess s V ν 2 = -ϱ ′ (0) 2 (1 + ν 2 2 )ν 2 2 ν 2 and consequently ν 2 2 = - 1 2 + ϱ ′ (0) 2 -4µ s 2ϱ ′ (0) so the statement follows.
Finally, Section 6 consists in using the sharp localization obtained in Theorem 1.7 in order to discuss the phenomena of return to equilibrium and metastability for the solutions of (1.4). More precisely, we are able to give a sharp rate of convergence of the semigroup e -tP h /h towards P 1 , the orthogonal projector on Ker P h : denoting λ * a non zero eigenvalue of P h whose real part is minimal, we establish that the rate of return to equilibrium is essentially given by Re λ * /h: Corollary 1.8. Under the assumptions of Theorem 1.7, there exists h 0 > 0 such that for all 0 < h ≤ h 0 , t ≥ 0 and N ≥ 1, there exists

C N > 0 such that ∥e -tP h /h -P 1 ∥ ≤ C N e -t Re λ * (1-C N h N )/h .
Moreover, if λ * does not share its expansion given by Theoerm 1.7 with another eigenvalue of P h (in particular it is a simple eigenvalue), then λ * is real and we even have

∥e -tP h /h -P 1 ∥ ≤ Ce -tλ * /h .
Besides, in the spirit of [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF], we also show the metastable behavior of the solutions of (1.4): Corollary 1.9. Suppose that the assumptions of Theorem 1.7 hold true. Let us consider some local minima

m 1 = m, m 2 , . . . , m K such that S U (0) = {+∞ = S(m 1 ) > S(m 2 ) > • • • > S(m K )}
for the map S from Definition 3.6. For 2 ≤ k ≤ K, denote P k the spectral projection associated to the eigenvalues that are O e -2 S(m k ) h . Then for any times (t ± k ) 1≤k≤K satisfying

t - K ≥ h -1 | ln(h ∞ )| and t - k ≥ | ln(h ∞ )|e 2 S(m k+1 ) h for k = 1, . . . , K -1 as well as t + 1 = +∞ and t + k = O h ∞ e 2 S(m k ) h for k = 2, . . . , K one has e -tP h /h = P k + O(h ∞ ) on [t - k , t + k ].
In other words, we have shown the existence of timescales on which, during its convergence towards the global equilibrium, the solution of (1.4) will essentially visit the metastable spaces associated to the small eigenvalues of P h . The results presented in this paper should be reasonably easy to adapt to the case of collision operators satisfying Hypothesis 1.3 with the space S 0 replaced by S κ for κ ∈ [0, 1/2[ (we should get some expansions in powers of h 1-2κ instead of just h). Another perspective would then be to study the critical case κ = 1/2 which should in particular cover the linear relaxation collision operator corresponding to the linear BGK model

Q h = h(1 -Π h ) (1.9)
where Π h denotes the orthogonal projection on

E h = µ h L 2 (R d x ) (1.10)
and for which Robbe gave a first localization of the small spectrum of the associated operator

X h 0 + Q h in [20].

Rough description of the small spectrum

Throughout the paper, we assume that Hypotheses 1.3 and 1.5 hold true. This implies in particular that Q h is bounded uniformly in h and self-adjoint in L 2 (R 2d ). Let us begin with a Lemma which consists in comparing our collision operator with the one introduced in (1.9) and studied in [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF]. This will in particular enable us to use some computations from [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF] later on. Lemma 2.1. There exists h 0 > 0 such that for all 0 < h < h 0 ,

Q h ≥ h C (1 -Π h )
where Π h is the projection introduced in (1.9). In particular, Q h is non negative.

Proof. Since the space E h defined in (1.10) is contained in Ker Q h , it is enough to prove that ⟨Q h u, u⟩

≥ h C ∥u∥ 2 for u ∈ E ⊥ h . Let u ∈ E ⊥
h and recall the notations H 0 and H 1 from (1.5) and (A.4). Let us consider an approximate square root A of (1 + H 1 ) given by

A = Op h 1 + v 2 /4 + η 2 + h(1 -d/2) 1/2 Id ∈ Ψ 0 ⟨(v, η)⟩ .
By symbolic calculus, we easily have

A 2 = 1 + H 1 + h 2 R 1 with R 1 ∈ Ψ 0 ⟨(v, η)⟩ 2 .
Besides, the symbol of A is clearly elliptic so A is invertible and its inverse is also a pseudo-differential operator satisfying

A -2 = (1 + H 1 ) -1 + h 2 R 2 with R 2 ∈ Ψ 0 ⟨(v, η)⟩ -2
(see for instance [START_REF] Dimassi | Spectral asymptotics in the semiclassical limit[END_REF], chapter 8). Thus, using the factorization from Hypothesis 1.3 and the self-adjointness of A, we get

⟨Q h u, u⟩ = A Op h (M h )A A -1 b h u , A -1 b h u .
Now according to Hypothesis 1.3 and symbolic calculus again, the principal symbol of A Op h (M h )A is elliptic so we can use the Gårding inequality to write

⟨Q h u, u⟩ ≥ 1 C A -2 b h u , b h u ≥ 1 C b * h (1 + H 1 ) -1 b h u , u - h 2 C b * h R 2 b h u , u .
Still using symbolic calculus, we get b * h R 2 b h = O(1) so applying (A.5) we finally have

⟨Q h u, u⟩ ≥ 1 C H 0 (1 + H 0 ) -1 u , u -O(h 2 )∥u∥ 2
and the conclusion comes from the fact that the spectrum of

H 0 (1 + H 0 ) -1 | E ⊥ h is contained in [h/C, +∞[. □
We can already prove that 0 is a simple eigenvalue of (P h , D) and that the other eigenvalues have positive real part. It is easy to check that M h defined in (1.2) is in Ker P h . Now let λ ∈ R and let us prove that for u ∈ Ker (P h -iλ), one has u ∈ C M h . Since X h 0 is skew-adjoint and Q h is self-adjoint and non-negative, we have 0 = Re⟨(P h -iλ)u, u⟩ = ∥Q

1/2 h u∥ 2 so in particular u ∈ Ker Q h = E h according to Lemma 2.1. Therefore, u = wµ h with w ∈ L 2 (R d
x ) and using that µ -1 h X h 0 u = iλw does not depend on v, we get in the sense of distributions ∂ x (e V /2h w) = 0 which yields the desired result.

Hypocoercivity. Let us now use the dilatation operators

S h :    L 2 (R 2d ) → L 2 (R 2d ) u → h -d/2 u . √ h , . √ h T h :    L 2 (R d x ) → L 2 (R d x )
u → h -d/4 u . √ h that were introduced in [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF] in which these were combined with a scaling of Π h to conjugate P h to a nonsemiclassical operator with h-dependent potential. In our case, it will enable us to use some computations and results already established in [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF].

Lemma 2.2. Denoting X 0 = v • ∂ x -∂ x V h (x) • ∂ v where V h = h -1 V ( √ h •), Q1 = h -1 S -1 h Q h S h and Dom (P ) = {u ∈ L 2 (R 2d ) ; X 0 u ∈ L 2 (R 2d )}, P = X 0 + Q1 , one has (hP , Dom(P )) = (S -1 h P h S h , S -1 h D). Moreover, (hP , Dom(P )) * = (S -1 h P * h S h , S -1 h D). Proof. We have for u ∈ L 2 (R 2d )
hX 0 u = S -1 h X h 0 S h u so using that S h is bounded we get Dom (P ) = S -1 h D. Consequently, (hP , Dom(P )) = (S -1 h P h S h , S -1 h D) and the result for the adjoint follows immediately. □

We also recall the notations of the following differential operators from [START_REF] Hérau | Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation boltzmann equation[END_REF] and [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF]:

a = ∂ x + ∂ x V h 2 ; b = ∂ v + v 2 and Λ 2 = a * a + b * b + 1. The operator (Λ 2 , C ∞ c (R 2d
)) is essentially self-adjoint. The Schwartz space S(R 2d ) is included in the domain of its self-adjoint extension (Λ 2 , D(Λ 2 )) which is invertible. We can then define the operator L = Λ -2 a * b, which is bounded uniformly in h (see [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF], Lemma 2.7), as well as the perturbation hε(L + L * ) = O(h) where ε > 0 will be chosen small enough later. Besides, notice that a * a = -∆

x + |∂ x V h | 2 /4 -∆V h /2 =: ∆ V h /2 is the Witten Laplacian in x associated to the potential V h /2 and that ∆ h V /2 := hT h a * aT -1 h = -h 2 ∆ x + |∂ x V | 2 /4 -h∆V /2
is the semi-classical Witten Laplacian associated to the potential V /2. The small spectrum of this operator was first studied by Helffer and Sjöstrand in [START_REF] Helffer | Puits multiples en mécanique semi-classique. iv. étude du complexe de witten[END_REF] and we now know that we can construct an orthonormal family (φ j ) 1≤j≤n0 ⊂ C ∞ c (R d x ) of quasimodes associated to this operator given by

φ j = χ j e -V -V (x j ) 2h
where x j is one of the local minima of V and χ j is a cut-off function localizing around x j . Recall the notation µ h from (1.2) and let us now define the families of functions g h j = φ j µ h and g j = S -1 h g h j for 1 ≤ j ≤ n 0 . These are actually quasimodes for our operators P h and P * h :

Lemma 2.3. The family (g h j ) 1≤j≤n0 is orthonormal and there exists α > 0 such that for all 1 ≤ j ≤ n 0 ,

P h g h j = O L 2 (e -α h ), P * h g h j = O L 2 (e -α h ).
Moreover, P h g h j and P * h g h j are in S(R 2d ) ⊆ D and we have

P * h P h g h j = O L 2 (e -α h ), P h P * h g h j = O L 2 (e -α h ).
Proof. The proof is the same as the one of Lemma 2.4 from [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF] since with the notation (1.10) and Lemma 2.1 we also have

E h = Ker Q h . □
One of the key results of this section is that the real part of the perturbation of our operator is bounded from below on a subspace of finite codimension given by the orthogonal of the quasimodes: Proposition 2.4. Denote N ± h,ε the bounded self-adjoint operator Id ± εh(L + L * ). There exists ε > 0 and h 0 > 0 such that for all h ∈]0, h 0 ] and u ∈ S(R 2d ) ∩ (g j ) ⊥

1≤j≤n0 , one has

Re⟨N + h,ε P u, u⟩ ≥ h C ∥u∥ 2
as well as

Re⟨N - h,ε P * u, u⟩ ≥ h C ∥u∥ 2 .
Proof. One has for u ∈ S(R 2d ), using the fact that X 0 is skew-adjoint:

Re⟨N + h,ε P u, u⟩ = Re⟨P u, N + h,ε u⟩ = Re⟨ Q1 u, N + h,ε u⟩ + Re⟨X 0 u, N + h,ε u⟩ = ∥ Q1/2 1 u∥ 2 + hεRe⟨ Q1 u, (L + L * )u⟩ + hεRe⟨X 0 u, (L + L * )u⟩ = ∥ Q1/2 1 u∥ 2 + hεRe⟨ Q1 u, (L + L * )u⟩ + hεRe⟨[L, X 0 ]u, u⟩ = I + hII + hIII
Note that if we replace P by P * and N + h,ε by N - h,ε , we get I -hII + hIII. Besides, it is also proven in [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF] that

[L, X 0 ] = A + Λ -2 a * a where A is also bounded uniformly in h. Since ∥Q h ∥ ≤ C and Q h ≥ h C (1 -Π h ) according to Lemma 2.1, we get ∥ Q1 ∥ ≤ C h and Q1 ≥ 1 C (1 -Π 1 ). Hence I ± hII ≥ I -h|II| ≥ ∥ Q1/2 1 u∥ 2 -hε∥ Q1 u∥∥(L + L * )u∥ ≥ ∥ Q1/2 1 u∥ 2 - √ Ch 1 2 ε∥ Q1/2 1 u∥∥(L + L * )u∥ ≥ 1 2 ∥ Q1/2 1 u∥ 2 -2Chε 2 ∥L∥ 2 ∥u∥ 2 ≥ 1 2C ∥(1 -Π 1 )u∥ 2 -2Chε 2 ∥L∥ 2 ∥u∥ 2 (2.1)
We can combine this with the following estimate from [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF] (proof of Proposition 2.5): there exists δ > 0 such that for u ∈ (g j ) ⊥ 1≤j≤n0 ,

III ≥ - 1 4 ∥(Id -Π 1 )u∥ 2 -ε 2 ∥A∥ 2 ∥u∥ 2 + εδ 4 ∥Π 1 u∥ 2 -ε∥(Id -Π 1 )u∥ 2 .
This yields for ε <

δ 4(∥A∥ 2 +C∥L∥ 2 ) that I ± hII + hIII ≥ 1 C ∥(Id -Π 1 )u∥ 2 + h εδ 4 ∥Π 1 u∥ 2 -hε 2 ∥A∥ 2 + C∥L∥ 2 ∥u∥ 2 ≥ h C ∥u∥ 2 . (2.2) so the proof is complete. □ This result extends to u ∈ (g j ) ⊥ 1≤j≤n0 ∩ Dom (P ) since S(R 2d
) is a core for both (P, Dom (P )) and (P * , Dom (P * )). It only differs from Proposition 2.5 in [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF] by a factor h in the estimate. This comes from the fact that in our case, Q1 = O(h -1 ) and not O(1) (because Q h = O(1) and not O(h)) so we have to use a perturbation of order h (the operator N ± h,ε ) to obtain the gain in ∥(1 -Π 1 )u∥ 2 in (2.1). As a consequence, the gain in ∥Π 1 u∥ 2 from (2.2) is of order h and not of order 1.

Corollary 2.5. There exists c > 0 and h

0 > 0 such that for all h ∈]0, h 0 ], u ∈ D ∩ (g h j ) ⊥ 1≤j≤n0 and z ∈ C uith Re z ≤ ch 2 ∥(P h -z)u∥ ≥ ch 2 ∥u∥ and ∥(P * h -z)u∥ ≥ ch 2 ∥u∥. Proof. Recall that N + h,ε = 1 + O(h). Hence, for u ∈ D ∩ (g h j ) ⊥ 1≤j≤n0
, we have by putting u = S h w and using that S h is unitary

∥(P h -z)u∥∥u∥ ≥ 1 2 ∥(P h -z)u∥∥N + h,ε w∥ ≥ 1 2 Re⟨(P h -z)u, S h N + h,ε w⟩ = 1 2 Re⟨N + h,ε (hP -z)w, w⟩ ≥ h 2 C ∥u∥ 2 -Re z∥N + h,ε ∥∥u∥ 2 ≥ h 2 2C ∥u∥ 2 if Re z ≤ h 2 /2C.
∥Au∥ 2 = O(e -2α h )∥u∥ 2 .
Now if we denote P the orthogonal projection on Span (g h j ) 1≤j≤n0 , we get by using Corollary 2.5 that for z ∈ C such that Re z ≤ ch 2 and u ∈ D

∥(P h -z)u∥ 2 = ∥(P h -z)(Id -P)u + (P h -z)Pu∥ 2 = ∥(P h -z)(Id -P)u∥ 2 + ∥(P h -z)Pu∥ 2 + 2Re⟨(P h -z)(Id -P)u, (P h -z)Pu⟩ ≥ c 2 h 4 ∥(Id -P)u∥ 2 + |z| 2 ∥Pu∥ 2 -O(e -α h )∥u∥ 2 + 2Re⟨(P h -z)(Id -P)u, (P h -z)Pu⟩.
The last term equals 2Re ⟨(Id -P)u,

P * h P h Pu⟩ -z⟨(Id -P)u, P h Pu⟩ -z⟨(Id -P)u, P * h Pu⟩ = (1 + |z|)O(e -α h )∥u∥ 2 .
Therefore choosing c ≤ c, there exists h 0 > 0 such that for h ≤ h 0 and z such that ch 2 ≤ |z| ≤ ch 2

∥(P h -z)u∥ 2 ≥ |z| 2 + O(e -α h ) ∥u∥ 2 ≥ c2 h 4 2 ∥u∥ 2 .
Once again, the same estimate holds with P * h instead of P h and since the annulus we are working on is invariant by complex conjugation we also have

∥(P h -z) * u∥ ≥ ch 2 2 ∥u∥ .
Therefore, we get the following resolvent estimate on the annulus centered in 0 and of radiuses ch 2 and ch 2 :

∥(P h -z) -1 ∥ = O(h -2 ) for ch 2 ≤ |z| ≤ ch 2 . (2.3)
We can now consider the spectral projection

Π 0 = 1 2iπ |z|=ch 2 (z -P h ) -1 dz (2.4)
and its range that we denote H. This operator will yield some information on Spec(P h ) ∩ B(0, ch 2 ) and therefore enable us to prove the main statement from Theorem 1.6.

The main point is that H is of dimension n 0 . It can be obtained by a direct adaptation of the proof of Proposition 3.1 from [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF]. Hence Spec(P h ) ∩ B(0, ch 2 ) which is the same as Spec(P h | H ) consists of n 0 eigenvalues (counted with algebraic multiplicity). Here again, our result slightly differs from the one in [START_REF]Small eigenvalues of the low temperature linear relaxation boltzmann equation with a confining potential[END_REF] as we do not rule out the possibilities that P h | H contains some Jordan blocks and that some of its eigenvalues are not real. It only remains to prove that these are exponentially small with respect to 1/h. We begin by noticing that thanks to Lemma 2.3, we have (z

-P h )g h j = zg h j + O(e -α h ) and (z -P * h )g h j = zg h j + O(e -α h
) from which we easily deduce

Π 0 g h j = g h j + O(e -α h ) and Π * 0 g h j = g h j + O(e -α h ). (2.5)
In particular, (Π 0 g h j ) 1≤j≤n0 is almost orthonormal so for u = u j Π 0 g h j ∈ H, we have

∥u∥ 2 = 1 + O(e -α/h ) n0 j=1 |u j | 2 .
Therefore it is enough to prove that P h is exponnentially small on (Π 0 g h j ) 1≤j≤n0 . But thanks to the resolvent estimate (2.3), it is easy to see that Π 0 = O(1) and since P h and Π 0 commute, we get the desired result.

To complete the proof of Theorem 1.6, it only remains to show the existence of the resolvent on {Re z ≤ ch 2 }\B(0, ch 2 ) as well as the estimate in O(h -2 ). Lemma 2.6. Denote Π0 = 1 -Π 0 . For all u ∈ L 2 (R 2d ), we have

Π0 u = w + r whith w ∈ (g h j ) ⊥ 1≤j≤n0 and r ∈ Span (g h j ) 1≤j≤n0 satisfying r = O(e -α h )∥ Π0 u∥.
Proof. First we take for r the orthogonal projection of Π0 u on Span (g h j ) 1≤j≤n0 . Then we notice that using (2.5), we get ⟨g h j , Π0 u⟩ = ⟨ Π * 0 g h j , Π0 u⟩ = O(e -α h )∥ Π0 u∥ which implies the announced estimate. □ Lemma 2.7. For all r ′ ∈ Span (g j ) 1≤j≤n0 , we have N ± h,ε r ′ ∈ Dom (P * ) = Dom (P ). Moreover, the restrictions to the finite dimensional subspace Span (g j ) 1≤j≤n0 of the operators P N ± h,ε and P * N ± h,ε are all O(1).

Proof. For the first statement, it is sufficient to show that for 1 ≤ j ≤ n 0 , the functions Lg j and L * g j are both in Dom (P ). But we have in the sense of distributions

X 0 Lg j = [X 0 , L]g j + LX 0 g j (2.6)
and we saw in the proof of Proposition 2.4 that [X 0 , L] is a bounded operator on L 2 (R 2d ) so it is then clear that X 0 Lg j ∈ L 2 (R 2d ) i.e Lg j ∈ Dom (P ). The same goes easily for L * g j . For the second statement, using Lemma 2.3 and the fact that Q1 = O(h -1 ), it suffices to notice that for 1 ≤ j ≤ n 0 , (2.6) implies that X 0 Lg j and X 0 L * g j are both O(1) as we saw that L and [X 0 , L] are O [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF]. □ Proposition 2.8. Consider Ph the restriction of P h to Π0 D acting on Π0 L 2 (R 2d ). Then for all z ∈ C such that Re z ≤ ch 2 , the resolvent ( Ph -z) -1 exists and we have the uniform estimate

( Ph -z) -1 = O(h -2 ).
Proof. We actually prove that the result of Proposition 2.4 remains true when replacing the set (g j ) ⊥ 1≤j≤n0 ∩ Dom (P ) by S -1 h Π0 D. We will deduce that the result of Corollary 2.5 also remains true when taking u ∈ Π0 D instead of (g h j ) ⊥ 1≤j≤n0 ∩ D, which is precisely the statement that we want to prove. Let u ∈ D, using the notations from Lemma 2.6 we have

Re ⟨P S -1 h Π0 u, N + h,ε S -1 h Π0 u⟩ = Re ⟨P S -1 h w, N + h,ε S -1 h w⟩ + Re ⟨P S -1 h w, N + h,ε S -1 h r⟩ + Re ⟨P S -1 h r, N + h,ε S -1 h w⟩ + Re ⟨P S -1 h r, N + h,ε S -1 h r⟩ . Now let us denote w ′ = S -1 h w ∈ (g j ) ⊥ 1≤j≤n0 ∩ Dom (P ) and r ′ = S -1 h r ∈ Span (g j ) 1≤j≤n0
. We can use Proposition 2.4 as well as Lemmas 2.6 and 2.7 to get

Re ⟨N + h,ε P S -1 h Π0 u, S -1 h Π0 u⟩ = Re ⟨P w ′ , N + h,ε w ′ ⟩ + Re ⟨w ′ , P * N + h,ε r ′ ⟩ + Re ⟨N + h,ε P r ′ , w ′ ⟩ + Re ⟨P r ′ , N + h,ε r ′ ⟩ ≥ h C ∥w∥ 2 -O ∥w∥ ∥r∥ -O(e -α h ∥r∥) ≥ h 2C ∥S -1 h Π0 u∥ 2 .
As usual, all of the above remains true with P * and N - h,ε instead of P and N + h,ε so the proof is now complete. □

End of Proof of Theorem 1.6 : Let z ∈ C satisfying Re z ≤ ch 2 and |z| ≥ ch 2 and recall the notation H = Ran Π 0 . We already know from Proposition 2.8 that Ph -z is invertible, but it is clearly also the case of

P h | H -z since P h | H = O(e -α/h
). Therefore P h -z is invertible and we have

(P h -z) -1 = ( Ph -z) -1 Π0 + (P h | H -z) -1 Π 0 . (2.7)
Besides, we easily have for such z that ∥(P h | H -z)u∥ ≥ 1 C h 2 ∥u∥ which combined with (2.7), Proposition 2.8 and the fact that

∥Π 0 ∥ = O(1) yields the estimate (P h -z) -1 = O(h -2 ). □ 3. Accurate quasimodes 3.1. General form. Let us denote W (x, v) = V (x) 2 + v 2
4 the global potential on R 2d . Before we can construct our quasimodes, we need to recall the general labeling of the minima which originates from [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a witten complex approach[END_REF] and was generalized in [START_REF]Tunnel effect and symmetries for kramers fokker-planck type operators[END_REF], as well as the topological constructions that go with it. In our case, it has to be done for the global potential, i.e the function W . However, by the definition of W , a strong connection between these constructions for W and the ones for V will appear, leading to simplifications. In order to give a proper statement about this connection, let us construct the labelings for both W and V . To this aim, we consider d ′ ∈ N * and a smooth Morse function Y on R d ′ bounded from below, having at least two local minima and such that |∇Y | ≥ 1/C outside of a compact. According to Hypothesis 1.5, one can for instance take Y = V /2 or Y = W and recall that as we discussed following Hypothesis 1.5, it implies that Y (X) ≥ |X|/C outside of a compact. We also denote U (k),Y the critical points of Y of index k. For shortness, we will write "CC" instead of "connected component".

Lemma 3.1. If X ∈ U (1)
,Y , then there exists r 0 > 0 such that for all 0 < r < r 0 , X has a connected neighborhood U r in B(X, r) such that U r ∩ {Y < Y (X)} has exactly 2 CCs.

Proof. Let X ∈ U (1),Y ; according to the Morse Lemma, there exists a connected neighborhood U r of X, r ′ > 0 and φ : U r → B(0, r ′ ) a smooth diffeomorphism such that

Y • φ -1 = Y (X) + 1 2 ⟨Hess X Y •, •⟩.
Besides, it is easy to see that

U r ∩ {Y < Y (X)} = φ -1 {y ∈ B(0, r ′ ) ; ⟨Hess X Y y, y⟩ < 0}
and {y ∈ B(0, r ′ ) ; ⟨Hess X Y y, y⟩ < 0} has exactly 2 CCs. □ Lemma 3.2. Let X ∈ R d ′ and suppose there exists r 0 > 0 such that for every neighborhood U of X in B(X, r 0 ), the set U ∩ {Y < Y (X)} is not connected. Then X ∈ U (1),Y .

Proof. First we clearly have that ∇Y (X) = 0 since otherwise one could use the implicit function theorem to find a neighborhood U of X in B(X, r 0 ) such that U ∩ {Y < Y (X)} is connected. It is also clear that X / ∈ U (0),Y so let us assume by contradiction that X ∈ U (k),Y with k ≥ 2. Then using the Morse Lemma as in the proof of Lemma 3.1, we would once again get that X has a neighborhood U in B(X, r 0 ) such that U ∩ {Y < Y (X)} has the same number of CCs as {y ∈ B(0, r) ; ⟨Hess X Y y, y⟩ < 0} which is connected since k ≥ 2. Hence X has to be in U (1),Y . □

In view of the result from Lemma 3.1 and following the approach from [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a witten complex approach[END_REF][START_REF]Tunnel effect and symmetries for kramers fokker-planck type operators[END_REF], we give the following definition:

Definition 3.3. a)
We say that X ∈ U (1),Y is a separating saddle point and we denote X ∈ V (1),Y if for every r > 0 small enough, the two CCs of and it is then clear that σ is well defined. Besides, if (σ n ) n≥1 is an increasing sequence in the set from (3.1) that converges towards σ and γ :

U r ∩ {Y < Y (X)} are contained in different CCs of {Y < Y (X)}. b) We say that σ ∈ R is a separating saddle value if σ ∈ Y (V (1),Y ). c) Finally, we say that a set E ⊂ R d ′ is critical if there exists σ ∈ Y (V (1),Y ) such that E is a CC of {Y < σ} satisfying ∂E ∩ V (1),Y ̸ = ∅.
[0, 1] → R d ′ is a continuous path linking m and m ′ , then γ([0, 1]) ∩ R d ′ \{Y < σ} = n≥1 γ([0, 1]) ∩ R d ′ \{Y < σ n }
is non empty by compactness so we can consider Ω ∋ m and Ω ′ ∋ m ′ two different CCs of {Y < σ}. To prove that σ is a separating saddle value, we will actually show that there exists a CC of {Y < σ} that we denote Ω ′′ which is not Ω and satisfies Ω ∩ Ω ′′ ̸ = ∅. Assume by contradiction that there exists ε > 0 such that (Ω + B(0, ε))\Ω is included in {Y ≥ σ}. In that case, the points of (Ω + B(0, ε))\Ω on which Y takes the value σ are local minima of Y which is a Morse function, so there are finitely many such points. Thus, up to taking ε smaller, we can assume that

Γ := dist(•, Ω) -1 ({ε}) ⊆ {Y > σ}.
Hence there exists δ > 0 such that the minimum of Y on Γ is σ + δ. Since any continuous path linking m and m ′ has to cross Γ, m and m ′ are in two different CCs of {Y < σ + δ/2}. This contradicts the maximality of σ and proves the existence of Ω ′′ . Hence, Lemma 3.2 implies that Ω ∩ Ω ′′ ⊆ U (1),Y and then Ω ∩ Ω ′′ ⊆ V (1),Y follows obviously from the definition of V (1),Y . □ Thanks to Lemma 3.4, we know that

V (1),Y ̸ = ∅. Let us then denote σ 2 > • • • > σ N
where N ≥ 2 the different separating saddle values of Y and for convenience we set σ 1 = +∞. We call labeling of the minima of Y any injection l :

U (0),Y → 1, N × N * . If l(m) = (k, j), we denote for shortness m = m k,j .
We are going to introduce the usual labeling of the minima for a potential Y (see for instance [START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a witten complex approach[END_REF][START_REF]Tunnel effect and symmetries for kramers fokker-planck type operators[END_REF][START_REF] Peutrec | Sharp spectral asymptotics for non-reversible metastable diffusion processes[END_REF]). We adopt a slightly unusual point of view in order to facilitate the establishment of the correspondence between the constructions for W and the ones for V /2 that we will state later on. For σ ∈ R ∪ {+∞}, let us denote C Y σ the set of all the CCs of {Y < σ}. Given a labeling l of the minima, we denote for

k ∈ 1, N U (0),Y k = l -1 ( 1, k × N * ) ∩ {Y < σ k }
and we say that the labeling is adapted to the separating saddle values if for all k ∈ 1, N , each element of l -1 ({k}×N * ) is a global minimum of Y restricted to some CC of {Y < σ k } and the map T Y k :

U (0),Y k → C Y σ k sending m ∈ U (0),Y k on the element of C Y σ k to which it belongs is bijective. In particular, l -1 ({k} × N * ) is contained in U (0),Y k
. Such labelings exist, one can for instance easily check that the usual labeling procedure presented in [START_REF]Tunnel effect and symmetries for kramers fokker-planck type operators[END_REF] is adapted to the separating saddle values. Lemma 3.5. Under an adapted labeling of the minima of Y , for any 2 ≤ k ≤ N , the elements of

T Y k l -1 ({k} × N * ) are critical. Proof. Let m k,j ∈ l -1 ({k} × N * ). There exists a CC of {Y < σ k-1 } that we call E which is such that T Y k (m k,j ) ⊆ E and E contains some m k ′ ,j ′ ∈ E for 1 ≤ k ′ ≤ k -1 and j ′ ∈ N * by bijectivity of T Y k-1 . Therefore, m k ′ ,j ′ and m k,j are in the same CC of {Y < σ k-1 } but are not both in T Y k (m k,j
) this time by bijectivity of T Y k . Applying Lemma 3.4 to m k ′ ,j ′ and m k,j , we obtain a separating saddle value σ which is the maximal real number such that m k ′ ,j ′ and m k,j are in two different CCs of {Y < σ}. Therefore we get σ = σ k so T Y k (m k,j ) is one of the CCs of {Y < σ} called Ω and Ω ′ in Lemma 3.4 and which are critical. □ Definition 3.6. Given an adapted labeling, we can now define the following mappings:

E Y : U (0),Y --→ P(R d ′ ) m k,j --→ T Y k (m k,j ). j Y : U (0),Y → P V (1),Y ∪ {s 1 }
given by j Y (m) = s 1 where s 1 is a fictive saddle point such that Y (s 1 ) = σ 1 = +∞; and for

2 ≤ k ≤ N , j Y (m k,j ) = ∂E Y (m k,j ) ∩ V (1)
,Y which is not empty according to Lemma 3.5 and included in

{Y = σ k }. σ Y : U (0),Y → Y (V (1),Y ) ∪ {σ 1 } m → Y (j Y (m))
where we allow ourselves to identify the set Y (j Y (m)) and its unique element in

Y (V (1),Y ) ∪ {σ 1 }. S Y : U (0),Y --→]0, +∞] m --→ σ Y (m) -Y (m).
Let us now state a Lemma that will enable us to show that, roughly speaking, the previous constructions for Y = V /2 are the projections on R d

x of the ones for Y = W . First, we give the following easy observation. Remark 3.7. By definition of W , we have

V /2 = W (•, 0). Moreover, if (x 0 , v 0 ) ∈ {W < σ}, then {x 0 } × B(0, |v 0 |) ⊆ {W < σ}.
For shortness, we denote

C σ = C V /2 σ and C σ = C W σ as well as U (k) = U (k),V /2 and U (k) = U (k),W (we do similarly with V or U k instead of U). Notice that U (k) = U (k) × {0}.
We introduce the natural projection π x : R 2d → R d

x sending (x, v) on x that we also consider as a map from P(R 2d ) to P(R d x ). Lemma 3.8. For all σ ∈ R, the projection π x sends C σ in C σ . Moreover, the map

π x : C σ → C σ is bijective.
Proof. The proof of the first statement is an easy consequence of Remark 3.7. For the second statement, let x ∈ E ∈ C σ and denote Ẽ the element of C σ containing (x, 0). By the first statement, we necessarily have π x ( Ẽ) = E so we have shown the surjectivity. Now let Ẽ1 , Ẽ2 ∈ C σ such that π x ( Ẽ1 ) = π x ( Ẽ2 ) = E 1 . Let also (x 1 , v 1 ) ∈ Ẽ1 and (x 2 , v 2 ) ∈ Ẽ2 . Since x 1 , x 2 ∈ E 1 , there exists a path (γ(t), 0) from (x 1 , 0) to (x 2 , 0) contained in {W < σ}. Thus, the concatenation of the paths (x 1 , (1 -t)v 1 ), (γ(t), 0) and (x 2 , tv 2 ) yields a path linking (x 1 , v 1 ) and (x 2 , v 2 ) in {W < σ}. Hence Ẽ1 = Ẽ2 and we get the injectivity. □ Proposition 3.9. a) We have V (1) = V (1) ×{0}. In particular, V /2 and W have the same separating saddle values. b) A set Ẽ ∈ C σ is critical if and only if π x ( Ẽ) is critical. c) A labeling ((m, 0) k,j ) k,j is adapted to W if and only if (m k,j ) k,j is adapted to V /2. Moreover, given an adapted labeling, the mappings from Definition 3.6 satisfy

E V /2 (m k,j ) = π x E W (m k,j , 0) and j W (m k,j , 0) = j V /2 (m k,j ) × {0}.
Proof. Let Ẽ ∈ C σ . Thanks to Remark 3.7, we easily have

(x, 0) ∈ ∂ Ẽ ⇐⇒ x ∈ ∂ π x ( Ẽ) . (3.2)
a): We already know that U (1) = U (1) × {0}. Besides, we easily deduce from (3.2) and Lemma 3.8 that (s, 0) ∈ U (1) is in the closure of two distinct CCs of {W < W (s, 0)} if and only if s ∈ U (1) is in the closure of two distinct CCs of {V < V (s)} so the first item is proven. b): This is also a straightforward consequence of (3.2) and Lemma 3.8 combined with item a). c): Let Ẽ ∈ C σ k . By Remark 3.7, we easily have

(m, 0) is a global minimum of W | Ẽ ⇐⇒ m is a global minimum of V | πx( Ẽ) . (3.3) Besides, since U (0) k = U (0) k × {0}, we have that π k defined as π x : U (0) k → U (0) k
is bijective. We can then conclude as

T W k = π -1 x • T V /2 k • π k (3.4)
where π x denotes the bijective map from Lemma 3.8. The last statement is a direct consequence of (3.4), (3.2) and item a).

□

From now on, we fix a labeling (m k,j ) k,j adapted to V . Note that σ V /2 (m) = σ W (m, 0) and S V /2 (m) = S W (m, 0). For shortness, we will denote in the rest of the paper j = j V /2 , σ = σ V /2 and S = S V /2 . However, be careful that we choose to denote E = π -1

x • E V /2 so that the range of E is in P(R 2d ). Following [START_REF] Bovier | Metastability in reversible diffusion processes ii. precise asymptotics for small eigenvalues[END_REF][START_REF] Helffer | Quantitative analysis of metastability in reversible diffusion processes via a witten complex approach[END_REF][START_REF]Tunnel effect and symmetries for kramers fokker-planck type operators[END_REF][START_REF] Peutrec | Sharp spectral asymptotics for non-reversible metastable diffusion processes[END_REF], we can now state our last assumption that allows us to treat the generic case. As mentionned in the introduction, this assumption could actually be omitted (see [START_REF] Michel | About small eigenvalues of witten laplacian[END_REF] or [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF]) but this would introduce additionnal difficulties that are not the main concern of this paper. Recall the notation (1.6) and let us extend our notions of asymptotic expansions to smooth functions that are not necessarily symbols. Throughout the paper, for d ′ ∈ N * , Ω ⊆ R d ′ and a ∈ C ∞ (Ω) a function depending on h and such that for all β ∈ N d ′ we have ∂ β a = O L ∞ (1), we will denote a ∼ h j≥0 h j a j , where (a j ) j≥0 ⊂ C ∞ (Ω) are allowed to depend on h, provided that for all β ∈ N d ′ and N ∈ N, there exists C β,N such that

∂ β a - N -1 j=0 h j a j ∞,Ω ≤ C β,N h N .
It implies in particular that ∂ β a j = O L ∞ (1). We will also say that a ∈ C ∞ (Ω) admits a classical expansion on Ω and we will denote a ∼ j≥0 h j a j if a ∼ h j≥0 h j a j and the (a j ) are independent of h. From now on, the letter r will denote a small universal positive constant whose value may decrease as we progress in this paper (one can think of r as 1/C). For x ∈ R d , we denote B 0 (x, r) = B(x, r) × B(0, r) ⊆ R 2d . We essentially follow the quasimodal construction from [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF]. We will also denote

H W = h -1 X h 0 = v -∂ x V .
Let m ∈ U (0) \{m}; for each s ∈ j(m) we introduce a function ℓ s,h that will appear in our quasimodes. Note that thanks to item b) from Hypothesis 3.10, each ℓ s,h corresponds to a unique m ∈ U (0) \{m}. Our goal will be to find some functions ℓ s,h such that our quasimodes are the most accurate possible. In order to begin the computations that will yield the equations that the function ℓ s,h should satisfy, we will for the moment assume that it satisfies the following:

(3.5) a) ℓ s,h is a smooth real valued function on R 2d whose support is contained in B 0 (s, 3r) b) ℓ s,h admits a classical expansion ℓ s,h (x, v) ∼ h j ℓ s j (x, v) on B 0 (s, 2r) c) ℓ s 0 vanishes at (s, 0) d) (s, 0) is a local minimum of the function W + (ℓ s 0 ) 2 /2
which is non degenerate e) the functions θ m,h (which depends on ℓ s,h ) and χ m that we will introduce in (3.7)- (3.10) are such that θ m,h is smooth on a neighborhood of supp χ m .

Once we will have found the desired function ℓ s,h , we will see in Proposition 5.2 that these assumptions are actually satisfied. Denote ζ ∈ C ∞ c (R, [0, 1]) an even cut-off function supported in [-γ, γ] that is equal to 1 on [-γ/2, γ/2] where γ > 0 is a parameter to be fixed later and

A h = 1 2 R ζ(s)e -s 2 2h ds = γ 0 ζ(s)e -s 2 2h ds = √ πh √ 2 (1 + O(e -α/h ))
for some α > 0. (3.6)

We now define for each m ∈ U (0) \{m} a function θ m,h as follows: if (x, v) ∈ B 0 (s, r) ∩ {|ℓ s,h | ≤ 2γ} for some s ∈ j(m),

θ m,h (x, v) = 1 2 1 + A -1 h ℓ s,h (x,v) 0 ζ(s)e -s 2 /2h ds (3.7)
whereas we set

θ m,h = 1 on E(m) + B(0, ε) \ s∈j(m) B 0 (s, r) ∩ {|ℓ s,h | ≤ 2γ} (3.
χ m ∈ C ∞ c (R 2d , [0, 1]) (3.10) such that χ m = 1 on {W ≤ σ(m) + ε} ∩ Ω + B(0, ε)
and supp χ m ⊂ Ω + B(0, ε) .

To sum up, we have the following picture:

• j(m) • m θ m,h = 1 θ m,h = 0 Ω s u p p χ m θ m,h given by (3.7)
We also denote

W m (x, v) = W (x, v) -V (m)/2
and it is clear that on the support of ∇χ m , we have

W m ≥ S(m) + ε.
Our quasimodes will be the L 2 -renormalizations of the functions

f m,h (x, v) = χ m (x, v)θ m,h (x, v)e -Wm(x,v)/h ; m ∈ U (0) \{m} (3.11)
and for m = m, f m,h (x, v) = e -Wm(x,v)/h ∈ Ker P h .

Note that these functions belong to C ∞ c (R 2d ) thanks to our assumption on the (ℓ s,h ) s∈j(m) and that for m ̸ = m, we have

supp f m,h ⊆ E(m) + B(0, ε ′ ) (3.12)
where ε ′ = max(ε, r).

3.2.

Action of the operator P h . Let us fix m ∈ U (0) \{m}. We will denote

W m,h = W m + s∈j(m) (ℓ s,h ) 2 /2 (3.13) and ψ m,h (x, v, v ′ ) = 1 0 ∂ v W m,h (x, v ′ + t(v -v ′ ))dt. (3.14) Remark 3.11. Using Hypothesis 1.3, it is easy to see that b * h Op h (M h ) = Op h (g h ), with g h = (-i t η + t v/2)M h - h 2 ( t ∇ v - i 2 t ∇ η )M h ∈ M 1,d S 0 τ (⟨(v, η)⟩ -1 )
where

t ∇ v M h = d k=1 ∂ v k m k,j 1≤j≤d
and t ∇ η is defined similarly.

Proposition 3.12. Let f m,h be the quasimode defined in (3.11). With the notations introduced in (3.6) and (3.13), one has

P h f m,h = h 2 A -1 h ω m,h e -W m,h h 1 j W (m)+B0(0,2r) + O L 2 h ∞ e -S(m) h
where ω m,h is a function bounded uniformly in h and defined on j W (m) + B 0 (0, 2r) by

ω m,h = s∈j(m) H W • ∇ℓ s,h + I s,h
with I s,h (x, v) given for (x, v) ∈ j W (m) + B 0 (0, 2r) by the oscillatory integral

(2πh) -d R d |v ′ |≤r e i h η•(v-v ′ ) g h x, v + v ′ 2 , η + iψ m,h (x, v, v ′ ) ∂ v ℓ s,h (x, v ′ ) dv ′ dη.
Proof. In order to lighten the notations, we will drop some of the exponents and indexes m, s and h in the proof. By (3.5), we have on the support of χ that θ is smooth and

∇θ = A -1 h 2 s∈j(m) e -(ℓ s ) 2 /2h ζ(ℓ s )∇ℓ s 1 B0(s,r) .
Here we have to put the indicator function because ζ(ℓ)∇ℓ might have some support in B 0 (s, 3r)\B 0 (s, r). We can then begin by computing

X h 0 f = hH W • ∇f = hH W • ∇θ χe -Wm/h + hH W • ∇χ θe -Wm/h (3.15) = h 2 A -1 h χe -W /h s∈j(m) ζ(ℓ s )H W • ∇ℓ s 1 B0(s,r) + O he -S(m)+ ε h .
since W m ≥ S(m) + ε on the support of ∇χ. Now we can use Remark 3.11 to write

Q h (f ) = hOp h (g) (∂ v θ)χe -Wm/h + (∂ v χ)θe -Wm/h (3.16) = h 2 A -1 h s∈j(m) Op h (g) ζ(ℓ s )χe -W /h ∂ v ℓ s 1 B0(s,r) + O he -S(m)+ ε h since g ∈ S(⟨(v, η)⟩ -1
) and thus Op h (g) is bounded uniformly in h. But since g does not depend on ξ, we have for s ∈ j(m)

(2πh) d Op h (g) ζ(ℓ)χe -W /h ∂ v ℓ 1 B0(s,r) (x, v) = R d |v ′ |≤r e i h η•(v-v ′ ) g x, v + v ′ 2 , η × χ(x, v ′ )ζ ℓ(x, v ′ ) e -W (x,v ′ )/h ∂ v ℓ(x, v ′ ) dv ′ dη 1 B(s,r) (x). (3.17)
Let us now treat separately the cases |v| ≥ 2r and |v| < 2r . When |v| ≥ 2r, we have |v -v ′ | ≥ r so we can apply the non stationnary phase to the integral in η to get that for all x ∈ B(s, r) and N ≥ 1, there exists C N > 0 such that

R d |v ′ |≤r e i h η•(v-v ′ ) g x, v + v ′ 2 , η χ(x, v ′ )ζ ℓ(x, v ′ ) e -W (x,v ′ )/h ∂ v ℓ(x, v ′ ) dv ′ dη ≤ C N h N |v| -N e -S(m) h
where we used item d) from (3.5), the fact that W m (s, 0) + ℓ 2 0 (s, 0)/2 = S(m) and the estimate |v -v ′ | ≥ |v|/2. Hence we have shown that

Q h f 1 {|v|≥2r} = O h ∞ e -S(m) h and P h f 1 {|v|≥2r} = O h ∞ e -S(m) h . (3.18)
Now for the case |v| < 2r, let us denote J s 1 (x, v) the RHS of (3.17). Proceeding as in [START_REF] Nakamura | Agmon-type exponential decay estimates for pseudo-differential operators[END_REF] in order to take the e -W (x,v ′ )/h in front of the oscillatory integral,we get that for any x ∈ B(s, r),

J s 1 (x, v) = e -W (x,v)/h J s 2 (x, v) (3.19)
where

J s 2 (x, v) = R d |v ′ |≤r e i h η-iψ(x,v,v ′ ) • v-v ′ g x, v + v ′ 2 , η χ(x, v ′ )ζ ℓ(x, v ′ ) ∂ v ℓ(x, v ′ ) dv ′ dη 1 B(s,r) (x)
and ψ is the function defined in (3.14). For K ⊂ {1, . . . , d} and z ∈ C d , denote z K = (z j ) j∈K . We also denote for d ′ ∈ N and 1

≤ j ≤ d ′ e j = (δ k,j ) 1≤k≤d ′ ∈ N d ′ (3.20)
the elements of the canonical basis of C d ′ . Now notice that ψ is a smooth function and that using the expansion of ℓ and (3.13), we get on B 0 (s, 2r)

× {|v ′ | ≤ 2r}, ψ(x, v, v ′ ) = v + v ′ 4 + 1 0 ℓ 0 ∂ v ℓ 0 (x, v ′ + t(v -v ′ ))dt + O(h).
In particular, we can choose r small enough so that |ψ| < τ on B 0 (s, 2r) × {|v ′ | ≤ 2r}. Besides, since g ∈ S 0 τ (⟨(v, η)⟩ -1 ), we have for all K ⊂ {1, . . . , d} and k ∈ {1, . . . , d}\K that the symbol

η k → g x, v + v ′ 2 , η + i j∈K [ψ(x, v, v ′ )] j e j
has an analytic continuation to {|η k | < τ } for any x ∈ B(s, r), v, v ′ ∈ B(0, 2r) and η ∈ R d . Hence, one can use the Cauchy formula which combined with the decay of g yields

R e i h η k -i[ψ(x,v,v ′ )] k (v k -v ′ k ) g x, v + v ′ 2 , η + i j∈K [ψ(x, v, v ′ )] j e j dη k = R e i h η k (v k -v ′ k ) g x, v + v ′ 2 , η+i j∈K∪{k} [ψ(x, v, v ′ )] j e j dη k .
Applying this successively for each component of η on the integrals in J s 2 finally gives J s 2 = J s 3 where

J s 3 (x, v) = R d |v ′ |≤r e i h η•(v-v ′ ) g x, v + v ′ 2 , η + iψ(x, v, v ′ ) χ(x, v ′ )ζ ℓ(x, v ′ ) ∂ v ℓ(x, v ′ ) dv ′ dη 1 B(s,r) (x).
Combined with (3.17) and (3.19), this yields for |v| < 2r

(2πh) d Op h (g) ζ(ℓ)χe -W /h ∂ v ℓ 1 B0(s,r) (x, v) = e -W (x,v)/h J s 3 (x, v). (3.21) Therefore, setting on j W (m) + B 0 (0, 2r) ω = s∈j(m) χζ(ℓ s )H W • ∇ℓ s 1 B0(s,r) + (2πh) -d J s 3 (x, v) ,
we have according to (3.15), (3.16), (3.18) and (3.21)

P h f = h 2 A -1 h ω e -W /h 1 j W (m)+B0(0,2r) + O h ∞ e -S(m) h .
Hence it is sufficient to check that on j W (m) + B 0 (0, 2r)

(ω -ω)e -W /h = O h ∞ e -S(m) h .
This can be done easily using again the non stationary phase on an h-independent neighborhood of (s, 0) on which χζ(ℓ) -1 vanishes since item d) from (3.5) implies that e -W /h = O(e -(S(m)+δ)/h ) outside of this neighborhood for some δ > 0. □ Remark 3.13. Since P * h = -X h 0 + Q h , it is clear from the previous proof that

P * h f m,h = h 2 A -1 h * ω m,h e -W m,h h 1 j W (m)+B0(0,2r) + O L 2 h ∞ e -S(m) h with * ω m,h = s∈j(m)
-H W • ∇ℓ s,h + I s,h .

Equations on ℓ s,h

From now on, we also fix s ∈ j(m).

Lemma 4.1. The function ω m,h admits the classical expansion ω m,h ∼ j≥0 h j ω m j on B 0 (s, 2r) where

ω m 0 = H W • ∇ℓ s 0 + M 0 x, v, i v 2 + ℓ s 0 ∂ v ℓ s 0 v + ℓ s 0 ∂ v ℓ s 0 • ∂ v ℓ s 0 and for j ≥ 1, ω m j = H W • ∇ℓ s j + M 0 x, v, i v 2 + ℓ s 0 ∂ v ℓ s 0 (v + 2ℓ s 0 ∂ v ℓ s 0 ) • ∂ v ℓ s j (4.1) + i ℓ s 0 t v + ℓ s 0 t (∂ v ℓ s 0 ) D η M 0 x, v, i(v/2 + ℓ s 0 ∂ v ℓ s 0 ) ∂ v ℓ s j ∂ v ℓ s 0 + M 0 x, v, i v 2 + ℓ s 0 ∂ v ℓ s 0 ∂ v ℓ s 0 • ∂ v ℓ s 0 ℓ s j + i t v + ℓ s 0 t (∂ v ℓ s 0 ) D η M 0 x, v, i(v/2 + ℓ s 0 ∂ v ℓ s 0 ) ∂ v ℓ s 0 ∂ v ℓ s 0 ℓ s j + R j (ℓ s 0 , . . . , ℓ s j-1 )
where R j : C ∞ (B 0 (s, 2r)) j → C ∞ (B 0 (s, 2r)) and D η denotes the partial differential with respect to the variable η.

Proof. Once again, we drop some of the exponents and indexes m, s and h in the proof. Denote B ∞ (0, 2r) = {v ′ , η ∈ R 2d ; max(|v ′ |, |η|) < 2r}. The first terms of ω 0 and ω j are both easily obtained thanks to the expansion of ℓ on B 0 (s, 2r). Hence it remains to get an expansion of g(x, v/2 + v ′ /2, η + iψ(x, v, v ′ )) that we will then be able to combine with the stationnary phase to get an expansion of the whole term I s,h of ω. Let us start with an expansion of ψ : the expansion of ℓ yields

∂ v W -v/2 ∼ j≥0 h j j k=0 ℓ k ∂ v ℓ j-k on B 0 (s, 2r) so using (3.14), we get ψ ∼ j≥0 h j ψ j on B 0 (s, 2r) × {|v ′ | ≤ 2r}
where

ψ 0 (x, v, v ′ ) = v + v ′ 4 + 1 0 ℓ 0 ∂ v ℓ 0 (x, v ′ + t(v -v ′ ))dt (4.2)
and for j ≥ 1,

ψ j (x, v, v ′ ) = 1 0 j k=0 ℓ k ∂ v ℓ j-k (x, v ′ + t(v -v ′ ))dt. (4.3) Besides, since M h ∼ n≥0 h n M n in M d S 0 τ (⟨(v, η)⟩ -2
) , we deduce thanks to Proposition C.2 and Remark 3.11 that g also has a classical expansion g

∼ n≥0 h n g n in M 1,d S 0 τ (⟨(v, η)⟩ -1
) , where the (g n ) are given by

g 0 (x, v, η) = -i t η + t v 2 M 0 (x, v, η) (4.4) and g n (x, v, η) = -i t η + t v 2 M n (x, v, η) - 1 2 ( t ∇ v - i 2 t ∇ η )M n-1 (x, v, η) (4.5)
for n ≥ 1. According to Corollary C.6, we have

g n x, v + v ′ 2 , η + iψ(x, v, v ′ ) ∼ j≥0 h j g n,j (x, v, v ′ , η) on B 0 (s, 2r) × B ∞ (0, 2r) with g n,0 (x, v, v ′ , η) = g n x, v + v ′ 2 , η + iψ 0 (x, v, v ′ ) (4.6)
and for j ≥ 1

g n,j (x, v, v ′ , η) = iD η g n x, v + v ′ 2 , η + iψ 0 (x, v, v ′ ) ψ j (x, v, v ′ ) + R 1 j (ℓ 0 , . . . , ℓ j-1 ) (4.7) where R 1 j : C ∞ (B 0 (s, 2r)) j → C ∞ (B 0 (s, 2r
)). Using the expansion of g itself and Proposition C.1, we get

g x, v + v ′ 2 , η + iψ(x, v, v ′ ) ∼ h n≥0 h n g n x, v + v ′ 2 , η + iψ(x, v, v ′ )
on B 0 (s, 2r) × B ∞ (0, 2r) so we can use Proposition C.3 which yields

g x, v + v ′ 2 , η + iψ(x, v, v ′ ) ∼ j≥0 h j j n=0 g n,j-n (x, v, v ′ , η) (4.8)
on B 0 (s, 2r) × B ∞ (0, 2r). Thus, using the expansion (4.8) that we just got, the one of ∂ v ℓ, and the one for an oscillatory integral given by the stationnary phase (see for instance [START_REF] Zworski | Semiclassical analysis[END_REF], Theorem 3.17) as well Proposition C.3, we finally get I s,h ∼ j≥0 h j I j on B 0 (s, 2r), (4.9) where

I j (x, v) = n1+n2+n3+n4=j 1 i n1 n 1 ! ∂ v ′ • ∂ η n1 g n2,n3 (x, v, v ′ , η)∂ v ℓ n4 (x, v ′ ) v ′ =v η=0
.

We can already use (4.6) to deduce the expression of ω 0 by noticing that according to (4.2),

ψ 0 (x, v, v) = v/2 + ℓ 0 ∂ v ℓ 0 .
For j ≥ 1, the terms of I j in which the function ℓ j appears are obviously the one given by n 4 = j, but also the one given by n 3 = j according to (4.7). Indeed, in that case, we have using (4.3) that

g 0,j (x, v, v, 0) = iℓ 0 D η g 0 x, v, i(v/2+ℓ 0 ∂ v ℓ 0 ) ∂ v ℓ j + iD η g 0 x, v, i(v/2 + ℓ 0 ∂ v ℓ 0 ) ∂ v ℓ 0 ℓ j + R 2 j (ℓ 0 , . . . , ℓ j-1 )
where R 2 j : C ∞ (B 0 (s, 2r)) j → C ∞ (B 0 (s, 2r)). We can now conclude as for any

X ∈ R d , D η g 0 x, v, i(v/2 + ℓ 0 ∂ v ℓ 0 ) (X) = -i t XM 0 x, v, i(v/2 + ℓ 0 ∂ v ℓ 0 ) + t v + ℓ 0 t (∂ v ℓ 0 ) D η M 0 x, v, i(v/2 + ℓ 0 ∂ v ℓ 0 ) (X)
according to (4.4). □ Denote (m n p,q ) p,q the entries of the matrix M n from Hypothesis 1.3. Since we have for

X ∈ R d D η M 0 x, v, i(v/2 + ℓ 0 ∂ v ℓ 0 ) X = ∂ η m 0 p,q x, v, i(v/2 + ℓ 0 ∂ v ℓ 0 ) • X 1≤p,q≤d
we get by putting

U (x, v) = M 0 x, v,i v 2 + ℓ 0 ∂ v ℓ 0 ∂ v ℓ 0 (4.10) + 1≤p,q≤d v p + ℓ 0 ∂ vp ℓ 0 i∂ η m 0 p,q x, v, i v 2 + ℓ 0 ∂ v ℓ 0 ∂ vq ℓ 0
that equation (4.1) reads

ω j = H W + 0 M 0 x, v, i v 2 + ℓ 0 ∂ v ℓ 0 (v + ℓ 0 ∂ v ℓ 0 ) + ℓ 0 U • ∇ℓ j + U • ∂ v ℓ 0 ℓ j + R j (ℓ 0 , . . . , ℓ j-1 ). Lemma 4.2. Let (x, v) ∈ B 0 (s, 2r) and |v ′ | < 2r. For any n ∈ N, β ∈ N d and 1 ≤ p, q ≤ d, we have ∂ β η m n p,q x, v + v ′ 2 , iψ m 0 (x, v, v ′ ) ∈ i |β| R and ∂ β η g n x, v + v ′ 2 , iψ m 0 (x, v, v ′ ) ∈ i |β| R d .
In particular, U defined in (4.10) sends B 0 (s, 2r) in R d .

Proof. Since ℓ 0 vanishes at (s, 0), we can suppose that r is such that iψ 0 (x, v, v ′ ) is in

D(0, τ ) d = {z ∈ C ; |z| < τ } d (4.11)
so by analyticity and using the parity of m n p,q , we have

∂ β η m n p,q x, v + v ′ 2 , iψ 0 (x, v, v ′ ) = γ∈N d ; |γ|+|β|∈2N i |γ| ∂ γ+β η m n p,q x, v+v ′ 2 , 0 γ! ψ 0 (x, v, v ′ ) γ ∈ i |β| R.
The result for g n follows easily using (4.4) and (4.5). □

We also have the following result whose proof is postponed to Appendix D as it involves tedious calculations.

Lemma 4.3. The term R j (ℓ s 0 , . . . , ℓ s j-1 ) from Lemma 4.1 is real valued. Moreover, it satisfies R j (ℓ s 0 , . . . , ℓ s j-1 ) = -R j (-ℓ s 0 , . . . , -ℓ s j-1 ).

In view of the results from Proposition 3.12 and Lemma 4.1, we want to find ℓ such that on B 0 (s, 2r),

H W • ∇ℓ 0 + M 0 x, v, i v 2 + ℓ 0 ∂ v ℓ 0 v + ℓ 0 ∂ v ℓ 0 • ∂ v ℓ 0 = 0 (4.12)
and for j ≥ 1

H W + 0 M 0 x, v, i v 2 + ℓ 0 ∂ v ℓ 0 (v + ℓ 0 ∂ v ℓ 0 ) + ℓ 0 U • ∇ℓ j (4.13) +∂ v ℓ 0 • U ℓ j + R j (ℓ 0 , . . . , ℓ j-1 ) = 0
where U was introduced in (4.10). Note that Lemmas 4.2 and 4.3 ensure that the fact that the (ℓ j ) j≥0 are real valued is compatible with equations (4.13).

Solving for

ℓ s 0 . Denote p(x, v, ξ, η) = iξ • v -iη • ∂ x V + (-i t η + t v/2)M 0 (x, v, η)(iη + v/2)
the principal symbol of the whole operator P h and p(x, v, ξ, η) = -p(x, v, iξ, iη) its complexification.

After computing the Hamiltonian of p which vanishes at (s, 0, 0, 0), we find that its linearization at this point is the matrix

F =     0 Id 0 0 -Hess s V 0 0 2M 0 (s, 0, 0) 0 0 0 Hess s V 0 1 2 M 0 (s, 0, 0) -Id 0     .
One can easily check that for any eigenvector (x, v, ξ, η) of F associated to an eigenvalue λ, the vector (-x, v, ξ, -η) is an eigenvector associated to -λ so the spectrum of F is centrally symmetric with respect to the origin. Moreover, writing

F =     0 0 Id 0 0 0 0 Id Id 0 0 0 0 Id 0 0         0 0 0 Hess s V 0 1 2 M 0 (s, 0, 0) -Id 0 0 Id 0 0 -Hess s V 0 0 2M 0 (s, 0, 0)    
and noticing that

F {v = η = 0} ∩ {v = η = 0} = Ker F ∩ {v = η = 0} = {0},
we see that F satisfies the assumptions of Lemma B.1. Therefore, F has no eigenvalues in iR so it has 2d eigenvalues (counted with algebraic multiplicity) in {Re z > 0} while the 2d others are in {Re z < 0}. Therefore we can apply the stable manifold theorem to get that the stable manifolds associated to H p given in a neighborhood of (s, 0, 0, 0) by Λ ± = (x, v, ξ, η) ; lim t→∓∞ e tHp (x, v, ξ, η) = (s, 0, 0, 0) are both of dimension 2d and for all ρ ± ∈ Λ ± , we have

H p(ρ ± ) ∈ T ρ± Λ ± (4.14)
and for t > 0, e ∓tHp ρ ± -(s, 0, 0, 0) ≤ Ce -t/C ∥ρ ± -(s, 0, 0, 0)∥.

Moreover, we have (see for instance [START_REF] Dimassi | Spectral asymptotics in the semiclassical limit[END_REF] Lemmas 3.2 and 3.3) that p(Λ ± ) = {0} (4.15) and Λ ± are Lagrangian manifolds. In order to get some parametrization for those manifolds, we follow the steps of [START_REF] Hérau | Tunnel effect for kramers-fokker-planck type operators[END_REF], Lemma 8.1.

Lemma 4.4. The tangent spaces T (s,0,0,0) Λ ± that we denote for shortness T s Λ ± are transverse to both {(s, 0)} × R 2d and R 2d × {(0, 0)}.

Proof. We provide an adaptation of the proof from [START_REF] Hérau | Tunnel effect for kramers-fokker-planck type operators[END_REF] as some simplifications appear in our case. Since we are working in the linearized case, we can assume that p coincides with its quadratic approximation at (s, 0, 0, 0) and for commodity we will work with the variable x s = x -s instead of x. Note that if a is a quadratic form, its Hamiltonian H a is then linear and we denote F a the associated matrix. We then decompose p = p 2 + p 1 -p 0 where

p 2 = M 0 (s, 0, 0)η • η, p 1 = v • ξ -Hess s V x s • η and p 0 = 1 4 M 0 (s, 0, 0)v • v.
It is clear that p 2 + p 0 is positive semi-definite, moreover, the subspace {v = η = 0} on which

p 2 + p 0 vanishes satisfies {v = η = 0} ∩ F -1 p1 {v = η = 0} = {0}. Thus the quadratic form q = (p 2 + p 0 ) + (p 2 + p 0 ) • F p1 is positive definite. Let us denote L ± = Λ ± ∩ {x s = v = 0}.
To prove that L ± = {0}, it is sufficient to establish that q = 0 on L ± . In order to do so, we will show that L ± is an F p1 -invariant subspace on which p 2 + p 0 = 0. Indeed, it is clear that p 0 = p 1 = 0 on L ± and thanks to (4.15) we deduce that p 2 also vanishes on L ± so in particular p 2 + p 0 = 0 on L ± . It also implies that L ± is included in {η = 0} so F p2 | L± = 0. Besides, we clearly have F p0 | L± = 0 so F p1 coincides on L ± with F p which leaves Λ ± invariant according to (4.14). Since it is easy to see that {x s = v = 0} is also invariant under F p1 , we can conclude as announced that L ± = {0}. The proof that Λ ± ∩ {ξ = η = 0} = {0} is similar. □ Since Λ ± are Lagrangian manifolds such that T s Λ ± are transverse to {(s, 0)} × R 2d , there exist ϕ ± ∈ C ∞ (B 0 (s, 2r), R) vanishing together with their gradients at (s, 0) and such that

Λ ± = (x, v, ∇ϕ ± (x, v) ; (x, v) ∈ B 0 (s, 2r) .
Therefore, T s Λ ± coincide with the graphs of the matrices Hess (s,0) ϕ ± which are then invertible according to Lemma 4.4. Now we need a result similar to the one of Proposition 8.2 in [START_REF] Hérau | Tunnel effect for kramers-fokker-planck type operators[END_REF].

Lemma 4.5. The Hessian matrix of ±ϕ ± at (s, 0) is definite positive.

Proof. The proof is simply an adaptation of the one found in [START_REF] Hérau | Tunnel effect for kramers-fokker-planck type operators[END_REF]. Here again we will assume that p coincides with its quadratic approximation at (s, 0, 0, 0) and work with the variable

x s = x -s instead of x. For δ ∈ [0, 1], let us denote pδ = (1 -δ)p + δ ξ 2 + η 2 -(x 2 s + v 2 ) = p δ 2 + (1 -δ)p 1 -p δ 0 where p δ 2 = (1 -δ)p 2 + δ(ξ 2 + η 2 ) and p δ 0 = (1 -δ)p 0 + δ(x 2 s + v 2 ).
Note in particular that p0 = p and that p1 = ξ 2 +η 2 -(x 2 s +v 2 ) corresponds to the well know Schrödinger case (see for instance [START_REF] Dimassi | Spectral asymptotics in the semiclassical limit[END_REF], chapter 3). Besides, we have that

F pδ =     0 0 Id 0 0 0 0 Id Id 0 0 0 0 Id 0 0         (1 -δ)     0 0 0 Hess s V 0 1 2 M 0 (s, 0, 0) -Id 0 0 Id 0 0 -Hess s V 0 0 2M 0 (s, 0, 0)     + 2δ Id    
so Lemma B.1 easily yields that the eigenvalues of F pδ cannot cross iR for some δ ∈ (0, 1]. Moreover, it is clear that for δ ∈ (0, 1], the quadratic form p δ 2 + p δ 0 is positive definite, so the results of Lemma 4.4 are true for the 2d-dimensional Lagrangian planes

Λ δ ± = (x s , v, ξ, η) ; lim t→∓∞ e tF pδ (x, v, ξ, η) = 0 for all δ ∈ [0, 1]. In particular, there exist ϕ δ ± ∈ C ∞ (B 0 (s, 2r), R) such that T s Λ δ ± = Λ δ ± = x s , v, Hess (s,0) ϕ δ ± x s v ; (x s , v) ∈ R 2d .
Hence the graph of Hess (s,0) ϕ δ ± is given by T s Λ δ ± which also corresponds to the sum of the generalized eigenspaces of F pδ associated to eigenvalues in {±Re z < 0} and therefore depends continuously on δ. Besides, by Lemma 4.4, Hess (s,0) ϕ δ ± is invertible for all δ ∈ [0, 1] and we know from the Schrödinger case that ±Hess (s,0) ϕ 1 ± > 0 so necessarily ±Hess (s,0) ϕ ± > 0. □ At this point, one can proceed as in [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF], Lemma 3.2 to establish the following Lemma.

Lemma 4.6. There exists ℓ s 0 ∈ C ∞ (B 0 (s, 2r), R) such that for (x, v) ∈ B 0 (s, 2r),

ϕ + (x, v) = W (x, v) -W (s, 0) + ℓ s 0 (x, v) 2 2 .
In particular, ℓ s 0 vanishes at (s, 0). Moreover, {ℓ s 0 ̸ = 0} is dense in B 0 (s, 2r).

This function also appears to solve (4.12) as we see in the next Proposition.

Proposition 4.7. The function ℓ s 0 from Lemma 4.6 is a solution of (4.12) in B 0 (s, 2r). Moreover, the vector ∇ℓ s 0 (s, 0) that we denote

ν s = ν s 1 ν s 2 is not 0 and satisfies Φ s ν s = -M 0 (s, 0, 0)ν s 2 • ν s 2 ν s ,
where

Φ s = 0 -Hess s V Id M 0 (s, 0, 0) . In particular, since Φ s is invertible, ν s 2 ̸ = 0. Finally, det Hess (s,0) W + (ℓ s 0 ) 2 2 = 2 -2d det(Hess s V ) .
Proof. The proof is the same as in [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF], Lemma 3.3 after matching the notations by setting Λ(s

) = Φ s , b 0 = H W , A 0 (s) = 0 0 0 M 0 (s, 0, 0) and B(s) = 0 Id -Hess s V 0 . □ 4.
2. Solving for ℓ s j j≥1 . Once again we drop some exponents s for shortness. Now that ℓ 0 is given by Lemma 4.6 and Proposition 4.7, we can solve the transport equations (4.13) by induction, so we suppose that ℓ 0 , . . . , ℓ j-1 are given and we want to find a solution ℓ j to (4.13). Denote

U = H W + 0 M 0 x, v, i v 2 + ℓ 0 ∂ v ℓ 0 (v + ℓ 0 ∂ v ℓ 0 ) + ℓ 0 U ∈ C ∞ (B 0 (s, 2r)) and α = ∂ v ℓ 0 • U ∈ C ∞ (B 0 (s, 2r))
where U was introduced in (4.10). The function ℓ j must satisfy ( U • ∇ + α)ℓ j = -R j (ℓ 0 , . . . , ℓ j-1 ) so we are intersted in the operaor L = U • ∇ + α that we decompose as

L = L s 0 + L > with L s 0 = U s 0 x -s v • ∇ + α s 0
where U s 0 is the differential of U at (s, 0) and α s 0 = α(s, 0), that is

U s 0 = 0 Id -Hess s V + 2M 0 (s, 0, 0)ν s 2 t ν s 1 M 0 (s, 0, 0)(Id + 2ν s 2 t ν s
2 ) and

α s 0 = M 0 (s, 0, 0)ν s 2 • ν s 2 . (4.16)
As usual, we will often omitt the exponents s in the notations. Notice that if we denote P n hom the space of homogeneous polynomials of degree n in the variables (x -s, v), we have L 0 ∈ L (P n hom ) and for

P ∈ P n hom , L > P (x, v) = O (x -s, v) n+1
near (s, 0). Lemma 4.8. The negative eigenvalue -α s 0 of the matrix Φ s from Proposition 4.7 is its only one (counting multiplicity) in {Re z ≤ 0}. Moreover, all the eigenvalues of U s 0 have positive real part and the operator L s 0 is invertible on P n hom . Proof. It is sufficient to prove the first statement. Indeed, if -α 0 is the only eigenvalue of Φ in {Re z ≤ 0}, we can then remark that

t U 0 = Φ + 2 0 ν 1 t ν 2 M 0 (s, 0, 0) 0 ν 2 t ν 2 M 0 (s, 0, 0)
and since the last term has its range included in Cν and sends ν on 2α 0 ν, the matrix of t U 0 in a basis (ν, b 2 , . . . , b 2d ) in which Φ becomes triangular is also triangular and has on its diagonal the eigenvalues of Φ except for -α 0 which is replaced by +α 0 . Hence Spec( U 0 ) = Spec( t U 0 ) ⊂ {Re z > 0} and we can conclude thanks to Lemma A.1 from [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF]. Let us then prove that -α 0 is the only eigenvalue (counting multiplicity) of Φ in {Re z ≤ 0}. We proceed as in [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF], Lemma 2.6. For t ∈ [0, 1], consider the matrix

Φ t = 2 Hess s W (1 -t)Id -tId tId tM 0 (s, 0, 0) + (1 -t)Id
which trivially satisfies the assumptions of Lemma B.1 for t ∈ [0, 1). It is also the case of Φ 1 as Φ 1 (x, 0) = (0, x). Hence for every t ∈ [0, 1], Φ t has no eigenvalues in iR and since these eigenvalues depend continuously on t, we get that

# Spec Φ 1 ∩ {Re z < 0} = # Spec Φ 0 ∩ {Re z < 0} .
But Φ 0 = 2 Hess s W has exactly one negative eigenvalue (with multiplicity) while all the others are positive since s ∈ U (1) , so we have indeed showed that -α 0 is the only eigenvalue of Φ = Φ 1 (counting multiplicity) in {Re z ≤ 0}. □

One can then proceed as in [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF], section 3.3 (see also [START_REF] Dimassi | Spectral asymptotics in the semiclassical limit[END_REF], chapter 3), i.e use Lemma 4.8 to find an approximate solution of (4.13) using formal power series and then refine it into an actual solution using again Lemma 4.8 as well as the characteristic method. We then get the following result.

T. NORMAND

Now we show that θ m,h is smooth on F 2 ∩ (Ω + B(0, ε)): let s ∈ j(m) and (x, v) ∈ B 0 (s, r) ∩ {ℓ s 0 = 2γ} ∩ (Ω + B(0, ε)). According to (5.3) and the fact that ℓ s,h = ℓ s 0 + O(h), there exists a small ball B centered in (x, v) such that

B ⊂ B 0 (s, r) ∩ {ℓ s,h > γ} ∩ E(m) + B(0, ε) .
Thus θ m,h = 1 on B and θ m,h is smooth at (x, v). Similarly, for (x, v) ∈ B 0 (s, r) ∩ {ℓ s,h = -2γ} ∩ (Ω + B(0, ε)), we can show that θ m,h = 0 in a neighborhood of (x, v). It only remains to prove that, as for F 1 , the set F 3 does not meet the support of χ m . First we remark that thanks to (5.2), we can forget the absolute value in the definition of F 3 :

F 3 = ∂ E(m) + B(0, ε) \ j(m) B 0 (s, r) ∩ {ℓ s 0 ≤ 2γ} . If (x, v) ∈ F 3 ∩B 0 (s, r), we have that ℓ s 0 (x, v) > 2γ so using (5.3), we see that (x, v) is outside Ω s +B(0, ε). Since it is not in (E(m) + B(0, ε)) either, it is outside Ω + B(0, ε) which contains the support of χ m . Now if (x, v) ∈ F 3 \ j W (m) + B 0 (0, r) , (5.4) implies that (x, v) is outside ∪ j(m) (Ω s + B(0, ε)
) so it is also outside Ω + B(0, ε) for ε small enough and the proof is complete. □ Lemma 5.3. Let m ∈ U (0) \{m} and denote fm,h = f m,h /∥f m,h ∥ where f m,h was defined in (3.11).

With the notation (4.16), we have that

⟨P h fm,h , fm,h ⟩ = he -2 S(m) h det(Hess m V ) 1/2 2π Bh (m) ∈ R
with Bh (m) admitting a classical expansion whose first term equals

s∈j(m) | det(Hess s V )| -1/2 α s 0 .
Proof. Since X h 0 is a skew-adjoint differential operator and f m,h is real valued, we have

⟨X h 0 f m,h , f m,h ⟩ = 0.
Besides, we know from (3.16

) that b h f m,h = h(∂ v θ)χe -Wm/h + O L 2 (h ∞ e -S(m)/h
) so we easily deduce from the fact that (∂ v θ)χe -Wm/h = O L 2 (e -S(m)/h ) and the boundedness of Op h (M h ) that

⟨Q h f m,h , f m,h ⟩ = h 2 Op h (M h ) (∂ v θ)χe -Wm/h , (∂ v θ)χe -Wm/h + O h ∞ e -2S(m) h .
Since we have with the notation (3.13)

(∂ v θ)χe -Wm/h = A -1 h 2 e -Wm/h χ s∈j(m) ζ(ℓ s )∂ v ℓ s 1 B0(s,r)
and using (3.21) with M instead of g, we get that

⟨P h f m,h , f m,h ⟩ = h 2 4 A -2 h s∈j(m) B0(s,r) e -2 Wm(x,v)/h χζ(ℓ s ) Ĩs (x, v) • ∂ v ℓ s d(x, v) (5.5) + O h ∞ e -2 S(m) h . where Ĩs (x, v) = (2πh) -d R d |v ′ |≤r e i h η•(v-v ′ ) χ(x, v ′ )ζ ℓ s (x, v ′ ) M x, v + v ′ 2 , η + iψ(x, v, v ′ ) ∂ v ℓ s (x, v ′ ) dv ′ dη.
Mimicking the proof of Proposition C.5, one can show that ζ(ℓ) admits a classical expansion whose first term is ζ(ℓ 0 ). Besides, since M and ψ also have a classical expansion, we could use the stationnary phase (see for instance [START_REF] Zworski | Semiclassical analysis[END_REF], Theorem 3.17) as well Proposition C.3 to get an expansion of Ĩ similar to the one obtained in (4.9). Thus, we get that Ĩ • ∂ v ℓ ∼ k≥0 h k a k where

a 0 (x, v) = χ(x, v)ζ ℓ 0 (x, v) M 0 x, v, i v 2 + ℓ 0 ∂ v ℓ 0 ∂ v ℓ 0 (x, v) • ∂ v ℓ 0 (x, v).
Hence, using the fact that on B 0 (s, r),

W -S(m) = W m + ℓ 2 0 2 -S(m) + ℓ 2 2 - ℓ 2 0 2 , it is clear that e 2S(m)/h B0(s,r) e -2 W (x,v)/h χζ(ℓ) Ĩ(x, v) • ∂ v ℓ d(x, v) ∼ h (5.6) k≥0 h k B0(s,r) e -2 Wm (x,v)+ℓ 2 0 (x,v)/2-S(m) h e -(ℓ 2 -ℓ 2 0 )(x,v) h χζ(ℓ)a k d(x, v).
We would like to apply Proposition C.7 so we need to check that the assumptions are satisfied. First, Hess (s,0) (W m + ℓ 2 0 /2) is definite positive by Lemma 4.5. Besides, h -1 (ℓ 2 -ℓ 2 0 ) admits a classical expansion whose first term is 2(ℓ 1 ℓ 0 ). Therefore, using the expansion of ζ(ℓ) as well as Proposition C.5, one easily gets that the function

e -(ℓ 2 -ℓ 2 0 ) h ζ • ℓ
admits a classical expansion whose first term is e -2(ℓ1ℓ0) ζ • ℓ 0 . Thus, according to Propositions C.7 and 4.7, there exists (b k,j ) such that

| det(Hess s V )| 1/2 (2πh) d B0(s,r) e -2 Wm (x,v)+ℓ 2 0 (x,v)/2-S(m) h e -(ℓ 2 -ℓ 2 0 )(x,v) h χζ(ℓ)a k d(x, v) ∼ j≥0 h j b k,j
where b k,0 = a k (s, 0). Hence, using (5.5), (5.6) and Proposition C.3, we deduce that

4A 2 h (2π) -d h -d-2 e 2S(m)/h ⟨P h f m,h , f m,h ⟩ ∼ k≥0 h k c k (5.7) with c 0 = s∈j(m) | det(Hess s V )| -1/2 M 0 (s, 0, 0)ν s 2 • ν s 2 = s∈j(m) | det(Hess s V )| -1/2 α s 0 .
Similarly, thanks to item a) from Hypothesis 3.10, one can use Proposition C.7 as we already did to see that there exists (c k ) k≥0 such that det(Hess m V)

R 2d \(j W (m)+B0(0,2r)) |P h f m,h (x, v)| 2 d(x, v) = O h ∞ e -2 S(m) h .
(5.9) Besides, we saw that thanks to Proposition C.7 and Lemma 4.5, we have for s ∈ j(m),

B0(s,2r) e -2 W (x,v) h d(x, v) = O h d e -2 S(m) h .
Moreover, the function ω from Proposition 3.12 is O L ∞ (B0(s,2r)) (h ∞ ) by Lemma 4.1 and the construction of the (ℓ s,h ) s∈j(m) . Hence, by Proposition 3.12, B0(s,2r)

|P h f m,h (x, v)| 2 d(x, v) = O h ∞ e -2 S(m) h .
(5.10)

The conclusion follows from (5.9), (5.10) as well as (5.8) and Lemma 5.3. The proof of ii) can be obtained similarly with the use of Proposition C.7 and Remark 3.13 after noticing that * ω also admits a classical expansion whose first term vanishes on j W (m). □ Lemma 5.5. For m and m ′ two distinct elements of U (0) , we have

i) ⟨P h fm,h , fm ′ ,h ⟩ = 0 ii) There exists c > 0 such that ⟨ fm,h , fm ′ ,h ⟩ = O(e -c/h ) Proof. i):
The result is obvious when one of the two minima is m. Recall the labeling of the minima that we introduced rigth before Hypothesis 3.10 as well as the map π x from Lemma 3.8. Let us first suppose that m = m k,j and m ′ = m k,j ′ with j ̸ = j ′ and k ̸ = 1 and denote E = E(m) and E ′ = E(m ′ ). In particular σ(m) = σ(m ′ ). Thanks to (3.12) and the fact that P h is local in x, we have

supp P h fm,h ⊆ π x (E) × R d v + B(0, ε ′ ) and supp fm ′ ,h ⊆ E ′ + B(0, ε ′ )
so up to taking ε ′ small enough, it is sufficient to show that π x (E) × R d v and E ′ do not intersect. Since our labeling is adapted, E and E ′ are two distinct CCs of {W < σ(m)} so by Lemma 3.8, π x (E) × R d v and E ′ are two disjoint open sets. Thus, using successively Remark 3.7 and (3.2), we get

π x (E) × R d v ∩ E ′ = ∂ π x (E) × R d v ∩ ∂E ′ ⊆ ∂ π x (E) × {0} ∩ ∂E ′ ⊆ ∂ π x (E) ∩ ∂ π x (E ′ ) × {0}.
which is empty thanks to Lemma 3.2 and item b) from Hypothesis 3.10.

Let us now treat the case m = m k,j and m ′ = m k ′ ,j ′ with k, k ′ ≥ 2 and k ̸ = k ′ . We can suppose that k < k ′ (i.e σ(m) > σ(m ′ )) because we can work with P * h instead of P h if needed. We decompose P h fm,h as in (3.15) and (3.16) and once again we use (3.12) to get supp fm ′ ,h ⊆ E ′ + B(0, ε ′ ) ⊆ W < σ(m) + σ(m ′ ) 2 as well as the fact that P h is local in x to get a localization of the support of the first term from (3.16):

supp Op h (g) (∂ v θ m )χ m e -Wm/h ⊆ j(m) + B(0, r) × R d v ⊆ W > σ(m) + σ(m ′ )
2 as W increases with the norm of v. Hence, the support of the first term from (3.16) does not meet the one of fm ′ ,h . The same goes easily for the first term of (3.15). For the second term of (3.15), its support is contained in the support of ∇χ m which is itself contained in {W ≥ σ(m) + ε} so it clearly does not meet the support of fm ′ ,h . It only remains to treat the second term from (3.16), i.e

h(2πh) -d R d |v ′ |≤r e i h η•(v-v ′ ) g x, v + v ′ 2 , η θ m ∂ v χ m (x, v ′ )e -Wm(x,v ′ )/h dv ′ dη. (5.11)
Using again the support properties of ∇χ m as well as the fact that

π x E ′ + B(0, ε ′ ) × {0} ⊆ {W < σ(m)}
we get that |v| is bounded from below by some positive constant on

supp ∇χ m ∩ π x E ′ + B(0, ε ′ ) × R d v .
Lemma 5.9. For all 1 ≤ j, k ≤ n 0 , it holds

⟨P h u j , u k ⟩ = δ j,k λj + O h ∞ λj λk .
In order to compute the small eigenvalues of P h , let us now consider the restriction P h | H : H → H. We denote ûj = u n0-j+1 , λj = λn0-j+1 and M the matrix of P h | H in the orthonormal basis (û 1 , . . . , ûn0 ). Since ûn0 = u 1 = f1 , we have

M = M ′ 0 0 0 where M ′ = ⟨P h ûj , ûk ⟩ 1≤j,k≤n0-1
and it is sufficient to study the spectrum of M ′ . We will also denote

{ Ŝ1 < • • • < Ŝp } the set {S(m j ) ; 2 ≤ j ≤ n 0 } and for 1 ≤ k ≤ p, E k the subspace of L 2 (R 2d
) generated by {û r ; S(m r ) = Ŝk }. Finally, we set ϖ k = e -( Ŝk -Ŝk-1 )/h for 2 ≤ k ≤ p and ε j (ϖ) = j k=2 ϖ k = e -( Ŝj -Ŝ1)/h for 2 ≤ j ≤ p (with the convention ε 1 (ϖ) = 1). Proposition 5.10. There exists a diagonal matrix M # h admitting a classical expansion whose first term is

M # 0 = diag s∈j(mn 0 -j+1) det(Hess mn 0 -j+1 V ) 1/2 2π| det(Hess s V )| 1/2 α s 0 ; 1 ≤ j ≤ n 0 -1 such that h -1 e 2 Ŝ1/h M ′ = Ω(ϖ) M # h + O(h ∞ ) Ω(ϖ)
where Ω(ϖ) = diag ε 1 (ϖ)Id E1 , . . . , ε p (ϖ)Id Ep .

Remark 5.11. In the words of Definition 6.7 from [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF], the last Proposition implies that h -1 e 2 Ŝ1/h M ′ is a classical graded symmetric matrix.

Proof. According to Lemma 5.9, we can decompose

M ′ = M ′ 1 + M ′ 2 with M ′ 1 = diag( λj ; 1 ≤ j ≤ n 0 -1) and M ′ 2 = O h ∞ λj λk 1≤j,k≤n0-1
.

We will take M # h = h -1 e 2 Ŝ1/h Ω(ϖ) -1 M ′ 1 Ω(ϖ) -1 which is clearly diagonal, so we just need to check that it has the proper classical expansion and that h -1 e 2 Ŝ1/h Ω(ϖ)

-1 M ′ 2 Ω(ϖ) -1 = O(h ∞ ). It is easy to compute h -1 e 2 Ŝ1/h Ω(ϖ) -1 M ′ 1 Ω(ϖ) -1 = h -1 diag e 2 Ŝj ′ /h λj ; 1 ≤ j ≤ n 0 -1
where 1 ≤ j ′ ≤ p is such that Ŝj ′ = S(m n0-j+1 ). Hence Lemma 5.3 yields

h -1 e 2 Ŝ1/h Ω(ϖ) -1 M ′ 1 Ω(ϖ) -1 = diag det(Hess mn 0 -j+1 V ) 1/2 2π Bh (m n0-j+1 ) ; 1 ≤ j ≤ n 0 -1
where Bh (m n0-j+1 ) was introduced in Lemma 5.3 and admits a classical expansion whose first term is s∈j(mn 0 -j+1)

| det(Hess s V )| -1/2 α s 0 so M # h has the desired expansion. Similarly, still using Lemma 5.3, one easily gets

Ω(ϖ) -1 M ′ 2 Ω(ϖ) -1 = O h ∞ λj λk ε j ′ (ϖ) -1 ε k ′ (ϖ) -1 1≤j,k≤n0-1
where 1 ≤ j ′ ≤ p and 1 ≤ k ′ ≤ p are such that λj ε j ′ (ϖ) -1 and λk ε k ′ (ϖ) -1 are both O( √ h e -Ŝ1/h ) so the proof is complete. □ Proof of Theorem 1.7. According to Remark 5.11, it now suffices to combine the result of Proposition 5.10 with Theorem 4 from [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF] which gives a description of the spectrum of classical graded almost symmetric matrices. Indeed, using the notations from this reference, we have for 1 ≤ j ≤ p that

J • R j M # h + O(h ∞ ) = J • R j M # h + O(h ∞ )
and the result comes easily since M # h is diagonal. Therefore, we have actually proved that B h (m) from Theorem 1.7 and Bh (m) from Lemma 5.3 have the same classical expansion. □

Return to equilibrium and metastability

The goal of this section is to prove Corollaries 1.8 and 1.9. We assume that the hypotheses of Theorem 1.7 are satisfied and we choose m * among the elements of U (0) \{m} for which S is maximal such that the expansion of det(Hess m * V ) 1/2 B h (m * ) is minimal. According to Lemma 5.3 and Theorem 1.7, one can think of λ m * ,h as the non zero eigenvalue of P h with the smallest real part modulo O(h ∞ e -2S(m * )/h ). We will denote P 1 the orthogonal projection on Ker P h and for shortness λ * instead of λ m * ,h .

Proof of Corollary 1.8. We follow the proof of Theorem 1.11 in [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible metastable diffusion processes[END_REF]. We have that

∥e -tP h /h -P 1 ∥ ≤ ∥e -tP h /h Π 0 -P 1 ∥ + ∥e -tP h /h (1 -Π 0 )∥.
and thanks to Proposition 2.8 and Proposition 2.1 from [START_REF]From resolvent bounds to semigroup bounds[END_REF], we easily get e -tP h /h (1 -Π 0 ) = O(e -cht ).

Thus it suffices for the first statement to prove that

∥e -tP h /h Π 0 -P 1 ∥ ≤ C N e -Re λ * (1-C N h N )t/h .
We recall that thanks to the resolvent estimates from Theorem 1.6, Π 0 = O(1) and since P 1 is an orthogonal projection on Ker P h , we have that e -tP h /h Π 0 -P 1 = e -tP h /h (Π 0 -P 1 ) and (Π 0 -P 1 ) = O(1). Therefore, it is sufficient to prove that

∥e -tP h /h | Ran(Π0-P1) ∥ ≤ C N e -Re λ * (1-C N h N )t/h . (6.1)
Besides, we saw in Section 2 that Ker P h = CM h where M h was defined in (1.2) and since the operator Π 0 from (2.4) satisfies Π * 0 M h = M h , we get that M ⊥ h is invariant under Π 0 so Ran(Π 0 -P 1 ) = H ∩ M ⊥ h . Thus, with the notations from Proposition 5.10 and according to (6.1), it only remains to show that

∥e -tM ′ /h ∥ ≤ C N e -Re λ * (1-C N h N )t/h .
This can be done following the steps of [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible metastable diffusion processes[END_REF], proof of Theorem 1.11 as with the notation (5.12) we have Re λ * ≤ λm * ,h (1 + C N h N ). The only difference is that here we have to apply the resolvent estimates given by Theorem 4 from [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF] instead of the ones given by Theorem A.4 from [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible metastable diffusion processes[END_REF]. For the last statement, we now asume that for m ∈ U (0) \{m * }, the expansion of λ(m, h) given by Theorem 1.7 differs from the one of λ * = λ(m * , h). In that case, it is clear that λ * is a simple eigenvalue but it also happens to be a real one. Indeed, using the fact that X h 0 and b h are differential operators with real coefficients and that M h is real valued and even in the variable η, we get that λ is an eigenvalue of P h if and only if λ is an eigenvalue of P h . The rest of the proof is then also similar to the end of the proof of Theorem 1.11 from [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible metastable diffusion processes[END_REF]. □

Finally, the proof of Corollary 1.9 is a straightforward adaptation of the one of Corollary 1.6 from [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF]. (Note that our notations t - k and t + k differ from that in [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF]).

Appendix A. Proof of Lemma 1.4

Let us begin by showing that there exists a self-adjoint operator A sucht that

ϱ(H 0 ) = b * h • A • b h . (A.1)
Since ϱ(0) = 0, there exists an analytic function ρ such that ϱ(z) = z ρ(z) and |ρ(z)| ≤ C⟨z⟩ -1 . Using Cauchy's formula, one easily gets that for all z 0 ∈ {Re z > -1 2C } and f an analytic function on {Re z > -1

C } satisfying f (z) = O(⟨z⟩ -β ) for some β > 0, we have that

f (z 0 ) = -1 2iπ {Re z=-1 2C } f (z)(z 0 -z) -1 dz. (A.2)
Working with a Hilbert basis of eigenfunctions of H 0 , this identity yields

f (H 0 ) = -1 2iπ {Re z=-1 2C } f (z)(H 0 -z) -1 dz. (A.3) Besides, denoting b h =    b 1 h . . . b d h    , we have b h H 0 = (b j h H 0 ) 1≤j≤d and using the identity b j h H 0 = b * h b h b j h + hb j h , we get b h H 0 = H 1 b h where H 1 =    H 0 + h . . . H 0 + h    . (A.4)
In particular, if u is an eigenfunction of H 0 associated to a positive eigenvalue, the function b h u is an eigenfunction of H 1 associated to the same eigenvalue and therefore

H 0 (H 0 -z) -1 = b * h (H 1 -z) -1 b h . (A.5)
It follows using (A.3) with f = ρ that (A.1) holds with A = ρ(H 0 + h) ⊗ Id:

ϱ(H 0 ) = H 0 ρ(H 0 ) = b * h • ρ(H 0 + h) ⊗ Id • b h .

We can improve the integrability in the integral representation of ρ(H

0 + h) by writing ρ(z) = ρ(z) 1 + z + ϱ(z) -ϱ ∞ 1 + z + ϱ ∞ 1 + z which yields always thanks to (A.3) ρ(H 0 + h) ⊗ Id = -1 2iπ {Re z=-1 2C } ρ(z) 1 + z (H 1 -z) -1 dz (A.6) + -1 2iπ {Re z=-1 2C } ϱ(z) -ϱ ∞ 1 + z (H 1 -z) -1 dz + ϱ ∞ (H 1 + 1) -1 .
Besides, it is well known (see for instance [START_REF] Dimassi | Spectral asymptotics in the semiclassical limit[END_REF]) that the resolvent (H 1 -z) -1 is a pseudo-differential operator and we denote its symbol R z (v, η). Thanks to [START_REF] Dereziński | On the weyl symbol of the resolvent of the harmonic oscillator[END_REF], we even have the explicit expression R

z (v, η) = G z (v 2 /2 + 2η 2 )
Id where G z is an entire function defined by

G z (µ) = 2h -1 1 0 (1 -s) -z h (1 + s) z h +d-2 e -s h µ ds = 2 h -1 0 (1 -hσ) -z h (1 + hσ) z h +d-2 e -σµ dσ.
Let us then set in view of (A.6)

M h (v, η) = -1 2iπ {Re z=-1 2C } ρ(z) 1 + z R z (v, η)dz + -1 2iπ {Re z=-1 2C } ϱ(z) -ϱ ∞ 1 + z R z (v, η)dz (A.7) + ϱ ∞ R -1 (v, η)
and we now want to show that M h is a matrix of symbols matching the properties listed in Hypothesis 1.3. To this purpose, we need to study more carefully the function R z for z fixed such that Re z ≤ -1/2C. We already saw that it is analytic in both variables v and η. Now if we take (v, η) ∈ R d × Σ τ and put µ = v 2 /2 + 2η 2 , we get that µ belongs to the sector

D τ = {µ ∈ C ; |Im µ| ≤ Re µ + 4dτ 2 }.
One can then easily adapt Theorem 10 from [START_REF] Dereziński | On the weyl symbol of the resolvent of the harmonic oscillator[END_REF] to show that for n ∈ N and µ ∈ D τ , we have

|∂ n µ G z (µ)| ≤ C h -1 0 σ n (1 -hσ) -Re z/h (1 + hσ) Re z/h e -Re µσ dσ (A.8) ≤ C +∞ 0 σ n e -(Re µ-2Re z)σ dσ ≤ C n ⟨µ⟩ -(n+1)
since Re µ -2Re z > 0 for τ small enough. From (A.8) we can already conclude that

M h ∈ M d S 0 τ (⟨(v, η)⟩ -2 ) . Thus ρ(H 0 + h) ⊗ Id = Op h (M h ) with M h sending R 2d in M d (R
) as H 0 is selfadjoint. Moreover, since R z is diagonal and even in the variable η, it is also the case of M h . It only remains to prove that M h satisfies items b) and d) from Hypothesis 1.3. In order to avoid some tedious computations, instead of proving the whole expansion from item b), we only show that M h admits a principal term M 0 in M d S 0 τ (⟨(v, η)⟩ -2 ) from which we will deduce that item d) is satisfied. One easily gets for Re z ≤ -1/2C and µ ∈ D τ fixed by dominated convergence that

lim h→0 G z (µ) = 2 ∞ 0 e σ(2z-µ) dσ = 1 µ/2 -z =: G 0 z (µ). (A.9)
We would like to get some estimates of the derivatives

∂ n µ (G z -G 0 z ) in O(h⟨µ⟩ -n-1
) on D τ uniformly in z ∈ {Re z ≤ -1/2C} in order to apply the formula (A.7) to those. We have

∂ n µ (G z -G 0 z )(µ) = 2 h -1 0 exp z 1 h ln 1 + hσ 1 -hσ -2σ + (d -2) ln(1 + hσ) -1 (-σ) n e σ(2z-µ) dσ -2 ∞ h -1 (-σ) n e σ(2z-µ) dσ = 2 h -1 /2 0 exp z 1 h ln 1 + hσ 1 -hσ -2σ + (d -2) ln(1 + hσ) -1 (-σ) n e σ(2z-µ) dσ (A.10) + O e Re (2z-µ) Ch . Let us denote g z,h (σ) = exp z 1 h ln 1 + hσ 1 -hσ -2σ + (d -2) ln(1 + hσ) -1 (-σ) n
and observe that for all 0 ≤ k ≤ n, one has

∂ k σ g z,h (0) = 0 and ∂ k σ g z,h (h -1 /2) = O(h -n ⟨z⟩ k ). (A.11) Besides, on σ ∈ [0, h -1 /2], it holds ∂ n+1 σ g z,h (σ) = n+1 j=1
O h⟨z⟩ j ⟨σ⟩ j σ j-1 . (A.12) Now, let us do n + 1 integrations by parts in the first term from (A.10). By (A.11), each boundary term is O(h -n ⟨z⟩ k ⟨2z -µ⟩ -(k+1) e Re (2z-µ)/Ch ) while the remaining integral term satisfies

2 (µ -2z) n+1 h -1 /2 0 ∂ n+1 σ g z,h (σ)e σ(2z-µ) dσ ≤ C n h n+1 j=1 ⟨z⟩ j |2z -µ| n+1 ∞ 0 σ j-1 ⟨σ⟩ j e σ Re(2z-µ) dσ ≤ C n h⟨µ⟩ -(n+1) Proof. Let N ∈ N and denote r N = g h - N -1 n=0 h n g n = O S 0 τ (1) (h N ). g h • ϕ h = N -1 n=0 h n g n + r N • ϕ h = N -1 n=0 h n (g n • ϕ h ) + r N • ϕ h .
But since all the derivatives of ϕ h are bounded uniformly in h, and the ones of r N are O L ∞ (Στ ) (h N ), we see that r N • ϕ h is O C ∞ ( m j=1 Kj ) (h N ) so we have the announced result. □ Proposition C.2. Since the matrix M h from Hypothesis 1.3 satisfies M h ∼ n≥0 h n M n in M d S 0 τ (⟨(v, η)⟩ -2 ) , the vector of symbols g h defined in Remark 3.11 also admits a classical expansion g h ∼ n≥0 h n g n in M 1,d S 0 τ (⟨(v, η)⟩ -1 ) , where the (g n ) are given by

g 0 (x, v, η) = -i t η + t v 2 M 0 (x, v, η)
and

g n (x, v, η) = -i t η + t v 2 M n (x, v, η) - 1 2 ( t ∇ v - i 2 t ∇ η )M n-1 (x, v, η)
for n ≥ 1.

Proof. We have

g h = (-i t η + t v/2)M h - h 2 ( t ∇ v - i 2 t ∇ η )M h
and the last term clearly admits the expansion

- n≥1 h n 1 2 ( t ∇ v - i 2 t ∇ η )M n-1
in S 0 τ (⟨(v, η)⟩ -2 ). For the first term of g h , it suffices to notice that for any N ∈ N,

-i t η + t v 2 O M d S 0 τ (⟨(v,η)⟩ -2 ) (h N ) = O M 1,d S 0 τ (⟨(v,η)⟩ -1 )
(h N ).

□

Proposition C.3. Let K a compact set in R d ′ and a ∼ h n≥0 h n a n in C ∞ (K) such that for all n ≥ 0, a n ∼ h j≥0 h j a n,j in C ∞ (K). Then

a ∼ h n≥0 h n n j=0
a j,n-j in C ∞ (K).

Proof. It suffices to write for N ∈ N

a = N -1 n=0 h n N -1-n j=0 h j a n,j + O C ∞ (K) (h N -n ) + O C ∞ (K) (h N ) = N -1 n=0 h n n j=0 a j,n-j + O C ∞ (K) (h N ).

□

Proposition C.4. Let K a compact set in R d ′ and a ∈ C ∞ (K) such that for all β ∈ N d ′ , there exists a β,j ∈ C ∞ (K) such that ∂ β a ∼ j≥0 h j a β,j in L ∞ (K). Then a β,j = ∂ β a 0,j , i.e a ∼ j≥0 h j a 0,j in C ∞ (K).

Proof. For simplicity, we take d ′ = 1. Let us denote a j = a 0,j . By induction, it is sufficient to prove the result for β = 1, i.e prove that a 1,j = a ′ j . Here again, it suffices to prove the case j = 0 which we can then apply to the function h -1 (a -a 0 ) and so on. Let x in the interior of K and t ∈ R * in a neighborhood of 0. We look at the differential fraction a 0 (x + t) -a 0 (x) t = a(x + t) -a(x) t + O(h) t = a ′ (x) + t Proof. We first prove that (C.1) holds in L ∞ (K). Doing a Taylor expansion of b, we have for and for j ≥ 1 g n,j (x, v, v ′ , η) = iD η g n x, v + v ′ 2 , η + iψ 0 (x, v, v ′ ) ψ j (x, v, v ′ ) + R 1 j (ℓ 0 , . . . , ℓ j-1 )

N ∈ N * that b • Ψ = b • Ψ 0 + N -1 |β|=1 ∂ β b • Ψ 0 β! (Ψ -Ψ 0 ) β + O (Ψ -Ψ 0 ) N = b • Ψ 0 + N -1 |β|=1 ∂ β b • Ψ 0 β! (Ψ -Ψ 0 ) β + O L ∞ (K) (h N ) (C.2) since Ψ -Ψ 0 = O C ∞ (K) (h).
where R 1 j : C ∞ (B 0 (s, 2r)) j → C ∞ (B 0 (s, 2r)).

Proof. Since ψ(s, 0, 0) = O(h), we can suppose that r was chosen small enough so that (x, v, v ′ , η) → η + iψ(x, v, v ′ ) sends B 0 (s, 2r) × B ∞ (0, 2r) in D(0, τ ) d Hence we can use Proposition C.5 to get that

g n x, v + v ′ 2 , η + iψ(x, v, v ′ ) ∼
iD η g n x, v + v ′ 2 , η + iψ 0 (x, v, v ′ ) ψ j (x, v, v ′ )
while the terms for which |β| > 1 only feature the functions ℓ 0 , . . . , ℓ j-1 . □

Finally, we state the version of Laplace's method for integral approximation that we use in this paper.

Proposition C.7. Let x 0 ∈ R d ′ , K a compact neighborhood of x 0 and φ ∈ C ∞ (K) such that x 0 is a non degenerate minimum of φ and its only global minimum on K. Let also a h ∼ j≥0 h j a j in C ∞ (K) and denote H ∈ M d ′ (R) the Hessian of φ at x 0 . The integral det(H) 1/2 (2πh) d ′ /2 K a h (x)e -φ(x)-φ(x 0 ) h dx admits a classical expansion whose first term is given by a 0 (x 0 ).

Appendix D. Proof of Lemma 4.3

According to the proof of Corollary C.6 and the end of the proof of Lemma 4.1 from which we keep the notations, we have the following expression for R j : R j (ℓ 0 , . . . , ℓ j-1 )(x, v) = n1+n2+n3+n4=j n3,n4̸ =j

1 i n1 n 1 ! ∂ v ′ • ∂ η n1 g n2,n3 (x, v, v ′ , η)∂ v ℓ n4 (x, v ′ ) v ′ =v η=0 (D.1) + j |β|=2 i |β| β! ∂ β η g 0 x, v + v ′ 2 , i v 2 + ℓ 0 (x, v) ∂ v ℓ 0 (x, v) × s∈S β,j k∈K β a∈A β,s,k β k l=1 ψ a l (x, v, v) k ∂ v ℓ 0 (x, v)
+ iD η g 0 x, v, i(v/2 + ℓ 0 (x, v) ∂ v ℓ 0 (x, v))

j-1 k=1 ℓ k ∂ v ℓ j-k (x, v) ∂ v ℓ 0 (x, v).
Using Lemma 4.2 and (C.3), it is clear that the last two terms of R j (ℓ 0 , • • • , ℓ j-1 ) given by (D.1) and the terms of the first sum for which n 1 = 0 are real valued. For the rest of the first term, we start by noticing that one can establish by induction that for n 1 ≥ 1,

∂ v ′ • ∂ η n1 = p∈ 1,d n 1
1 i n1 n 1 ! ∂ v ′ • ∂ η n1 g n2,n3 (x, v, v ′ , η)∂ v ℓ n4 (x, v ′ ) v ′ =v η=0
from (D.1) is also real so R j (ℓ s 0 , . . . , ℓ s j-1 ) is real valued. For the last statement, it suffices to use the formula (D.1) after noticing that ψ (and hence the (g n2,n3 )) remain unchanged when ℓ is replaced by -ℓ.

20 4 . 1 .

 41 Solving for ℓ s 1. Introduction 1.1. Motivations. We are interested in the linear Boltzmann equation: (1.1)

Lemma 3 . 4 .

 34 Let m, m ′ two distinct local minima of Y . The real number σ = sup a ∈ R ; m and m ′ are in two different CCs of {Y < a} is well defined and {Y < σ} has at least two CCs Ω ∋ m and Ω ′ ∋ m ′ . Moreover, σ is a separating saddle value and Ω, Ω ′ are critical. Proof. We can assume that Y (m) ≤ Y (m ′ ) so taking a := inf A Y where A is a well chosen annulus centered in m ′ , we see that a ∈ R ; m and m ′ are in two different CCs of {Y < a} ̸ = ∅ (3.1)

Hypothesis 3 . 10 .

 310 For all m ∈ U (0) , we have a) m is the only global minimum of V | E V /2 (m) b) for any m ′ ∈ U (0) \{m}, the sets j(m) and j(m ′ ) do not intersect.According to Proposition 3.9 and (3.3), this hypothesis is equivalent to the facts that (m, 0) is the only global minimum of W | E(m) and j W (m, 0) ∩ j W (m ′ , 0) = ∅ which is what we use in practice.

8 )

 8 with ε(r) > 0 to be fixed later and θ m,h = 0 everywhere else.(3.9) Note that θ m,h takes values in [0, 1]. Denote Ω the CC of {W ≤ σ(m)} containing m. The CCs of {W ≤ σ(m)} are separated so for ε > 0 small enough, there exists ε > 0 such thatmin W (x, v) ; d (x, v), Ω = ε = σ(m) + 2ε.Thus the distance between {W ≤ σ(m) + ε} ∩ Ω + B(0, ε) and ∂ Ω + B(0, ε) is positive and we can consider a cut-off function

1 0( 1 - 1 0( 1 -( 1 -

 11111 s)a ′′ (x + st)ds + O(h) t= a 1,0 (x) + O(h) + t s)a ′′ (x + st)ds + Os)a 2,0 (x + st)ds.Taking now the limit t → 0, we get a ′ 0 (x) = a 1,0 (x) which was the desired result. □ Proposition C.5. Recall the notation (4.11) and letK ⊂ R d ′ a compact set, Ψ : K → D(0, τ ) d a smooth function such that Ψ ∼ j≥0 h j Ψ j in C ∞ (K) and b an analytic function on Σ τ . Then b • Ψ ∼ j≥0 h j b j (C.1) in C ∞ (K), with b 0 = b • Ψ 0 and for j ≥ 1, b j = j |β|=1 ∂ β b • Ψ 0 β! s∈S β,j k∈K β a∈A β,s,k β k l=1 Ψ a l k ,whereK β = supp β = {k ∈ 1, d ; β k ̸ = 0},S β,j = {s ∈ N d ; supp s = K β , |s| = j and s ≥ β} and A β,s,k = {a ∈ (N * ) β k ; |a| = s k }.

∂

  Now one can see that(Ψ -Ψ 0 ) β ∼ j≥|β| h j s∈S β,j k∈K β a∈A β,s,k j k∈K β a∈A β,s,k β k l=1 Ψ a l k + O C ∞ (K) (h N ) + O L ∞ (K) (h N ) = b • Ψ 0 + β b • Ψ 0 β! s∈S β,j k∈K β a∈A β,s,k β k l=1 Ψ a l k + O L ∞ (K) (h N ) which proves that (C.1) holds in L ∞ (K). Besides,the derivatives of b • Ψ are linear combinations of products of some derivatives of Ψ with some ∂ γ b • Ψ where γ is a integer multi-index. Hence the expansion of Ψ in C ∞ (K) and the result that we just proved applied to ∂ γ b • Ψ instead of b • Ψ yield that for all β ∈ N d ′ , ∂ β (b • Ψ) admits a classical expansion in L ∞ (K) whose coefficients are smooth. Therefore, Proposition C.4 enables us to conclude that (C.1) holds in C ∞ (K). □ Corollary C.6. Using the notations from the proof of Lemma 4.1, we haveg n x, v + v ′ 2 , η + iψ(x, v, v ′ ) ∼ j≥0 h j g n,j (x, v, v ′ , η) on B 0 (s, 2r) × B ∞ (0, 2r) with g n,0 (x, v, v ′ , η) = g n x, v + v ′ 2 , η + iψ 0 (x, v, v ′ )

j≥0 h j g

  n,j (x, v, v ′ , η) on B 0 (s, 2r) × B ∞ (0, 2r) withg n,0 (x, v, v ′ , η) = g n x, v + v ′ 2 , η + iψ 0 (x, v, v ′ )and for j ≥ 1g n,j (x, v, v ′ , η) = j |β|=1 i |β| β! ∂ β η g n x, v + v ′ 2 , η + iψ 0 (x, v, v ′ ) s∈S β,j k∈K β a∈A β,s,k β k l=1 ψ a l k (C.3) where K β = supp β = {k ∈ 1, d ; β k ̸ = 0}, S β,j = {s ∈ N d ; supp s = K β , |s| = jand s ≥ β} and A β,s,k = {a ∈ (N * ) β k ; |a| = s k }. Now, we see thanks to (C.3) that the terms of g n,j (x, v, v ′ , η) for which |β| = 1 yield

g

  notation (3.20), we define γ(p) = n1 k=1 e p k (note that |γ(p)| = n 1 ). Besides, we have for 0 ≤ n 2 ≤ j and p ∈ 1, d n1∂γ(p) η g n2,0 (x, v, v ′ , 0) = ∂ γ(p) η g n2 x, v + v ′ 2 , iψ 0 (x, v, v ′ ) ∈ i n1 R d (D.3)according to Lemma 4.2 and in the case j ≥ 2, for 1 ≤ n 3 ≤ j -1∂ γ(p) η g n2,n3 (x, v, v ′ , n2 x, v + v ′ 2 , iψ 0 (x, v, v ′ ) s∈S β,n 3 k∈K β a∈A β,s,k β k l=1 ψ a l k ∈ i n1 R dwhere we used (C.3) and Lemma 4.2 once again. The combination of (D.2), (D.3) and (D.4) enables us to conclude that the term n1+n2+n3+n4=j n1̸ =0; n3,n4̸ =j

  Resolvent estimates and first localization of the small eigenvalues. Using Lemma 2.3, it is clear that for u ∈ Span (g h j ) 1≤j≤n0 and A ∈ {P h , P * h , P * h P h , P h P * h } we have

The same proof holds when replacing P by P * and N + h,ε by N - h,ε . □ 2.2.

  Let m ∈ U (0) \{m}. Using the notations from Lemma 5.3, we havei) ∥P h fm,h ∥ 2 = O(h ∞ ⟨P h fm,h , fm,h ⟩) ii) ∥P * h fm,h ∥ 2 = O(h⟨P h fm,h , fm,h ⟩). Proof.To prove i), first remark that thanks to (3.15)-(3.18) we have

		1/2
	(5.8)	(2πh) d	∥f m,h ∥ 2 ∼	k≥0	h k ck
	with c0 = 1. The conclusion follows from (5.7), (3.6) and (5.8).	□
	Lemma 5.4.

Acknowledgements. The author is grateful to Laurent Michel for his advice through this work and especially for his suggestions in the proof of Lemma 1.4, as well as Jean-François Bony for helpful discussions. This work is supported by the ANR project QuAMProcs 19-CE40-0010-01.

Proposition 4.9. For all j ≥ 1, there exists ℓ s j ∈ C ∞ (B 0 (s, 2r)) solving (4.13). Moreover, ℓ s j is real valued in view of Lemmas 4.2 and 4.3.

Computation of the small eigenvalues

Now that we have found (ℓ j ) j≥0 ⊂ C ∞ (B 0 (s, 2r), R) solving (4.12) and (4.13) with ℓ 0 vanishing at (s, 0), we can use a Borel procedure to construct ℓ ∈ C ∞ (R 2d , R) supported in B 0 (s, 3r) and satisfying ℓ ∼ j≥0 ℓ j on B 0 (s, 2r).

Remark 5.1. The properties a)-c) from (3.5) are satisfied by both the functions ℓ s,h and -ℓ s,h . Moreover, by Lemma 4.3, (-ℓ s j ) j≥0 also solve (4.12) and (4.13).

We are now in position to prove that all the properties from (3.5) are satisfied.

Proposition 5.2. We can choose the signs of the functions (ℓ s,h ) j(m) such that (3.5) holds true and the coefficients from the classical expansion of ℓ s,h solve (4.12) and (4.13).

Proof. Recall that by item b) from Hypothesis 3.10, each function ℓ s,h corresponds to a unique m ∈ U (0) \{m}. Thanks to Lemmas 4.5 and 4.6, it is clear that item d) from (3.5) is satisfied by both ℓ s,h and -ℓ s,h . Hence according to Remark 5.1, it is sufficient to prove that the signs of (ℓ s,h ) j(m) can be chosen so that θ m,h is smooth on a neighborhood of supp χ m . From (3.7), (3.8) and (3.9) we see that the only parts on which it is not clear that θ m,h is smooth are

and

Let s ∈ j(m) and (x, v) ∈ B 0 (s, r)\{(s, 0)} such that ℓ s 0 (x, v) = 0. Using Lemma 4.6, we see that if r > 0 is small enough,

because (s, 0) is a non degenerate local minimum of ϕ + . Hence, {ℓ s 0 = 0} ∩ B 0 (s, r) ⊂ {W ≥ σ(m)}. Now assume by contradiction that for any r > 0, the function ℓ s 0 takes both positive and negative values on E(m) ∩ B 0 (s, r). Then according to Lemma 3.1, the two CCs of U r ∩ {W < σ(m)} are both included in E(m) (the one on which ℓ s 0 > 0 and the one where ℓ s 0 < 0). This is a contradiction with the fact that s ∈ V (1) . Therefore ℓ s 0 has a sign on E(m) ∩ B 0 (s, r) and we can choose it so that ℓ s 0 is a positive function on E(m) ∩ B 0 (s, r). By uniform continuity, we can then choose ε(γ) > 0 small enough so that

Similarly, if we denote Ω s the other CC of {W < σ(m)} which contains (s, 0) on its boundary, we have since (s, 0) is not a critical point of ℓ s 0 that this function is negative on Ω s ∩ B 0 (s, r) and

Choosing once again ε(r) small enough, we can even assume that

We first prove that F 1 does not meet the support of χ m . Recall that Ω denotes the CC of {W ≤ σ(m)} containing m. For s ∈ j(m), we can deduce from (5.1) that if (x, v) ∈ ∂B 0 (s, r) such that ℓ s 0 (x, v) = 0, then (x, v) / ∈ Ω. Hence |ℓ s 0 | must attain a positive minimum on ∂B 0 (s, r) ∩ Ω, so we can choose γ(r) > 0 such that ∂B 0 (s, r) ∩ {|ℓ s 0 | ≤ 2γ} does not intersect Ω. It follows that we can choose ε(γ) > 0 such that

Hence, choosing r small enough, the support of (5.11) is contained in {x / ∈ π x E ′ + B(0, ε ′ ) } × R d v so in particular it does not intersect with the support of fm ′ ,h . ii): Here we can suppose that V (m) ≥ V (m ′ ). Let us first treat the case where V (m) = V (m ′ ). Then according to item a) from Hypothesis 3.10, E and E ′ are two disjoint open sets. Hence, as we saw earlier, Lemma 3.2 and item b) from Hypothesis 3.10 imply that E ∩ E ′ = ∅. The conclusion then follows from (3.12). If V (m) > V (m ′ ), then item a) from Hypothesis 3.10 implies that (m, 0) is the only global minimum of W | E+B(0,ε ′ ) . Therefore using (3.12), we can easily compute

The conclusion immediately follows from (5.8). □ Remark 5.6. For the first item, we have actually proven that

From now on, we denote λm,h = ⟨P h fm,h , fm,h ⟩ = ⟨Q h fm,h , fm,h ⟩ (5.12) for which we computed a classical expansion in Lemma 5.3. Let us consider once again the spectral projection introduced in (2.4). We saw in particular that Π 0 = O(1). Lemma 5.7. For any m ∈ U (0) , we have

Proof. We simply recall the proof from [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible metastable diffusion processes[END_REF]: we write

We can then conclude using Lemma 5.4 and the resolvent estimate from Theorem 1.6. The proof for the adjoint is almost identical. □ Lemma 5.8. The family (Π 0 fm,h ) m∈U (0) is almost orthonormal: there exists c > 0 such that

In particular, it is a basis of the space H = Ran Π 0 introduced in (2.4). Moreover, we have

Proof. The proof is the same as the one of Proposition 4.10 in [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible metastable diffusion processes[END_REF]. □

Let us re-label the local minima m 1 , . . . , m n0 so that (S(m j )) j=1,...,n0 is non increasing in j. For shortness, we will now denote fj = fmj,h and λj = λmj,h which still depend on h. Note in particular that according to Lemma 5.3, λj = O( λk ) whenever 1 ≤ j ≤ k ≤ n 0 . We also denote (ũ j ) j=1,...,n0 the orthogonalization by the Gram-Schmidt procedure of the family (Π 0 fj ) j=1,...,n0 and

In this setting and with our previous results, we get the following (see [START_REF] Peutrec | Sharp spectral asymptotics for non-reversible metastable diffusion processes[END_REF], Proposition 4.12 for a proof).

thanks to (A.12). Thus, we have shown that for n ∈ N, µ ∈ D τ and Re z ≤ -1/2C,

Id and defining M 0 (v, η) as in (A.7) with R z replaced by R 0 z , we deduce that

so item b) from Hypothesis 1.3 holds true. Finally, by definition of M 0 and thanks to (A.9) and (A.2), we have

by assumption on ϱ. Therefore item d) from Hypothesis 1.3 holds true and the proof is complete.

Appendix B. Linear algebra Lemma

We use the following lemma which is inspired by [START_REF] Bony | Eyring-kramers law for fokker-planck type differential operators[END_REF], Lemma 2.6. Then M has no spectrum in iR.

Proof. Let λ ∈ R and X ∈ Ker [M -iλ], we first show that X ∈ Ker T . Since T is hermitian positive semidefinite, it is sufficient to show that ⟨T X, X⟩ = 0. Using the properties of S, A and T we have ⟨T X, X⟩ = Re (A + T )X, X = Re S -1 S(A + T )X, X = Re iλ S -1 X, X = 0 so X ∈ Ker T . Thanks to the assumption, it only remains to prove that X ∈ Ker M . This can be done easily by noticing that

Here we use the convention -1 j=0 a j = 0 for any sequence (a j ) j≥0 in a vector space. For

means that for all β ∈ N d ′ , there exists C β,N such that ∥∂ β a∥ ∞,K ≤ C β,N h N (resp. there exists C N such that ∥a∥ ∞,K ≤ C N h N ). We will also use the notations from Definition 1.2 and (1.6).

Proposition C.1. Let m ∈ N * ; d 1 , . . . , d m ∈ N * and for 1 ≤ j ≤ m, K j ⊂ R dj some compact sets. Let a smooth function

. Consider g h ∼ h n≥0 h n g n in S 0 τ (1) or in C ∞ (K) if ϕ h actually takes values in R d . Then

Thomas Normand, Institut de Mathématiques de Bordeaux, Université de Bordeaux

Email address: thomas.normand@math.u-bordeaux.fr