Sharp spectral asymptotics for non-reversible metastable diffusion processes

TL;DR

This paper analyzes the spectral properties of non-reversible diffusion processes in low temperature regimes, revealing the number and behavior of small eigenvalues related to metastable states and providing Eyring-Kramers type formulas.

Contribution

It establishes the existence and asymptotic behavior of small eigenvalues of the diffusion operator for non-reversible processes with Morse potential barriers.

Findings

Exactly $n_0$ eigenvalues in the low-temperature limit

Eigenvalues have exponentially small moduli

Asymptotic behavior described by Eyring-Kramers formulas

Abstract

Let $U_h:\mathbb R^{d}\to \mathbb R^{d}$ be a smooth vector field and consider the associated overdamped Langevin equation $$dX_t=-U_h(X_t)\,dt+\sqrt{2h}\,dB_t$$ in the low temperature regime $h\rightarrow 0$. In this work, we study the spectrum of the associated diffusion $L=-h\Delta+U_h\cdot\nabla$ under the assumptions that $U_h=U_{0}+h\nu$, where the vector fields $U_{0}:\mathbb R^{d}\to \mathbb R^{d}$ and $\nu:\mathbb R^{d}\to \mathbb R^{d}$ are independent of $h\in(0,1]$, and that the dynamics admits $e^{-\frac Vh}$ as an invariant measure for some smooth function $V:\mathbb{R}^d\rightarrow\mathbb{R}$. Assuming additionally that $V$ is a Morse function admitting $n_0$ local minima, we prove that there exists $\epsilon>0$ such that in the limit $h\to 0$, $L$ admits exactly $n_0$ eigenvalues in the strip $\{0\leq \operatorname{Re}(z)< \epsilon\}$, which have moreover exponentially small moduli. Under a generic assumption on the potential barriers of the Morse function $V$, we also prove that the asymptotic behaviors of these small eigenvalues are given by Eyring-Kramers type formulas.

Equations

$$d{X}_t=-U_h(X_t)dt+\sqrt{2h}d{B}_t$$

$$d{X}_t=-U_h(X_t)dt+\sqrt{2h}d{B}_t,$$

$${\partial }_{t}u+L(u)=0\text{and}{\partial }_t\rho +L^{†}(\rho )=0,$$

$$L=-h\Delta +U_h\cdot \nabla$$

$$L^{†}=-\mathrm{div}\circ (h\nabla +U_h)$$

$$U_h=U_0+h\nu$$

$$b\cdot \nabla V=0,\mathrm{div}(\nu )=0,\text{and}\mathrm{div}(b)=\nu \cdot \nabla V.$$

$$L_{V,b,\nu }:=L=-h\Delta +\nabla V\cdot \nabla +b_h\cdot \nabla ,$$

$$b_h:=b+h\nu =U_0-\nabla V+h\nu =U_h-\nabla V.$$

$$b\cdot \nabla V=0,\mathrm{div}b=0\text{and}\nu =0,$$

$$b\ =\ J\,\nabla V\quad\text{and}\quad\nu\ =\ \Big{(}\,\sum_{i=1}^{d}\partial_{i}J_{ij}\,\Big{)}_{1\leq j\leq d}\,,$$

$$L_{V,b,\nu}\ =\ -h\,e^{\frac{V}{h}}\,\operatorname{div}\circ\big{(}\,e^{-\frac{V}{h}}\big{(}I_{d}-J\big{)}\nabla\,\big{)}\,,$$

$$d{X}_t=-\nabla V(X_t)dt+\sqrt{2h}d{B}_t$$

$$m_h(dx):=Z_h^{-1}{e}^{-\frac{V(x)}{h}}dx\text{and}{Z}_h:={\int }_{{\mathbb{R}}^d}{e}^{-\frac{V(x)}{h}}dx.$$

$$\forall u,v\in {\mathcal{C}}_c^{\infty }({\mathbb{R}}^d),⟨L_{V,b,\nu }u,v{⟩}_{L^2(m_h)}=⟨u,L_{V,-b,-\nu }v{⟩}_{L^2(m_h)}.$$

$$\mathrm{Re}⟨L_{V,b,\nu }u,u{⟩}_{L^2(m_h)}$$

$$=h\Vert \nabla u{\Vert }_{L^2(m_h)}^2\ge 0$$

$$V\ge -C\text{on Rd}$$

$$|\nabla V(x)|\ge \frac{1}{C}\text{and}|\mathrm{Hess}V(x)|\le C|\nabla V(x){|}^2.$$

$$|b(x)|+|\nu (x)|\le C(1+|\nabla V(x)|).$$

$$\mathsf{U}:L^2({\mathbb{R}}^d,dx)⟶L^2({\mathbb{R}}^d,m_h),\mathsf{U}(u)=m_h^{-\frac{1}{2}}u=Z_h^{\frac{1}{2}}{e}^{\frac{V}{2h}}u.$$

$${\mathsf{U}}^{\ast }h{L}_{V,b,\nu }\mathsf{U}$$

$$={\Delta }_{\varphi }+b_h\cdot d_{\varphi },$$

$$d_{\varphi }:=d_{\varphi ,h}:=h\nabla +\nabla \varphi =h{e}^{-\frac{\varphi }{h}}\nabla e^{\frac{\varphi }{h}}$$

$${\Delta }_{\varphi }:={\Delta }_{\varphi ,h}:=-h^2\Delta +|\nabla \varphi {|}^2-h\Delta \varphi =-h^2{e}^{\frac{\varphi }{h}}\mathrm{div}{e}^{-\frac{\varphi }{h}}{d}_{\varphi }$$

$$P_{\varphi }:=P_{\varphi ,b,\nu }:={\Delta }_{\varphi }+b_h\cdot d_{\varphi }$$

$$L_{V,0,0}=-h\Delta +\nabla V\cdot \nabla =h{\nabla }^{\ast }\nabla$$

$${\sigma }_{ess}({\Delta }_{\varphi })\subset [c_0,+\infty [.$$

$$D({\Delta }_{\varphi })\subset D(P_{\varphi })\cap D(P_{\varphi }^{\ast })\subset D(P_{\varphi })\cup D(P_{\varphi }^{\ast })\subset \{u\in L^2({\mathbb{R}}^d),d_{\varphi }u\in L^2({\mathbb{R}}^d)\},$$

$$\forall u\in D({\mathbf{P}}_{\varphi }),\mathrm{Re}⟨{\mathbf{P}}_{\varphi }u,u⟩=\Vert d_{\varphi }u{\Vert }^2.$$

$$\Gamma_{\Lambda_{0}}\ :=\ \big{\{}\operatorname{Re}(z)\geq 0\ \text{and}\ |\operatorname{Im}z|\leq\Lambda_{0}\max\big{(}\operatorname{Re}(z),\sqrt{\operatorname{Re}(z)}\big{)}\big{\}}\ \subset\ {\mathbb{C}}\,,$$

Full text

[TABLE]

in the low temperature regime $h\rightarrow 0$. In this work, we study the spectrum of the associated diffusion $L=-h\Delta+U_{h}\cdot\nabla$ under the assumptions that $U_{h}=U_{0}+h\nu$, where the vector fields $U_{0}:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ and $\nu:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ are independent of $h\in(0,1]$, and that the dynamics admits $e^{-\frac{V}{h}}dx$ as an invariant measure for some smooth function $V:\mathbb{R}^{d}\rightarrow\mathbb{R}$. Assuming additionally that $V$ is a Morse function admitting $n_{0}$ local minima, we prove that there exists $\epsilon>0$ such that in the limit $h\to 0$, $L$ admits exactly $n_{0}$ eigenvalues in the strip $\{0\leq\operatorname{Re}(z)<\epsilon\}$, which have moreover exponentially small moduli. Under a generic assumption on the potential barriers of the Morse function $V$, we also prove that the asymptotic behaviors of these small eigenvalues are given by Eyring-Kramers type formulas.

D. Le Peutrec : Institut Denis Poisson, Université d’Orléans, Université de Tours, CNRS, Orléans, France. E-mail: dorian.le-peutrec@univ-orleans.fr

L. Michel : Université de Bordeaux, Institut Mathématiques de Bordeaux, Talence, France. E-mail: laurent.michel@math.u-bordeaux.fr

MSC 2010: 60J60, 35Q82, 81Q12, 35P15, 81Q20.

Keywords: Non-reversible overdamped Langevin dynamics, Metastability, Spectral theory, Semiclassical analysis, Eyring-Kramers formulas.

1. 1 Introduction

2. 1.1 Setting and motivation

3. 1.2 Preliminary analysis

4. 1.3 Generic Morse-type hypotheses and labelling procedure

5. 1.4 Main results and comments

6. 2 General spectral estimates

7. 2.1 Proof of Proposition 1.1

8. 2.2 Spectral analysis near the origin

9. 3 Geometric preparation

10. 4 Spectral analysis in the case of Morse functions

11. 4.1 Construction of accurate quasimodes

12. 4.2 Quasimodal estimates

13. 4.3 Proof of Theorem 1.9

14. 4.4 Proof of Theorem 1.11

15. A Some results in linear algebra

1. Introduction

1.1. Setting and motivation

Let $d\geq 2$, $U_{h}:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}^{d}$ be a smooth vector field depending on a small parameter $h\in(0,1]$, and consider the associated overdamped Langevin equation

[TABLE]

where $X_{t}\in{\mathbb{R}}^{d}$ and $(B_{t})_{t\geq 0}$ is a standard Brownian motion in ${\mathbb{R}}^{d}$. The associated Kolmogorov (backward) and Fokker-Planck equations are then the evolution equations

[TABLE]

where the elliptic differential operator

[TABLE]

is the infinitesimal generator of the process (1.1),

[TABLE]

denotes the formal adjoint of $L$, and for $x\in{\mathbb{R}}^{d}$ and $t\geq 0$: $u(t,x)=\mathbb{E}^{x}[f(X_{t})]$ is the expected value of the observable $f(X_{t})$ when $X_{0}=x$ and $\rho(t,\cdot)$ is the probability density (with respect to the Lebesgue measure on ${\mathbb{R}}^{d}$) of presence of $(X_{t})_{t\geq 0}$. In this setting, the Fokker–Planck equation, that is the second equation of (1.2), is also known as the Kramers-Smoluchowski equation.

Throughout this paper, we assume that the vector field $U_{h}$ decomposes as

[TABLE]

for some real smooth vector fields $U_{0}$ and $\nu$ independent of $h$. Moreover, we consider the case where the above overdamped Langevin dynamics admits a specific stationary distribution satisfying the following assumption:

Assumption 1**.**

There exists a smooth function $V:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$ such that $L^{\dagger}(e^{-\frac{V}{h}})=0$ for every $h\in(0,1]$.

A straightforward computation shows that Assumption 1 is satisfied if and only if the vector field $U_{h}=U_{0}+h\nu$ satisfies the following relations, where we denote $b:=U_{0}-\nabla V$,

[TABLE]

Using this decomposition, the generator $L$ writes

[TABLE]

where

[TABLE]

Note moreover that the two following particular cases enter in the framework of Assumption 1:

The case where

[TABLE]

which is in particular satisfied when $\nu=0$ and $b$ is the matrix product $b=J\,\nabla V$, where $J$ is a smooth map from ${\mathbb{R}}^{d}$ into the set of real antisymmetric matrices of size $d$ such that \operatorname{div}\big{(}J\,\nabla V\big{)}=0. For instance, this later condition holds if $J(x)=\tilde{J}\circ V(x)$ for some antisymmetric matrices $\tilde{J}(y)$ depending smoothly on $y\in{\mathbb{R}}$.

• 2.

The case where

[TABLE]

where $J$ is a smooth map from ${\mathbb{R}}^{d}$ into the set of real antisymmetric matrices of size $d$.

In the case of (1.7), $L_{V,b,\nu}$ has in particular the following supersymmetric-type structure,

[TABLE]

and both cases coincide when $b_{h}$ has the form $b_{h}=b=J\,\nabla V$ for some constant antisymmetric matrix $J$. In the case of (1.6), the structure (1.8) fails to be true in general and we refer to [20] for more details on these questions. Let us also point out that under Assumption 1, the vector field $b_{h}$ defined in (1.5) is very close to the transverse vector field introduced in [1] and next used in [14].

In this paper, we are interested in the spectral analysis of the operator $L_{V,b,\nu}$ and in its connections with the long-time behaviour of the dynamics (1.1) when $h\to 0$. In this regime, the process $(X_{t})_{t\geq 0}$ solution to (1.1) is typically metastable, which is characterized by a very slow return to equilibrium. We refer especially in this connection to the related works [1, 14] dealing with the mean transition times between the different wells of the potential $V$ for the process $(X_{t})_{t\geq 0}$. Our setting is also motivated by the question of accelerating the convergence to equilibrium, which is of interest for computational purposes. It is indeed known that non-gradient perturbations of the overdamped gradient Langevin dynamics

[TABLE]

which preserve the invariant measure $e^{-\frac{V}{h}}dx$ cannot converge slower to equilibrium than the associated gradient dynamics (1.9). See in particular [18] on this topic, where the authors considered linear drifts and computed the optimal rate of return to equilibrium in this case, and references therein.

1.2. Preliminary analysis

In view of Assumption 1, we look at $L_{V,b,\nu}$ acting in the natural weighted Hilbert space $L^{2}({\mathbb{R}}^{d},m_{h})$, where

[TABLE]

Note that we assume here that $e^{-\frac{V}{h}}\in L^{1}({\mathbb{R}}^{d})$ for every $h\in(0,1]$, which will be a simple consequence of our further hypotheses. In this setting, a first important consequence of (1.3) is the following identity, easily deduced from the relation $\operatorname{div}(b_{h}e^{-\frac{V}{h}})=0$,

[TABLE]

In particular, using (1.4), it holds

[TABLE]

for all $u\,\in\,{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d})$ and the operator $L_{V,b,\nu}$ acting on ${\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d})$ in $L^{2}({\mathbb{R}}^{d},m_{h})$ is hence accretive.

Let us now introduce the following confining assumptions at infinity on the functions $V$, $b$, and $\nu$ that we will consider in the rest of this work.

Assumption 2**.**

There exist $C>0$ and a compact set $K\subset{\mathbb{R}}^{d}$ such that it holds

[TABLE]

and, for all $x\in{\mathbb{R}}^{d}\setminus K$,

[TABLE]

Moreover, there exists $C>0$ such that the vector fields $b=U_{0}-\nabla V$ and $\nu$ satisfy the following estimate for all $x\in{\mathbb{R}}^{d}$:

[TABLE]

One can show that when $V$ is bounded from below and the first estimate of (1.12) is satisfied, it also holds, for some $C>0$, $V(x)\geq C|x|$ outside a compact set (see for example [19, Lemma 3.14]). In particular, when Assumption 2 is satisfied, then $e^{-\frac{V}{h}}\in L^{1}({\mathbb{R}}^{d})$ for all $h\in(0,1]$ (which justifies the definition of $Z_{h}$ in (1.10)).

In order to study the operator $L_{V,b,\nu}$ in $L^{2}({\mathbb{R}}^{d},m_{h})$, it is often useful to work with its counterpart in the flat space $L^{2}({\mathbb{R}}^{d},dx)$ by using the unitary transformation

[TABLE]

Defining $\phi:=\frac{V}{2}$, we then have the unitary equivalence

[TABLE]

where

[TABLE]

and

[TABLE]

denotes the usual semiclassical Witten Laplacian acting on functions. It is thus equivalent to study $L_{V,b,\nu}$ acting in the weighted space $L^{2}({\mathbb{R}}^{d},m_{h})$ or

[TABLE]

acting in the flat space $L^{2}({\mathbb{R}}^{d},dx)$.

The Witten Laplacian $\Delta_{\phi}=P_{\phi,0,0}$, which is the counterpart of the weighted Laplacian

[TABLE]

(the adjoint is considered here with respect to $m_{h}$) acting in the flat space $L^{2}({\mathbb{R}}^{d},dx)$, is moreover essentially self-adjoint on ${\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{n})$ (see [7, Theorem 9.15]). We still denote by $\Delta_{\phi}$ its unique self-adjoint extension and by $D(\Delta_{\phi})$ the domain of this extension. In addition, it is clear that for every $h\in(0,1]$, it holds $\Delta_{\phi}e^{-\frac{\phi}{h}}=0$ in the distribution sense. Hence, under Assumption 2, since $\phi=\frac{V}{2}$ satisfies the relation (1.12), it holds $e^{-\frac{\phi}{h}}\in L^{2}({\mathbb{R}}^{d})$ and the essential self-adjointness of $\Delta_{\phi}$ then implies that $e^{-\frac{\phi}{h}}\in D(\Delta_{\phi})$ so that $0\in{\rm\,Ker\,}\Delta_{\phi}$. It follows moreover from (1.12) and from [8, Proposition 2.2] that there exists $h_{0}>0$ and $c_{0}>0$ such that for all $h\in(0,h_{0}]$, it holds

[TABLE]

Coming back to the more general operator $P_{\phi}=P_{\phi,b,\nu}$ defined in (1.16), or equivalently to the operator $L_{V,b,\nu}$ according to the relation (1.14), the following proposition gathers some of its basic properties which specify in particular the preceding properties of $\Delta_{\phi}$ (and their equivalents concerning the weighted Laplacian $L_{V,0,0}$). It will be proven in Section 2.1.

Proposition 1.1**.**

Under Assumption 1, the operator $P_{\phi}$ with domain ${\mathcal{C}}_{c}^{\infty}({\mathbb{R}}^{d})$ is accretive. Moreover, assuming in addition Assumption 2, there exists $h_{0}\in(0,1]$ such that the following hold true for every $h\in(0,h_{0}]$:

1. i)

The closure of $(P_{\phi},{\mathcal{C}}_{c}^{\infty}({\mathbb{R}}^{d}))$, that we still denote by $P_{\phi}$, is maximal accretive, and hence its unique maximal accretive extension. 2. ii)

The operator $P^{*}_{\phi}$ is maximal accretive and ${\mathcal{C}}_{c}^{\infty}({\mathbb{R}}^{d})$ is a core for $P^{*}_{\phi}$. We have moreover the inclusions

[TABLE]

where, for any unbounded operator $A$, $D(A)$ denotes the domain of $A$. In addition, for ${\bf P}_{\phi}\in\{P_{\phi},P^{*}_{\phi}\}$, we have the equality

[TABLE] 3. iii)

There exists $\Lambda_{0}>0$ such that, defining

[TABLE]

the spectrum $\sigma(P_{\phi})$ of $P_{\phi}$ is included in $\Gamma_{\Lambda_{0}}$ and

[TABLE] 4. iv)

There exists $c_{1}>0$ such that the map $z\mapsto(P_{\phi}-z)^{-1}$ is meromorphic in $\{\operatorname{Re}(z)<c_{1}\}$ with finite rank residues. In particular, the spectrum of $P_{\phi}$ in $\{\operatorname{Re}(z)<c_{1}\}$ is made of isolated eigenvalues with finite algebraic multiplicities. 5. v)

It holds ${\rm\,Ker\,}P_{\phi}={\rm\,Ker\,}P^{*}_{\phi}={\rm Span}\{e^{-\frac{\phi}{h}}\}$ and [math] is an isolated eigenvalue of $P_{\phi}$ (and then of $P^{*}_{\phi}$) with algebraic multiplicity one.

From (1.14) and the last item of Proposition 1.1, note that ${\rm\,Ker\,}L_{V,b,\nu}={\rm Span}\{1\}$ and that [math] is an isolated eigenvalue of $L_{V,b,\nu}$ with algebraic multiplicity one. Moreover, according to Proposition 1.1 and to the Hille-Yosida theorem, the operators $L_{V,b,\nu}$ and its adjoint $L^{*}_{V,b,\nu}$ (in $L^{2}({\mathbb{R}}^{d},m_{h})$) generate, for every $h>0$ small enough, contraction semigroups $(e^{-tL_{V,b,\nu}})_{t\geq 0}$ and $(e^{-tL^{*}_{V,b,\nu}})_{t\geq 0}$ on $L^{2}({\mathbb{R}}^{d},m_{h})$ which permit to solve (1.2).

1.3. Generic Morse-type hypotheses and labelling procedure

In order to describe precisely, in particular by stating Eyring-Kramers type formulas, the spectrum around [math] of $L_{V,b,\nu}$ (or equivalently of $P_{\phi}$) in the regime $h\to 0$, we will assume from now on that $V$ is a Morse function:

Assumption 3**.**

The function $V$ is a Morse function.

Under Assumption 3 and thanks to Assumption (1.12), the set ${\mathcal{U}}$ made of the critical points of $V$ is finite. In the following, the critical points of $V$ with index [math] and with index $1$, that is its local minima and its saddle points, will play a fundamental role, and we will respectively denote by ${\mathcal{U}}^{(0)}$ and ${\mathcal{U}}^{(1)}$ the sets made of these points. Throughout the paper, we will moreover denote

[TABLE]

From the pioneer work by Witten [24], it is well-known that for every $h\in(0,1]$ small enough, there is a correspondence between the small eigenvalues of $\Delta_{\phi}$ and the local minima of $\phi=\frac{V}{2}$. More precisely, we have the following result (see in particular [11, 6, 8] or more recently [22]).

Proposition 1.2**.**

Assume that (1.12) and Assumption 3 hold true. Then, there exist $\epsilon_{0}>0$ and $h_{0}>0$ such that for every $h\in(0,h_{0}]$, $\Delta_{\phi}$ has precisely $n_{0}$ eigenvalues (counted with multiplicity) in the interval $[0,\epsilon_{0}h]$. Moreover, these eigenvalues are actually exponentially small, that is live in an interval $[0,Che^{-2\frac{S}{h}}]$ for some $C,S>0$ independent of $h\in(0,h_{0}]$.

Since the operator $P_{\phi}=\Delta_{\phi}+b_{h}\cdot d_{\phi}$ is not self-adjoint (when $b_{h}\neq 0$), the analysis of its spectrum is more complicated than the one of the spectrum of $\Delta_{\phi}$. The following result states a counterpart of Proposition 1.2 in this setting. In this statement and in the sequel, for any $a\in\mathbb{C}$ and $r>0$, we will denote by $D(a,r)\subset\mathbb{C}$ the open disk of center $a$ and radius $r$.

Theorem 1.3**.**

Assume that Assumptions 1 to 3 hold true, and let $\epsilon_{0}>0$ be given by Proposition 1.2. Then, for every $\epsilon_{1}\in(0,\epsilon_{0})$, there exists $h_{0}>0$ such that for all $h\in(0,h_{0}]$, the set $\sigma(P_{\phi})\cap\{\operatorname{Re}z<\epsilon_{1}h\}$ is finite and consists in

[TABLE]

eigenvalues counted with algebraic multiplicity. Moreover, there exists $C>0$ such that for all $h\in(0,h_{0}]$,

[TABLE]

where $S$ is given by Proposition 1.2. Eventually, for every $\epsilon\in(0,\epsilon_{1})$, one has, uniformly with respect to $z$,

[TABLE]

Lastly, all the above conclusions also hold for $P^{*}_{\phi}$.

This theorem will be proved in Section 2.2 using Proposition 1.2 and a finite dimensional reduction. In order to give sharp asymptotics of the small eigenvalues of $P_{\phi}$, that is the ones in $D(0,Ch^{\frac{1}{2}}e^{-\frac{S}{h}})$, we will introduce some additional, but generic, topological assumptions on the Morse function $V$ (see Assumption 4 below). To this end, we first recall the general labelling of [12] (see in particular Definition 4.1 there) generalizing the labelling of [8] (and of [2, 3]). The main ingredient is the notion of separating saddle point, defined in Definition 1.5 below (see also an illustration in Figure 1.1) after the following observation. Here and in the sequel, we define, for $a\in\mathbb{R}$,

[TABLE]

and $\{V>a\}$, $\{V\geq a\}$ in a similar way. The following lemma recalls the local structure of the sublevel sets of a Morse function. A proof can be found in [8].

Lemma 1.4**.**

Let $z\in{\mathbb{R}}^{d}$ and $V:{\mathbb{R}}^{d}\to{\mathbb{R}}$ be a Morse function. For any $r>0$, we denote by $B(z,r)\subset{\mathbb{R}}^{d}$ the open ball of center $z$ and radius $r$. Then, for every $r>0$ small enough, $B(z,r)\cap\{V<V(z)\}$ has at least two connected components if and only if $z$ is a saddle point of $V$, i.e. if and only if $z\in\mathcal{U}^{(1)}$. In this case, $B(z,r)\cap\{V<V(z)\}$ has precisely two connected components.

Definition 1.5**.**

i) We say that the saddle point $\mathbf{s}\in{\mathcal{U}}^{(1)}$ is a separating saddle point of $V$ if, for every $r>0$ small enough, the two connected components of $B(\mathbf{s},r)\cap\{V<V(\mathbf{s})\}$ (see Lemma 1.4) are contained in different connected components of $\{V<V(\mathbf{s})\}$. We will denote by ${\mathcal{V}}^{(1)}$ the set made of these points.

ii) We say that $\sigma\in{\mathbb{R}}$ is a separating saddle value of $V$ if it has the form $\sigma=V(\mathbf{s})$ for some $\mathbf{s}\in{\mathcal{V}}^{(1)}$.

iii) Moreover, we say that $E\subset{\mathbb{R}}^{d}$ is a critical component of $V$ if there exists $\sigma\in V(\mathcal{V}^{(1)})$ such that $E$ is a connected component of $\{V<\sigma\}$ satisfying $\partial E\cap{\mathcal{V}}^{(1)}\neq\emptyset$.

Let us now describe the general labelling procedure of [12]. We will omit details when associating local minima and separating saddle points below, but the following proposition (cf. [5, Proposition 18]) can be helpful to well understand the construction.

Proposition 1.6**.**

Assume that $V$ is a Morse function with a finite number of critical points and such that $V(x)\to+\infty$ when $|x|\to+\infty$. Let $\lambda\in{\mathbb{R}}$ and let $\mathcal{C}$ be a connected component of $\{V<\lambda\}$. Then, it holds

[TABLE]

Let us also define

[TABLE]

with the convention $\sigma:=\min_{\mathcal{C}}V$ when $\mathcal{C}\cap{\mathcal{V}}^{(1)}=\emptyset$. It then holds:

1. i)

For every $\mu\in(\sigma,\lambda]$, the set $\mathcal{C}\cap\{V<\mu\}$ is a connected component of $\{V<\mu\}$. 2. ii)

If $\mathcal{C}\cap{\mathcal{V}}^{(1)}\neq\emptyset$, then $\mathcal{C}\cap\mathcal{U}^{(0)}\subset\{V<\sigma\}$ and all the connected components of $\mathcal{C}\cap\{V<\sigma\}$ are critical.

Under the hypotheses of Proposition 1.6, $V({\mathcal{V}}^{(1)})$ is finite. We moreover assume that $n_{0}\geq 2$, so that, under the hypotheses of Proposition 1.1 and of Theorem 1.3, [math] is not the only exponentially small eigenvalue of $P_{\phi}$ (or equivalently of $L_{V,b,\nu}$) and ${\mathcal{V}}^{(1)}\neq\emptyset$ by Proposition 1.6. We then denote the elements of $V({\mathcal{V}}^{(1)})$ by $\sigma_{2}>\sigma_{3}>\ldots>\sigma_{N}$, where $N\geq 2$. For convenience, we also introduce a fictive infinite saddle value $\sigma_{1}=+\infty$. Starting from $\sigma_{1}$, we will recursively associate to each $\sigma_{i}$ a finite family of local minima $(\mathbf{m}_{i,j})_{j}$ and a finite family of critical components $(E_{i,j})_{j}$ (see Definition 1.5).

Let $N_{1}:=1$, $\underline{\mathbf{m}}=\mathbf{m}_{1,1}$ be a global minimum of $V$ (arbitrarily chosen if there are more than one), and $E_{1,1}:={\mathbb{R}}^{d}$. We now proceed in the following way:

1. –

Let us denote, for some $N_{2}\geq 1$, by $E_{2,1},\dots,E_{2,N_{2}}$ the connected components of $\{V<\sigma_{2}\}$ which do not contain $\mathbf{m}_{1,1}$. They are all critical by the preceding proposition and we associate to each $E_{2,j}$, where $j\in\{1,\dots,N_{2}\}$, some global minimum $\mathbf{m}_{2,j}$ of $V|_{E_{2,j}}$ (arbitrarily chosen if there are more than one). 2. –

Let us then consider, for some $N_{3}\geq 1$, the connected components $E_{3,1},\dots,E_{3,N_{3}}$ of $\{V<\sigma_{3}\}$ which do not contain the local minima of $V$ previously labelled. These components are also critical and included in the $E_{2,j}\cap\{V<\sigma_{3}\}$’s, $j\in\{1,\dots,N_{2}\}$, such that $E_{2,j}\cap\{V=\sigma_{3}\}\cap{\mathcal{V}}^{(1)}\neq\emptyset$ (and $\sigma_{3}=\max_{E_{2,j}\cap{\mathcal{V}}^{(1)}}V$ for such a $j$). We then again associate to each $E_{3,j}$, $j\in\{1,\dots,N_{3}\}$, some global minimum $\mathbf{m}_{3,j}$ of $V|_{E_{3,j}}$. 3. –

We continue this process until having considered the connected components of $\{V<\sigma_{N}\}$, after which all the local minima of $V$ have been labelled.

Next, we define two mappings

[TABLE]

where, for any set $A$, $\mathcal{P}(A)$ denotes the power set of $A$, and $\mathbf{s}_{1}$ is a fictive saddle point such that $V(\mathbf{s}_{1})=\sigma_{1}=+\infty$, as follows: for every $i\in\{1,\dots,N\}$ and $j\in\{1,\dots,N_{i}\}$,

[TABLE]

and

[TABLE]

In particular, it holds $E(\underline{\mathbf{m}})={\mathbb{R}}^{d}$ and

[TABLE]

Lastly, we define the mappings

[TABLE]

by

[TABLE]

where, with a slight abuse of notation, we have identified the set $V({\bf j}(\mathbf{m}))$ with its unique element. Note that $S(\mathbf{m})=+\infty$ if and only if $\mathbf{m}=\underline{\mathbf{m}}$. An example of the preceding labelling is given in Figure 1.2 below.

Our generic topological assumption is the following one. Assume that $V$ is a Morse function with a finite number $n_{0}\geq 2$ of critical points such that $V(x)\to+\infty$ when $|x|\to+\infty$, and let ${E}:\mathcal{U}^{(0)}\to\mathcal{P}({\mathbb{R}}^{d})$ and ${\bf j}:\mathcal{U}^{(0)}\to\mathcal{P}({\mathcal{V}}^{(1)}\cup\{\mathbf{s}_{1}\})$ be the mappings defined in (1.17) and in (1.18).

Assumption 4**.**

For every $\mathbf{m}\in\mathcal{U}^{(0)}$, the following hold true:

• i)

the local minimum $\mathbf{m}$ is the unique global minimum of $V|_{E(\mathbf{m})}$,

• ii)

for all $\mathbf{m}^{\prime}\in{\mathcal{U}}^{(0)}\setminus\{\mathbf{m}\}$, ${\bf j}(\mathbf{m})\cap{\bf j}(\mathbf{m}^{\prime})=\emptyset$.

In particular, $V$ uniquely attains its global minimum, at $\underline{\mathbf{m}}\in\mathcal{U}^{(0)}$.

Note that the example of Figure 1.2 does not satisfy Assumption 4 since neither item i) nor ii) holds there. See also Figure 1.3 below for a similar example satisfying Assumption 4.

Let us moreover underline that this assumption is a little more general than the one considered in the generic case in [8, 12] (see also [2, 3]) where, for instance, each set ${\bf j}(\mathbf{m})$, $\mathbf{m}\in{\mathcal{U}}^{(0)}\setminus\{\mathbf{m}\}$, is assumed to only contain one element.

Remark 1.7**.**

One can also show that Assumption 4 implies that for every $\mathbf{m}\in{\mathcal{U}}^{(0)}$ such that $\mathbf{m}\neq\underline{\mathbf{m}}$, there is precisely one connected component $\widehat{E}(\mathbf{m})\neq E(\mathbf{m})$ of $\{f<\boldsymbol{\sigma}(\mathbf{m})\}$ such that $\overline{\widehat{E}(\mathbf{m})}\cap\overline{E(\mathbf{m})}\neq\emptyset$. In other words, there exists a connected component $\widehat{E}(\mathbf{m})\neq E(\mathbf{m})$ of $\{f<\boldsymbol{\sigma}(\mathbf{m})\}$ such that ${\bf j}(\mathbf{m})\subset\partial\widehat{E}(\mathbf{m})$. Moreover, the global minimum $\mathbf{m}^{\prime}$ of $V|_{\widehat{E}(\mathbf{m})}$ is unique and satisfies $\boldsymbol{\sigma}(\mathbf{m}^{\prime})>\boldsymbol{\sigma}(\mathbf{m})$ and $V(\mathbf{m}^{\prime})<V(\mathbf{m})$ (see examples of such sets in Figure 1.3). We refer to [21] or [5] for more details on the geometry of the sublevel sets of a Morse function.

1.4. Main results and comments

In order to state our main results, we also need the following lemma which is fundamental in our analysis.

Lemma 1.8**.**

For $x\in{\mathbb{R}}^{d}$, let $B(x):={\rm Jac}_{x}b$ denote the Jacobian matrix of $b=U_{0}-\nabla V$ at $x$, and consider a saddle point $\mathbf{s}\in\mathcal{U}^{(1)}$.

1. i)

The matrix ${\rm Hess\,}V(\mathbf{s})+B^{*}(\mathbf{s})\in\mathcal{M}_{d}({\mathbb{R}})$ admits precisely one negative eigenvalue $\mu=\mu(\mathbf{s})$, which has moreover geometric multiplicity one. 2. ii)

Denote by $\xi=\xi(\mathbf{s})$ one of the two (real) unitary eigenvectors of ${\rm Hess\,}V(\mathbf{s})+B^{*}(\mathbf{s})$ associated with $\mu$. The real symmetric matrix

[TABLE]

is then positive definite and its determinant satisfies:

[TABLE] 3. iii)

Lastly, denoting by $\lambda_{1}=\lambda_{1}(\mathbf{s})$ the negative eigenvalue of ${\rm Hess\,}V(\mathbf{s})$, it holds $|\mu|\geq|\lambda_{1}|$, with equality if and only if $B^{*}(\mathbf{s})\xi=0$, and

[TABLE]

Note that the real matrix ${\rm Hess\,}V(\mathbf{s})+B^{*}(\mathbf{s})$ of Lemma 1.8 is in general non symmetric. Let us also point out that the statements of Lemma 1.8 already appeared in the related work [14] (see in particular the beginning of Section 8 there), and in [15], where proofs are given (see indeed Section 4.1 there). We will nevertheless give a proof in Section 3 for the sake of completeness.

We can now state our main results.

Theorem 1.9**.**

Suppose that Assumptions 1 to 4 hold true, and let $\epsilon_{0}>0$ be given by Proposition 1.2. Then, for all $\epsilon_{1}\in(0,\epsilon_{0})$, there exists $h_{0}>0$ such that for all $h\in(0,h_{0}]$, one has, counting the eigenvalues with algebraic multiplicity,

[TABLE]

where, denoting by $\underline{\mathbf{m}}$ the unique absolute minimum of $V$, $\lambda(\underline{\mathbf{m}},h)=0$ and, for all $\mathbf{m}\neq\underline{\mathbf{m}}$, $\lambda(\mathbf{m},h)$ satisfies the following Eyring-Kramers type formula:

[TABLE]

where $S:{\mathcal{U}}^{(0)}\rightarrow(0,+\infty]$ is defined in (1.19) and, for every $\mathbf{m}\in{\mathcal{U}}^{(0)}\setminus\{\underline{\mathbf{m}}\}$,

[TABLE]

*where ${\bf j}:\mathcal{U}^{(0)}\to\mathcal{P}({\mathcal{V}}^{(1)}\cup\{\mathbf{s}_{1}\})$ is defined in (1.18) and the $\mu(\mathbf{s})$’s are defined in Lemma 1.8.

In addition, it holds*

[TABLE]

Remark 1.10**.**

In the case where $V$ has precisely two minima $\underline{\mathbf{m}}$ and $\mathbf{m}$ such that $V(\underline{\mathbf{m}})=V(\mathbf{m})$, the above result can be easily generalized. In this case, using the definitions of $S$ and ${\bf j}$ given in (1.19) and in (1.18) (note that the choice of $\underline{\mathbf{m}}$ among the two minima of $V$ is arbitrary in this case), we have, counting the eigenvalues with algebraic multiplicity, for every $h>0$ small enough,

[TABLE]

where

[TABLE]

with

[TABLE]

Moreover, since $\sigma(L_{V,b,\nu})=\overline{\sigma(L_{V,b,\nu})}$, the eigenvalue $\lambda(\mathbf{m},h)$ is real.

Let us make a few comments on the above theorem.

First, observe that if we assume that $U_{h}=\nabla V$, that is if $b_{h}=0$ (see (1.5)), we obtain the precise asymptotics of the small eigenvalues of $L_{V,0,0}$ (or equivalently of $\Delta_{\phi}$ after multiplication by $\frac{1}{h}$, see (1.14)) and hence recover the results already proved in this reversible setting in [3, 8] (see also [19] for an extension to logarithmic Sobolev inequalities). In this case, for every saddle point $\mathbf{s}$ appearing in (1.21), the real number $\mu(\mathbf{s})$ is indeed the negative eigenvalue of ${\rm Hess\,}V(\mathbf{s})$ according to the first item of Lemma 1.8. Let us also point out that under the hypotheses made in [3, 8], the set ${\bf j}(\mathbf{m})$ actually contains one unique element for every $\mathbf{m}\in{\mathcal{U}}^{(0)}\setminus\{\underline{\mathbf{m}}\}$. Moreover, our analysis permits in this case to recover that the error term ${\mathcal{O}}(\sqrt{h})$ is actually of order ${\mathcal{O}}(h)$, as proven in [8]. However, it does not permit to prove that this ${\mathcal{O}}(h)$ actually admits a full asymptotic expansion in $h$ as proven in [8].

To the best of our knowledge, the above theorem is the first result giving sharp asymptotics of the small eigenvalues of the generator $L_{V,b,\nu}$ in the non-reversible case. Similar results were obtained by Hérau-Hitrik-Sjöstrand for the Kramers-Fokker-Planck (KFP) equation in [12]. Compared to our framework, they deal with non-self-adjoint and non-elliptic operators, which makes the analysis more complicated. However, the KFP equation enjoys several symmetries which are crucial in their analysis. First of all, the KFP operator has a supersymmetric structure (for a non-symmetric skew-product $\langle.,.\rangle_{\text{KFP}}$) which permits to write the interaction matrix associated with the small eigenvalues as a square $M=A^{*}A$, where the adjoint $A^{*}$ is taken with respect to $\langle.,.\rangle_{\text{KFP}}$. Using this square structure, the authors can then follow the strategy of [8] to construct accurate approximations of the matrices $A$ and $A^{*}$. However, since $\langle.,.\rangle_{\text{KFP}}$ is not a scalar product, they cannot identify the squares of the singular values of $A$ with the eigenvalues of $M$. This difficulty is solved by using an extra symmetry (the PT-symmetry), which permits to modify the skew-product $\langle.,.\rangle_{KFP}$ into a new product $\langle.,.\rangle_{KFPS}$, which is a scalar product when restricted to the “small spectral subspace", and for which the identity $M=A^{*}A$ remains true with an adjoint taken with respect to $\langle.,.\rangle_{KFPS}$. This permits to conclude as in [8], using in particular the Fan inequalities to estimate the singular values of $A$.

In the present case, none of these two symmetries are available in general ($L_{V,b,\nu}$, or equivalently $P_{\phi}$, enjoys however a supersymmetric structure when $b$ and $\nu$ satisfy the relation (1.7), see indeed (1.8) or Remark 3.2 below in this connection). We then developed an alternative approach based on the construction of very accurate quasimodes and partly inspired by [4] (see also the related constructions made in [2, 14, 17]). This permits the construction of the interaction matrix $M$ as above. However, since we cannot write $M=A^{*}A$ and use the Fan inequalities as in [8, 12] (and e.g. in [10, 16, 21, 5, 17]), we have to compute directly the eigenvalues of $M$. To this end, we use crucially the Schur complement method. This leads to Theorem A.4 in appendix, which permits to replace the use of the Fan inequalities to perform the final analysis in our setting. We believe that these two arguments are quite general and may be used in other contexts.

Though it is generic, one may ask if Assumption 4 is necessary to get Eyring-Kramers type formulas as in Theorem 1.9. In the reversible setting, the full general (Morse) case was recently treated by the second author in [21], but applying the methods developed there to our non-reversible setting was not straightforward and we decided to postpone this analysis to future works. Let us point out in this connection that in the general (Morse) case, some tunneling effect between the characteristic wells of $V$ defined by the mapping $E$ (see (1.17)) mixes their corresponding prefactors, see indeed Remark 1.10, or [21] for more intricate situations in the reversible setting.

Note that Theorem 1.9 does not state that the operator $L_{V,b,\nu}$ is diagonalizable when restricted to the spectral subspace associated with its small eigenvalues. Indeed, since $L_{V,b,\nu}$ is not self-adjoint, we cannot exclude the existence of Jordan’s blocks. We cannot neither exclude the existence of non-real eigenvalues, but the spectrum of $L_{V,b,\nu}$ is obviously stable by complex conjugation since $L_{V,b,\nu}$ is a partial differential operator with real coefficients. However, in the case where for every $\mathbf{m}\in{\mathcal{U}}^{(0)}\setminus\{\underline{\mathbf{m}}\}$, the prefactors $\zeta(\mathbf{m}^{\prime})$ defined in (1.21) are all distinct for $\mathbf{m}^{\prime}\in S^{-1}(S(\mathbf{m}))$, the $\lambda(\mathbf{m},h)$’s, $\mathbf{m}\in{\mathcal{U}}^{(0)}$, are then real eigenvalues of multiplicity one of $L_{V,b,\nu}$, and its restriction to its small spectral subspace is diagonalizable.

Coming back to the contraction semigroups $(e^{-tL_{V,b,\nu}})_{t\geq 0}$ and $(e^{-tL^{*}_{V,b,\nu}})_{t\geq 0}$ on $L^{2}({\mathbb{R}}^{d},m_{h})$ introduced just after Proposition 1.1, Theorem 1.9 has the following consequences on the rate of convergence to equilibrium for the process (1.1).

Theorem 1.11**.**

Assume that the hypotheses of Theorem 1.9 hold and let $\mathbf{m}^{*}\in{\mathcal{U}}^{(0)}\setminus\{\underline{\mathbf{m}}\}$ be such that

[TABLE]

where the prefactors $\zeta(\mathbf{m})$’s, $\mathbf{m}\in{\mathcal{U}}^{(0)}\setminus\{\underline{\mathbf{m}}\}$, are defined in (1.21), and $S:{\mathcal{U}}^{(0)}\rightarrow(0,+\infty]$ is defined in (1.19). Let us then define, for any $h>0$,

[TABLE]

Then, there exist $h_{0}>0$ and $C>0$ such that for every $h\in(0,h_{0}]$, it holds

[TABLE]

where $\Pi_{0}$ denotes the orthogonal projector on ${\rm\,Ker\,}L_{V,b,\nu}={\rm Span}\{1\}$:

[TABLE]

Assume moreover that $(X_{t})_{t\geq 0}$ is solution to (1.1) and that the probability distribution $\varrho_{0}$ of $X_{0}$ admits a density $\mu_{0}\in L^{2}({\mathbb{R}}^{d},m_{h})$ with respect to the probability measure $m_{h}$. Then, for every $t\geq 0$, the probability distribution $\varrho_{t}$ of $X_{t}$ admits the density $\mu_{t}=e^{-tL^{*}_{V,b,\nu}}\mu_{0}\in L^{2}({\mathbb{R}}^{d},m_{h})$ with respect to $m_{h}$, and for every $h\in(0,h_{0}]$, it holds

[TABLE]

*where $\|\cdot\|_{TV}$ denotes the total variation distance.

Finally, when there exists one unique $\mathbf{m}^{*}$ satisfying (1.22), the eigenvalue $\lambda(\mathbf{m}^{*},h)$ associated with $\mathbf{m}^{*}$ (see (1.20)) is real and simple, and the estimates (1.23) and (1.24) remain valid if one replaces $\lambda(h)(1-C\sqrt{h})$ by $\lambda(\mathbf{m}^{*},h)$ in the exponential terms.*

Theorems 1.9 and 1.11 describe the metastable behaviour of the dynamics (1.1) from a spectral perspective.

Concerning the question of accelerating the convergence to equilibrium mentioned at the end of Section 1.1, the exponential rate of convergence to equilibrium appearing in the estimates (1.23) and (1.24) is generically strictly larger than the optimal rate for the associated gradient dynamics (1.9). To be more precise, let us assume, as in the last part of the statement of Theorem 1.11, that there exists one unique $\mathbf{m}^{*}$ satisfying (1.22). The exponential rate of return to equilibrium appearing in (1.23) and (1.24) is then given by the spectral gap $\lambda(\mathbf{m}^{*},h)$ of $L_{V,b,\nu}$. Moreover, denoting by $\lambda^{\!\nabla}(\mathbf{m}^{*},h)$ the spectral gap of the generator $L_{V,0,0}$ of the associated gradient dynamics (1.9), that is the optimal rate of return to equilibrium in the gradient setting, it follows from Theorem 1.9 and item iii) in Lemma 1.8 that, as soon as $B^{*}(\mathbf{s}^{*})\neq 0$ for at least one $\mathbf{s}^{*}\in{\bf j}(\mathbf{m}^{*})$, the ratio of the rates $\frac{\lambda(\mathbf{m}^{*},h)}{\lambda^{\!\nabla}(\mathbf{m}^{*},h)}$ converges to some constant $c>1$ when $h\to 0$.

In addition, it is not difficult to see that playing with $b_{h}$, one can make $\lim\limits_{h\to 0}\frac{\lambda(\mathbf{m}^{*},h)}{\lambda^{\!\nabla}(\mathbf{m}^{*},h)}$ arbitrarily big. Taking for example $b_{h}=b=aJ\,\nabla V$ around $\mathbf{s}^{*}$ for $a\in{\mathbb{R}}$ and some constant antisymmetric and invertible matrix $J$, it holds

[TABLE]

Nevertheless, making this limit too big will deteriorate the constant $C$ appearing in (1.23) and (1.24), as well as the interval $(0,h_{0}]\ni h$ for which these estimates remain relevant. A more interesting problem is the computation of the optimal rate when $h_{0}$ is small but fixed, that is when the preceding $J$ has a constant size (see [18] in the case of linear drifts). We did not make the whole computation, but a partial one seems to indicate that the optimal (or at least a reasonable) choice for $J$ is given when it sends the unstable direction of ${\rm Hess\,}V(\mathbf{s}^{*})$ onto one of its stable directions corresponding to a maximal eigenvalue.

A closely related point of view to ours is to study the mean transition times between the different wells of the potential $V$ for the process $(X_{t})_{t\geq 0}$ solution to (1.1). In the non-reversible case, this question has been studied recently e.g. in [1, 14], to which we also refer for more details and references on this subject.

In [1], an Eyring-Kramers type formula (for the mean transition times) is derived from formal computations relying on the study of the appropriate quasi-potential. In the case of a double-well potential $V$ and under the assumption that $U_{h}=\nabla V+b$ (that is that $\nu=0$, see (1.5)) for some vector field $b$ only satisfying $b\cdot\nabla V=0$ (that is without assuming $\operatorname{div}b=0$ as we do when $\nu=0$, see (1.3)), the authors derived formula (5.65), where, in comparison with a formula such as (the inverse of) (1.20) in Theorem 1.9, appears in the prefactor some extra term measuring the non-Gibbsianness of their situation. In this general setting, the measure $m_{h}$ is indeed invariant for the dynamics if and only if $\operatorname{div}b=0$, and this extra term involves the integral of the function $F:=\operatorname{div}(b)$ along the so-called instanton trajectory. Under the additional assumption that $m_{h}$ is invariant (that is that $F=0$), this extra term equals $1$, which leads to the formula (5.66) in [1], which is similar to (the inverse of) (1.20) in Theorem 1.9 (see more precisely Corollary 1.12 below, which clarifies the relation between eigenvalues of $L_{V,b,\nu}$ and mean transition times). In the present paper, we restrict ourselves to the Gibbsian case, so that our formulas do not contain any extra prefactor as discussed above. It would be of great interest to study the general case of a drift of the form $\nabla V+b$, where $b\cdot\nabla V=0$ but without assuming $\operatorname{div}b=0$, by mixing our approach and quasi-potential constructions.

In [14], the authors use a potential theoretic approach to prove an Eyring-Kramers type formula similar to the formula (5.66) of [1] in the case of a double-well potential $V$, when $b$ and $\nu$ satisfy the relation (1.7) in such a way that $L_{V,b,\nu}$ has the form (1.8). Though the mathematical objects considered in [14] and in the present paper are not the same, these two works share some similarities. Nevertheless, we would like to emphasize that our approach permits to go beyond the supersymmetric assumption (1.7) and to treat the case of multiple-well potentials.

To be more precise on the connections between the present paper and [14] (and also [1]), let us conclude this introduction with the corollary below which combines the results given by Theorem 1.9 when $V$ is a double-well potential and [14, Theorem 5.2 and Remarks 5.3 and 5.6]. This result generalizes in particular, in this non-reversible double-well setting, the results obtained in the reversible case in [2, 3] on the relations between the small eigenvalues of $L_{V,b,\nu}$ and the mean transition times of (1.1) when $b=\nu=0$.

Corollary 1.12**.**

Assume that the hypotheses of Theorem 1.9 hold with moreover

[TABLE]

*and that $V$ admits precisely two local minima $\underline{\mathbf{m}}$ and $\mathbf{m}$ such that $V(\underline{\mathbf{m}})<V(\mathbf{m})$ (it then holds ${\mathcal{V}}^{(1)}={\bf j}(\mathbf{m})$). Assume in addition that $b$ and $\nu$ satisfy the relation (1.7), and hence that $b=J\,\nabla V$ for some smooth map $J$ from ${\mathbb{R}}^{d}$ into the set of real antisymmetric matrices of size $d$, and that $J$ is uniformly bounded on ${\mathbb{R}}^{d}$.

Let $\mathcal{O}(\underline{\mathbf{m}})$ be a smooth open connected set containing $\underline{\mathbf{m}}$ such that $\overline{\mathcal{O}(\underline{\mathbf{m}})}\subset\{V<\boldsymbol{\sigma}(\mathbf{m})\}$. Let then $(X_{t})_{t\geq 0}$ be the solution to (1.1) such that $X_{0}=\mathbf{m}$ and let*

[TABLE]

be the first hitting time of $\mathcal{O}(\underline{\mathbf{m}})$. The expectation of $\tau_{\mathcal{O}(\underline{\mathbf{m}})}$ and the non-zero small eigenvalue $\lambda(\mathbf{m},h)$ of $L_{V,b,\nu}$ are then related by the following formula in the limit $h\to 0$:

[TABLE]

Let us mention here that the hypotheses of Corollary 1.12 are simply the minimal hypotheses permitting to apply at the same time Theorem 1.9 and [14, Theorem 5.2] in its refinement specified in [14, Remark 5.6].

Acknowledgements. The authors thank the anonymous referees for their remarks who permitted to improve the quality of the paper. Both authors are members of the ANR project QuAMProcs 19-CE40-0010-01.

2. General spectral estimates

2.1. Proof of Proposition 1.1

The unbounded operator $(P_{\phi},{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d}))$ is accretive, since, according to (1.11), one has:

[TABLE]

In order to prove that its closure is maximal accretive, it then suffices to show that ${\rm Ran}(P_{\phi}+1)$ is dense in $L^{2}({\mathbb{R}}^{d})$ (see for example [7, Theorem 13.14]). The proof of this fact is rather standard but we give it for the sake of completeness (see in particular the proof of [9, Proposition 5.5] for a similar proof). Suppose that $f\in L^{2}({\mathbb{R}}^{d})$ is orthogonal to ${\rm Ran}(P_{\phi}+1)$. It then holds $(P_{\phi}^{*}+1)f=0$ in the distribution sense and, since $P_{\phi}$ is real, one can assume that $f$ is real. In particular, since $P_{\phi}^{*}=\Delta_{\phi}-b_{h}\cdot d_{\phi}$ is elliptic with smooth coefficients, $f$ belongs to ${\mathcal{C}}^{\infty}({\mathbb{R}}^{d})$. Thus, for every $\zeta\in{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d},{\mathbb{R}})$, one has

[TABLE]

Take now $\zeta$ such that $0\leq\zeta\leq 1$, $\zeta=1$ on $B(0,1)$ and $\mathop{\rm supp}\nolimits\zeta\subset B(0,2)$, and define $\zeta_{k}:=\zeta(\frac{\cdot}{k})$ for $k\in{\mathbb{N}}^{*}$. According to (1.13) and to the above relation, there exists $C>0$ such that for every $k\in{\mathbb{N}}^{*}$, it holds

[TABLE]

where $\varepsilon>0$ is arbitrary. Choosing $\varepsilon=\frac{1}{2C}$ and using (1.12), it follows that for every $h>0$ small enough, it holds

[TABLE]

which implies, taking the limit $k\to+\infty$, that $f=0$. Hence, the closure of $P_{\phi}$, that we still denote by $P_{\phi}$, is maximal accretive. Note moreover, that (2.1) implies that $D(P_{\phi})\subset\{u\in L^{2}({\mathbb{R}}^{d}),\;d_{\phi}u\in L^{2}({\mathbb{R}}^{d})\}$ and that $\operatorname{Re}\langle P_{\phi}u,u\rangle=\|d_{\phi}u\|^{2}$ for every $u\in D(P_{\phi})$.

Let us now prove that $D(\Delta_{\phi})\subset D(P_{\phi})$, which amounts to show that for every $u\in D(\Delta_{\phi})$, there exists a sequence $(u_{n})_{n\in{\mathbb{N}}}$ of ${\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d})$ such that $u_{n}\to u$ in $L^{2}({\mathbb{R}}^{d})$ and $(P_{\phi}u_{n})_{n\in{\mathbb{N}}}$ is a Cauchy sequence. Since $(\Delta_{\phi},{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d}))$ is essentially self-adjoint, for any such $u$, there exists a sequence $(u_{n})_{n\in{\mathbb{N}}}$ in ${\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d})$ such that $u_{n}\to u$ in $L^{2}({\mathbb{R}}^{d})$ and $(\Delta_{\phi}u_{n})_{n\in{\mathbb{N}}}$ is a Cauchy sequence, and it thus suffices to show that $(b_{h}\cdot d_{\phi}u_{n})_{n\in{\mathbb{N}}}$ is also a Cauchy sequence. For this purpose, we introduce the exterior derivative $d$ acting from [math]-forms into $1$-forms and the twisted semiclassical derivative $d_{\phi}=e^{-\phi/h}\circ hd\circ e^{\phi/h}$. Note that the notation $d_{\phi}$ has actually already been defined in (1.15) with a different meaning; we are thus making here a slight abuse of notation, by identifying the exterior derivative $d$ acting on functions with $\nabla$. Thanks to (1.12) and to (1.13), there exists $C>0$ such that for every $h>0$ small enough and every $u\in{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d})$, one has

[TABLE]

where $\Delta_{\phi}^{(1)}$ denotes the Witten Laplacian acting on $1$-forms, that is

[TABLE]

Combined with the intertwining relation $\Delta_{\phi}^{(1)}d_{\phi}=d_{\phi}\Delta_{\phi}^{(0)}$, we get

[TABLE]

for every $u\in{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d})$. This implies that for any Cauchy sequence $(u_{n})_{n\in{\mathbb{N}}}$ in $L^{2}({\mathbb{R}}^{d})$ such that $(\Delta_{\phi}u_{n})_{n\in{\mathbb{N}}}$ is a Cauchy sequence, $(b_{h}\cdot d_{\phi}u_{n})_{n\in{\mathbb{N}}}$ is also a Cauchy sequence, and thus that $D(\Delta_{\phi})\subset D(P_{\phi})$.

The statement about $P_{\phi}^{*}$ is then a straightforward consequence of the above analysis. Indeed, since $P_{\phi}^{*}=\Delta_{\phi}-b_{h}\cdot d_{\phi}$ on ${\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d})$, the above arguments imply that the closure of $(P_{\phi}^{*},{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d}))$ is maximal accretive and that its domain contains $D(\Delta_{\phi})$. Moreover, $P_{\phi}^{*}$ is maximal accretive since $P_{\phi}$ is, and hence coincides with the closure of $(P_{\phi}^{*},{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d}))$.

Let us now prove the statement on the spectrum of $P_{\phi}$. Throughout, we will denote ${\mathbb{C}}_{+}=\{\operatorname{Re}(z)\geq 0\}$. It follows from (1.12) and from (1.13) that for every $u\in{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d})$, it holds, for some $C>0$ and every $h>0$ small enough,

[TABLE]

Let us set $\Lambda_{0}=5C$ for some $C\geq 1$ satisfying (2.3), and let $z\in{\mathbb{C}}_{+}$ be such that $|\operatorname{Im}(z)|\geq\Lambda_{0}\max(\operatorname{Re}(z),\sqrt{\operatorname{Re}(z)})$. Suppose first that $\operatorname{Re}(z)\|u\|^{2}\geq\frac{1}{2}\|d_{\phi}u\|^{2}$. Then, thanks to the estimate (2.3), we have

[TABLE]

Since $|\langle(b_{h}\cdot d_{\phi}-i\operatorname{Im}(z))u,u\rangle|\leq|\langle(P_{\phi}-z)u,u\rangle|$, this implies that

[TABLE]

Suppose now that $\operatorname{Re}(z)\|u\|^{2}\leq\frac{1}{2}\|d_{\phi}u\|^{2}$. One then directly obtains

[TABLE]

which, combined with (2.4), implies that

[TABLE]

for every $z\in{\mathbb{C}}_{+}\setminus\Gamma_{\Lambda_{0}}$ and $u\in{\mathcal{C}}^{\infty}_{c}({\mathbb{R}}^{d})$. Since $P_{\phi}$ is closed, it follows that $P_{\phi}-z$ is injective with closed range, and hence semi-Fredholm, for every $z\in{\mathbb{C}}_{+}\setminus\Gamma_{\Lambda_{0}}$ such that $\operatorname{Re}(z)\neq 0$. Assume now for a while that the fourth item in Proposition 1.1, which is proved independently just below, is satisfied, and let $\lambda\in{\mathbb{R}}$ be such that $i\lambda\in\sigma(P_{\phi})$. By assumption, $i\lambda$ is then an eigenvalue of $P_{\phi}$ and there exists some $u\in D(P_{\phi})\setminus\{0\}$ such that $P_{\phi}u=i\lambda u$. In particular, it holds

[TABLE]

which implies $u\in{\rm Span}\{e^{-\frac{\phi}{h}}\}$ and then $\lambda=0$. This shows that $\sigma(P_{\phi})\cap i{\mathbb{R}}\subset\{0\}$ and thus, $P_{\phi}$ being maximal accretive, that $\sigma(P_{\phi})\cap\{\operatorname{Re}(z)\leq 0\}\subset\{0\}$. It follows that $P_{\phi}-z$ is semi-Fredholm for every $z\in{\mathbb{C}}\setminus\Gamma_{\Lambda_{0}}$, and has index [math] on $\{\operatorname{Re}z\leq 0\}\setminus\{0\}$. But the open set ${\mathbb{C}}\setminus\Gamma_{\Lambda_{0}}$ being connected, the index of $P_{\phi}-z$ is constant, and then equal to [math], on ${\mathbb{C}}\setminus\Gamma_{\Lambda_{0}}$ (see [13, Theorem 5.17 in Chap. 4]). Hence, $P_{\phi}-z$ being injective on ${\mathbb{C}}\setminus\Gamma_{\Lambda_{0}}$, it is invertible from $D(P_{\phi})$ onto $L^{2}({\mathbb{R}}^{d})$ on ${\mathbb{C}}\setminus\Gamma_{\Lambda_{0}}$ and the resolvent estimate stated in Proposition 1.1 becomes a direct consequence of (2.5).

Let us now prove the fourth item of Proposition 1.1. Thanks to (1.12), there exist $c>0$ and $R>0$ such that

[TABLE]

Take $c_{1}\in(0,c)$ and let $W$ be a nonnegative smooth function such that $\mathop{\rm supp}\nolimits(W)\subset B(0,R)$ and $W(x)+|\nabla\phi(x)|^{2}\geq\frac{c+c_{1}}{2}$ for all $x\in{\mathbb{R}}^{d}$. There exists consequently $h_{0}>0$ such that for all $h\in(0,h_{0}]$, one has

[TABLE]

on ${\mathbb{R}}^{d}$. Introduce the operator

[TABLE]

with domain $D(P_{\phi})$. Since $P_{\phi}$ is maximal accretive and $W\in{\mathcal{C}}_{c}^{\infty}({\mathbb{R}}^{d},{\mathbb{R}}^{+})$, $\tilde{P}_{\phi}$ is also maximal accretive (see for example [7, Theorem 13.25]). Moreover, for every $u\in{\mathcal{C}}_{c}^{\infty}({\mathbb{R}}^{d})$ and then for every $u\in D(P_{\phi})$, one has

[TABLE]

which implies as above that for every $z$ in $\{\operatorname{Re}(z)<c_{1}\}$, $\tilde{P}_{\phi}-z$ is invertible from $D(P_{\phi})$ onto $L^{2}({\mathbb{R}}^{d})$. Hence, for every $z$ in $\{\operatorname{Re}(z)<c_{1}\}$, we can write

[TABLE]

Of course, $z\mapsto(\tilde{P}_{\phi}-z)^{-1}$ is holomorphic on $\{\operatorname{Re}z<c_{1}\}$ and thanks to the analytic Fredholm theorem, it then suffices to prove that

[TABLE]

is compact for every $z$ in $\{\operatorname{Re}(z)<c_{1}\}$. This follows from the compactness of the embedding $H^{1}_{R}\subset L^{2}({\mathbb{R}}^{d})$ and from the fact that for every $z\in\{\operatorname{Re}z<c_{1}\}$, $K(z)$ acts continuously from $L^{2}({\mathbb{R}}^{d})$ into $H_{R}^{1}$, where

[TABLE]

Indeed, for any $z$ in $\{\operatorname{Re}(z)<c_{1}\}$, the operator $d_{\phi}(\tilde{P}_{\phi}-z)^{-1}:L^{2}({\mathbb{R}}^{d})\to L^{2}({\mathbb{R}}^{d})$ is continuous thanks to (2.1) and hence, since $W$ is smooth and supported in $B(0,R)$, $K(z):L^{2}({\mathbb{R}}^{d})\to H^{1}_{R}$ is also continuous.

To conclude, it remains to prove the last statement of Proposition 1.1. To this end, note first that $P_{\phi}e^{-\frac{\phi}{h}}=0$ according to (1.16) and let us recall that, according to (1.12), $e^{-\frac{\phi}{h}}\in D(\Delta_{\phi})\subset D(P_{\phi})$. Thus, ${\rm Span}\{e^{-\frac{\phi}{h}}\}\subset{\rm\,Ker\,}P_{\phi}$ and [math] is an eigenvalue of $P_{\phi}$. It has moreover finite algebraic multiplicity according to the preceding analysis. Conversely, the relation

[TABLE]

leads to ${\rm\,Ker\,}P_{\phi}\subset{\rm Span}\{e^{-\frac{\phi}{h}}\}$ and the same arguments also show that ${\rm\,Ker\,}P^{*}_{\phi}={\rm Span}\{e^{-\frac{\phi}{h}}\}$. This implies that [math] is an eigenvalue of $P_{\phi}$ with algebraic multiplicity one. Indeed, if it was not the case, there would exist $u\in D(P_{\phi})$ such that $u\notin{\rm\,Ker\,}P_{\phi}$ and $P_{\phi}u=e^{-\frac{\phi}{h}}$, and hence such that

[TABLE]

2.2. Spectral analysis near the origin

Let us denote by $(e_{k}^{W})_{k\geq 1}$ the eigenfunctions of $\Delta_{\phi}$ associated with the non-decreasing sequence of eigenvalues $(\lambda_{k}^{W})_{k\geq 1}$. Let $\epsilon_{0}$ and $h_{0}>0$ be given by Proposition 1.2. We recall that for every $h\in(0,h_{0}]$, it holds

[TABLE]

where $n_{0}$ is the number of local minima of $\phi$. We define

[TABLE]

and $R_{+}:=R_{-}^{*}$, i.e.

[TABLE]

Note in particular the relations

[TABLE]

where $\Pi$ denotes the orthogonal projection onto {\rm Ran}(R_{-})={\rm Span}\big{(}e_{k}^{W},k\in\{1,\dots,n_{0}\}\big{)}. We also define the spectral projector

[TABLE]

For $z\in{\mathbb{C}}$, let us then consider on the Hilbert space $\hat{E}:={\rm Ran}(\hat{\Pi})$ the following unbounded operator which will be useful in the rest of this section:

[TABLE]

Hence $D(\hat{P}_{\phi,z})$ is dense in $\hat{E}$ and, since ${\rm Ran}\,\Pi\subset D(\Delta_{\phi})\subset D(P_{\phi})$, it holds $\hat{\Pi}(D(P_{\phi}))\subset D(P_{\phi})$ and $\hat{P}_{\phi,z}$ is well and densely defined.

Lemma 2.1**.**

Let $\epsilon_{0}$ and $h_{0}>0$ be given by Proposition 1.2. Then, for every $h\in(0,h_{0}]$, the operator $\hat{P}_{\phi,z}:D(\hat{P}_{\phi,z})\to\hat{E}$ defined in (2.7) is invertible on $\{\operatorname{Re}z<\epsilon_{0}h\}$. Moreover, for any $\epsilon_{1}\in(0,\epsilon_{0})$ it holds:

[TABLE]

uniformly with respect to $z$.

*Proof. *We begin by the following observation: the unbounded operator

[TABLE]

is well and densely defined on $\hat{E}$, and satisfies moreover

[TABLE]

Indeed, the relation $\langle\hat{\Pi}(P_{\phi}-z)\hat{\Pi}v,w\rangle=\langle v,\hat{\Pi}(P^{*}_{\phi}-z)\hat{\Pi}w\rangle$, valid for every $v\in D(P_{\phi})$ and $w\in D(P^{*}_{\phi})$, implies that $\hat{\Pi}(P^{*}_{\phi}-z)\hat{\Pi}\subset\hat{P}^{*}_{\phi,z}$. Moreover, for every $v\in D(P_{\phi})$ and $w\in D(\hat{P}_{\phi,z}^{*})$, one has

[TABLE]

Since $P_{\phi}\Pi$ is continuous, $\Pi$ being continuous with finite rank, one has $|\langle P_{\phi}\Pi v,w\rangle|\leq C\|v\|\|w\|$ for some $C>0$ independent of $(v,w)$, which implies that $w\in D(P_{\phi}^{*})$. Hence $D(\hat{P}_{\phi,z}^{*})\subset D(P_{\phi}^{*})$ and since ${\rm Ran}(\Pi)\subset D(\Delta_{\phi})\subset D(P_{\phi}^{*})$, this implies $\hat{\Pi}(P_{\phi}^{*}-z)\hat{\Pi}=\hat{P}_{\phi,z}^{*}$.

Let now consider $z$ in $\{\operatorname{Re}z<\epsilon_{0}h\}$ and let us prove that $\hat{P}_{\phi,z}$ is invertible from $D(\hat{P}_{\phi,z})$ onto $\hat{E}$. First, according to Proposition 1.2, we have for every $u\in D(\Delta_{\phi})$,

[TABLE]

and the inequality (2.8) is also true when $u\in D(P_{\phi})$. Indeed, for any $u\in D(P_{\phi})$, there exists a sequence $(u_{n})_{n\in{\mathbb{N}}}$ in $D(\Delta_{\phi})$ such that $u_{n}\to u$ and $P_{\phi}u_{n}\to P_{\phi}u$ in $L^{2}({\mathbb{R}}^{d})$. Hence $\hat{\Pi}u_{n}\to\hat{\Pi}u$ and, since $P_{\phi}\Pi$ is continuous, it also holds $P_{\phi}\hat{\Pi}u_{n}\to P_{\phi}\hat{\Pi}u$. In particular, it follows that $\hat{P}_{\phi,z}$ is injective. Note that a similar analysis shows that $\hat{P}_{\phi,z}^{*}$ is also injective.

Second, let us show that $\hat{P}_{\phi,z}$ is closed, which will in particular imply that ${\rm Ran}(\hat{P}_{\phi,z})$ is closed according to (2.8). For shortness, we denote $\hat{P}=\hat{P}_{\phi,z}$ and $P=P_{\phi}$. Suppose that $(u_{n})_{n\in{\mathbb{N}}}$ is a sequence in $D(\hat{P})\subset D(P)$ such that $u_{n}\rightarrow u$ and $\hat{P}u_{n}\rightarrow v$ in $\hat{E}$. Since ${\rm Ran}\,\Pi\subset D(\Delta_{\phi})\subset D(P^{*})$, it holds

[TABLE]

and thus $(P--z)u_{n}=\hat{P}u_{n}+\Pi(P-z)u_{n}$ converges. Since $P$ is closed, this implies that $u\in D(P)\cap{\rm Ran}\hat{\Pi}=\hat{\Pi}(D(P))$ and that

[TABLE]

Multiplying this relation by $\hat{\Pi}$, we get $v=\hat{P}u$, which proves that $\hat{P}$ is closed.

To prove that $\hat{P}$ is invertible from $D(\hat{P})$ onto $\hat{E}$, it is thus enough to prove that ${\rm Ran}(\hat{P})$ is dense in $\hat{E}$. Let then $v\in\hat{E}$ be such that $\langle\hat{P}u,v\rangle=0$ for all $u\in D(\hat{P})$. Then $v\in D(\hat{P}^{*})$ and $\hat{P}^{*}v=0$. By injectivity of $\hat{P}^{*}$, it thus holds $v=0$, which proves the invertibility of $\hat{P}:D(\hat{P}_{\phi,z})\to\hat{E}$.

The relation (2.8) then implies that for all $z\in\{\operatorname{Re}z\leq\epsilon_{1}h\}$, one has

[TABLE]

with $\delta=\epsilon_{0}-\epsilon_{1}>0$. Hence, for the operator norm on $\hat{E}\subset L^{2}({\mathbb{R}}^{d})$, one has

[TABLE]

uniformly with respect to $z\in\{\operatorname{Re}z<\epsilon_{1}h\}$. $\square$

For $z\in{\mathbb{C}}$, we now consider the Grushin operator ${\mathcal{P}}_{\phi}(z):D(P_{\phi})\times{\mathbb{C}}^{n_{0}}\rightarrow L^{2}({\mathbb{R}}^{d})\times{\mathbb{C}}^{n_{0}}$ defined by

[TABLE]

Lemma 2.2**.**

Let $\epsilon_{0}$ and $h_{0}>0$ be given by Proposition 1.2. Then, the operator ${\mathcal{P}}_{\phi}(z)$ is invertible on $\{\operatorname{Re}z<\epsilon_{0}h\}$. More precisely, for every $z\in\{\operatorname{Re}z<\epsilon_{0}h\}$, $(u,u_{-})\in D(P_{\phi})\times{\mathbb{C}}^{n_{0}}$ and $(f,y)\in L^{2}({\mathbb{R}}^{d})\times{\mathbb{C}}^{n_{0}}$, it holds

[TABLE]

if and only if

[TABLE]

where

[TABLE]

*Proof. *Let $(f,y)\in L^{2}({\mathbb{R}}^{d})\times{\mathbb{C}}^{n_{0}}$ and assume that $(u,u_{-})\in D(P_{\phi})\times{\mathbb{C}}^{n_{0}}$ satisfies

[TABLE]

Applying $R_{+}$ to the first equation and $R_{-}$ to the second one, we get, according to (2.6):

[TABLE]

with $v\in{\rm Ran}\,\hat{\Pi}\cap D(P_{\phi})=\hat{\Pi}(D(P_{\phi}))$ solution to

[TABLE]

Then, applying $\hat{\Pi}$ to the latter equation, we get, using $\hat{\Pi}R_{-}=0$,

[TABLE]

Conversely, note that if $v\in{\rm Ran}\,\hat{\Pi}\cap D(P_{\phi})$ is solution to (2.11), then according to (2.6),

[TABLE]

is solution to (2.10).

Hence, the statement of Lemma 2.2 simply follows from Lemma 2.1 which implies that, for every $z\in\{\operatorname{Re}z<\epsilon_{0}h\}$,

[TABLE]

is the unique solution to (2.11). $\square$

Proof of Theorem 1.3. Let $\epsilon_{0}$ and $h_{0}$ be as in Lemmata 2.1 and 2.2, and take $\epsilon_{1}\in(0,\epsilon_{0})$. For $z\in\{\operatorname{Re}z<\epsilon_{0}h\}$, let ${\mathcal{E}}_{\phi}(z)={\mathcal{P}}_{\phi}(z)^{-1}$. According to Lemma 2.2, it thus holds

[TABLE]

where $E,E_{-},E_{+},E_{-+}$ are holomorphic in $\{\operatorname{Re}z<\epsilon_{0}h\}$ and satisfy the following formulas:

[TABLE]

[TABLE]

and

[TABLE]

Moreover, $P_{\phi}-z$ is invertible if and only if $E_{-+}(z)$ is, in which case it holds

[TABLE]

We refer in particular to [23] for more details in this connection.

We now want to use these formulas to compute the number of poles of $(P_{\phi}-z)^{-1}$. Thanks to (2.2), one has, for some $C>0$ and all $k\in\{1,\ldots,n_{0}\}$,

[TABLE]

Using the bound $\lambda_{k}^{W}\leq Che^{-2\frac{S}{h}}$ given by Proposition 1.2, this yields the existence of some $C>0$ such that for every $k\in\{1,\ldots,n_{0}\}$,

[TABLE]

This shows that $R_{+}\Delta_{\phi}R_{-}={\mathcal{O}}(he^{-2\frac{S}{h}})$ and $R_{+}b\cdot d_{\phi}R_{-}={\mathcal{O}}(\sqrt{h}e^{-\frac{S}{h}})$. Hence, for all $z\in{\mathbb{C}}$, it holds

[TABLE]

On the other hand, we deduce from (2.16) and from the related relation

[TABLE]

valid for any $u\in D(P_{\phi})$ and $k\in\{1,\ldots,n_{0}\}$, that

[TABLE]

Moreover, we know from Lemma 2.1 that, uniformly on $\{\operatorname{Re}z<\epsilon_{1}h\}$, it holds $\hat{P}_{\phi,z}^{-1}={\mathcal{O}}(h^{-1})$. Therefore, injecting this estimate and (2.17), (2.18) into (2.13) and (2.12), we obtain respectively, uniformly on $\{\operatorname{Re}z<\epsilon_{1}h\}$,

[TABLE]

and

[TABLE]

According to (2.19), $E_{-+}(z)$ is then invertible when $z\in\{\operatorname{Re}z<\epsilon_{1}h\}$ satisfies $|z|\geq Ch^{\frac{1}{2}}e^{-\frac{S}{h}}$ for $C$ large enough and the spectrum of $P_{\phi}$ in $\{\operatorname{Re}z<\epsilon_{1}h\}$ is then of order $\mathcal{O}(h^{\frac{1}{2}}e^{-\frac{S}{h}})$. Moreover, for $|z|=\frac{\epsilon_{1}}{2}h$, it holds

[TABLE]

and injecting (2.21) and (2.20) into (2.15) shows that

[TABLE]

Thus, the spectral projector on the open disk $D(0,\frac{\epsilon_{1}}{2}h)$ satisfies

[TABLE]

where we recall that $\Pi$ is a projector of rank $n_{0}$. This implies that for every $h>0$ small enough, the rank of $\Pi_{D(0,\frac{\epsilon_{1}}{2}h)}$, which is the number of eigenvalues of $P_{\phi}$ in $D(0,\frac{\epsilon_{1}}{2}h)$ counted with algebraic multiplicity, is precisely $n_{0}$.

In order to achieve the proof of Theorem 1.3, it just remains to prove the resolvent estimate stated there. On the one hand, it follows easily from (2.14), (2.20), and Lemma 2.1 that

[TABLE]

uniformly with respect to $z\in\{\operatorname{Re}z<\epsilon_{1}h\}$. On the other hand, taking $\epsilon\in(0,\epsilon_{1})$, it follows from (2.19) that $E_{-+}^{-1}(z)={\mathcal{O}}(h^{-1})$, uniformly with respect to $z\in\{\operatorname{Re}z<\epsilon_{1}h\}\cap\{|z|>\epsilon h\}$. Plugging all these estimates into (2.15), we obtain the announced result.

Eventually, since $\sigma(P_{\phi}^{*})=\overline{\sigma(P_{\phi})}$ and, for all $z\notin\sigma(P_{\phi})$, $\|(P^{*}_{\phi}-\overline{z})^{-1}\|=\|(P_{\phi}-z)^{-1}\|,$ it follows easily that the conclusions of Theorem 1.3 also hold true for $P_{\phi}^{*}$. $\square$

3. Geometric preparation

Let us begin this section by observing that the identity $b\cdot\nabla V=0$ arising from (1.3) implies that ${\mathcal{U}}\subset\{x\in{\mathbb{R}}^{d},\ b(x)=0\}$, where we recall that ${\mathcal{U}}$ denotes the set of critical points of the Morse function $V$, as it can be easily proved using a Taylor expansion. Moreover, we have the following

Lemma 3.1**.**

Suppose that Assumptions 1 and 3 hold true and let $\mathbf{u}\in{\mathcal{U}}$ be a critical point of $V$. Then, there exists a smooth map $J_{\mathbf{u}}:{\mathbb{R}}^{d}\rightarrow{\mathcal{M}}_{d}({\mathbb{R}})$ such that $J_{\mathbf{u}}(\mathbf{u})$ is antisymmetric and $b(x)=J_{\mathbf{u}}(x)\nabla V(x)$ for all $x$ in some neighborhood of $\mathbf{u}$. Moreover, it holds

[TABLE]

where $B(\mathbf{u})={\rm Jac}_{\mathbf{u}}b$ is the Jacobian matrix of $b$ at $\mathbf{u}$.

*Proof. *Let $\mathbf{u}\in{\mathcal{U}}$ that we assume to be [math] to lighten the notation. Thanks to the Taylor formula, there exists a smooth map $G:{\mathbb{R}}^{d}\rightarrow{\mathcal{M}}_{d}({\mathbb{R}})$ such that $b(x)=G(x)x$ for all $x\in{\mathbb{R}}^{d}$ and $G(0)={\rm Jac}_{0}\,b$. The same construction works for $\nabla V$ and denoting by ${\mathcal{S}}_{d}$ the set of symmetric matrices, there exists a smooth map $A:{\mathbb{R}}^{d}\rightarrow{\mathcal{S}}_{d}$ such that $\nabla V(x)=A(x)x$ for all $x\in{\mathbb{R}}^{d}$ and $A(0)={\rm Hess\,}V(0)$. The equation $\langle b(x),\nabla V(x)\rangle=0$ for all $x\in{\mathbb{R}}^{d}$ then yields $\langle G(x)x,A(x)x\rangle=0$ and hence, since $A(x)$ is symmetric, $\langle A(x)G(x)x,x\rangle=0$ for all $x\in{\mathbb{R}}^{d}$. Expanding $A(x)G(x)$ in powers of $x$, this implies that

[TABLE]

Hence, the matrix $A(0)G(0)$ is antisymmetric. Since $A(0)$ is symmetric and invertible (since $V$ is a Morse function), this implies that $G(0)A(0)^{-1}$ is antisymmetric. Moreover, $A(x)$ is then also invertible in a neighborhood $\mathcal{V}$ of [math] and we can thus define $J_{0}(x)=G(x)A(x)^{-1}$ on $\mathcal{V}$. One then has

[TABLE]

for all $x\in\mathcal{V}$ and $J_{0}(0)=G(0)A(0)^{-1}$ is antisymmetric thanks to the above analysis. $\square$

Remark 3.2**.**

It is not clear from the above proof that the relation $b\cdot\nabla V=0$ implies the existence of a smooth map $J:{\mathbb{R}}^{d}\rightarrow{\mathcal{M}}_{d}({\mathbb{R}})$ with antisymmetric matrices values such that $b=J\,\nabla V$. However, it follows from (1.3) that for such a map $J$, the vector fields of the form $b_{h}=J\,\nabla V+h\nu$ enter in our framework as soon as

[TABLE]

This is for instance the case when \nu=\big{(}\sum_{i=1}^{d}\partial_{i}J_{ij}\big{)}_{j=1,\dots,d}, which is in particular satisfied when $J$ appears to be constant. Moreover, when \nu=\big{(}\sum_{i=1}^{d}\partial_{i}J_{ij}\big{)}_{j=1,\dots,d}, $L_{V,b,\nu}$ (or equivalently $P_{\phi}$) admits a supersymmetric structure according to (see indeed (1.8))

[TABLE]

where the adjoint is considered with respect to $m_{h}$ (or equivalently

[TABLE]

where the adjoint is now considered with respect to the Lebesgue measure). Using this structure, we may follow the general approach of [12] to analyse the spectrum of $P_{\phi}$. Nevertheless, the operator $P_{\phi}$ still does not have any PT-symmetry and following this approach would again require to replace the use of the Fan inequalities by the one of Theorem A.4 in the final part of the analysis. We believe that this approach may yield complete asymptotic expansions of the small eigenvalues of $P_{\phi}$ (or $L_{V,b,\nu}$) in this setting.

However, when $J$ has antisymmetric matrices values and (3.1) holds but \nu\neq\big{(}\sum_{i=1}^{d}\partial_{i}J_{ij}\big{)}_{j=1,\dots,d}, the operator $P_{\phi}$ is not supersymmetric anymore (see **[20]** for related results).

We are now in position to prove Lemma 1.8. Throughout the rest of this section, we denote

[TABLE]

the eigenvalues of ${\rm Hess\,}V(\mathbf{s})$ counted with multiplicity. For shortness, we will denote

[TABLE]

We recall from Lemma 3.1 that $J$ is antisymmetric.

Step 1 : Let us first prove that $\det({\rm Hess\,}V(\mathbf{s})+B^{*})<0$. Since the matrix ${\rm Hess\,}V(\mathbf{s})+B^{*}$ is real, it thus admits at least one negative eigenvalue.

Since ${\rm Hess\,}V(\mathbf{s})$ is real and symmetric, there exists $P\in\mathcal{M}_{d}({\mathbb{R}})$ such that

[TABLE]

where $D:={\rm Diag}(-\mu_{1},\mu_{2},\dots,\mu_{d})$. It then holds:

[TABLE]

Since $(P^{-1}\,J\,P)^{*}=-P^{-1}\,J\,P$, there exist moreover $p\in\{0,\dots,\lfloor{\frac{d}{2}}\rfloor\}$, $\eta_{1},\dots,\eta_{p}>0$, and $Q\in\mathcal{M}_{d}({\mathbb{R}})$ satisfying $Q^{*}=Q^{-1}$ such that

[TABLE]

where, for every $k\in\{1,\dots,p\}$,

[TABLE]

Here, the rank of the matrix $J$ is $2p$ and its nonzero eigenvalues are the $\pm i\eta_{k}$, $k\in\{1,\dots,p\}$. Therefore, it holds

[TABLE]

where, for every $k\in\{1,\dots,p\}$,

[TABLE]

We then deduce from (3.2) and (3.3) that

[TABLE]

which concludes this first step.

Step 2 : Let us denote by $\mu$ a negative eigenvalue of ${\rm Hess\,}V(\mathbf{s})+B^{*}$ and let us show that $\mu$ is its only negative eigenvalue and has geometric multiplicity one.

Assume first by contradiction that $\mu$ has geometric multiplicity two and denote by $\xi_{1},\xi_{2}$ two associated unitary eigenvectors such that $\langle\xi_{1},\xi_{2}\rangle=0$. Let us also define $\xi^{\prime}_{i}:=P^{-1}\xi_{i}$ for $i\in\{1,2\}$ so that $\xi^{\prime}_{1}$ and $\xi^{\prime}_{2}$ are orthogonal and unitary. According to (3.2), it holds moreover for $i\in\{1,2\}$,

[TABLE]

In particular, since $(P^{-1}\,J\,P)^{*}=-P^{-1}\,J\,P$, it holds for every $(a,b)\in{\mathbb{R}}^{2}$ satisfying $a^{2}+b^{2}=1$:

[TABLE]

Applying the Max-Min principle to the symmetric matrix $D^{-1}$, this shows that the second eigenvalue $\mu_{2}(D^{-1})$ of the matrix $D^{-1}$ satisfies $\mu_{2}(D^{-1})\leq\frac{1}{\mu}<0$, contradicting $D^{-1}={\rm Diag}(-\frac{1}{\mu_{1}},\frac{1}{\mu_{2}},\dots,\frac{1}{\mu_{d}})$.

Hence the negative eigenvalue $\mu$ has geometric multiplicity one and we have to show that it is the only negative eigenvalue of ${\rm Hess\,}V(\mathbf{s})+B^{*}$. We reason again by contradiction, assuming that ${\rm Hess\,}V(\mathbf{s})+B^{*}$ admits another negative eigenvalue that we denote by $\eta$. Note in particular that it follows from the relation (see indeed (3.2))

[TABLE]

that $\eta$ is also an eigenvalue of ${\rm Hess\,}V(\mathbf{s})-B^{*}(\mathbf{s})={\rm Hess\,}V(\mathbf{s})\,(I_{d}+J)$. Denote now by $\xi_{1}$ a unitary eigenvector of ${\rm Hess\,}V(\mathbf{s})+B^{*}$ associated with $\mu$ and by $\xi_{2}$ a unitary eigenvector of ${\rm Hess\,}V(\mathbf{s})-B^{*}$ associated with $\eta$. Defining again $\xi^{\prime}_{i}:=P^{-1}\xi_{i}$ for $i\in\{1,2\}$, we have thus

[TABLE]

It follows that

[TABLE]

The vectors $\xi_{1}^{\prime}$ and $\xi_{2}^{\prime}$ are in particular linearly independent and it holds for some positive constant $c$ and every $(a,b)\in{\mathbb{R}}^{2}\setminus\{(0,0)\}$,

[TABLE]

Applying again the Max-Min principle to the symmetric matrix $D^{-1}$ leads to $\mu_{2}(D^{-1})\leq-c<0$ and hence to a contradiction. This concludes the proof of the second step.

Step 3 : Let us now prove the relation

[TABLE]

which is equivalent to

[TABLE]

where $\xi$ denotes a unitary eigenvector of ${\rm Hess\,}V(\mathbf{s})+B^{*}$ associated with $\mu$ and $\xi^{\prime}:=P^{-1}\xi$. To this end, note first that it obviously holds

[TABLE]

Moreover, since $D^{-1}\xi^{\prime}=\frac{1}{\mu}(I_{d}-P^{-1}\,J\,P)\xi^{\prime}$, it also holds

[TABLE]

Since $P^{-1}\,J\,P\xi^{\prime}$ belongs to $(\xi^{\prime})^{\perp}$, we deduce (3.5) and then (3.4) from (3.6) and (3.7).

Step 4 : To conclude the proof of the second item of Lemma 1.8, it only remains to show that the real symmetric matrix $M_{V}:={\rm Hess\,}V(\mathbf{s})+2|\mu|\,\xi\,\xi^{*}$ is positive definite, where we recall that $\xi$ denotes a unitary eigenvector of ${\rm Hess\,}V(\mathbf{s})+B^{*}$ associated with $\mu$. This is an easy consequence of the Max-Min principle and of the relation $\det M_{V}=-\det D>0$ obtained in the previous step. We have indeed, defining again $\xi^{\prime}:=P^{-1}\xi$,

[TABLE]

which implies that the second eigenvalue of $D+2|\mu|\,\xi^{\prime}\,\xi^{\prime*}$, that is the second eigenvalue of $M_{V}$, is greater than or equal to $\mu_{2}$, and hence positive. The first eigenvalue of $M_{V}$ is then positive according to $\det M_{V}>0$. This concludes this step of the proof.

Step 5 : We now prove the third item of Lemma 1.8. Since ${\rm Hess\,}V(\mathbf{s})(I_{d}-J)\xi=\mu\xi$ and $J^{*}=-J$, it first holds

[TABLE]

which proves the second part of the third item of Lemma 1.8. Defining again $\xi^{\prime}:=P^{-1}\xi$, this also means

[TABLE]

This implies that $\frac{1}{\mu}\geq-\frac{1}{\lambda_{1}}$, i.e. that $|\mu|\geq\mu_{1}$, with equality if and only if $\xi^{\prime}=\pm(1,0,\dots,0)^{*}$, that is if and only if $\xi$ is a unitary eigenvector of $({\rm Hess\,}V(\mathbf{s}))^{-1}$ associated with $-\frac{1}{\mu_{1}}$, which is equivalent to the relation $J\xi=0$ by (3.8), and hence to $B^{*}\xi=0$ since $J=-({\rm Hess\,}V(\mathbf{s}))^{-1}B^{*}$.

4. Spectral analysis in the case of Morse functions

4.1. Construction of accurate quasimodes

In the following, we assume that Assumption 4 is satisfied. Let us then consider some arbitrary $\mathbf{m}\in\mathcal{U}^{(0)}\setminus\{\underline{\mathbf{m}}\}$, that is, according to Assumption 4, a local minimum of $V$ which is not the global minimum $\underline{\mathbf{m}}$ of $V$. According to the labelling procedure of Section 1.3 leading to the definitions (1.17)–(1.19), it holds in particular $\mathbf{m}=\mathbf{m}_{i,j}$ and $\boldsymbol{\sigma}(\mathbf{m})=\sigma_{i}$ for some $i\in\{2,\dots,N\}$ and $j\in\{1,\dots,N_{i}\}$. For every $\mathbf{s}\in{\bf j}(\mathbf{m})$ and $\rho,\delta>0$, where we recall that the mapping ${\bf j}:\mathcal{U}^{(0)}\to\mathcal{P}({\mathcal{V}}^{(1)}\cup\{\mathbf{s}_{1}\})$ has been defined in (1.18) and that $V(\mathbf{s})=\boldsymbol{\sigma}(\mathbf{m})$, we define the set

[TABLE]

and the set $\mathcal{C}_{\mathbf{s},\rho,\delta}$ by:

[TABLE]

where $\xi(\mathbf{s})$ has been defined in Lemma 1.8. We recall that $\xi(\mathbf{s})$ is an unitary eigenvector of the matrix ${\rm Hess\,}V(\mathbf{s})+B^{*}(\mathbf{s})$ associated with its only negative eigenvalue $\mu(\mathbf{s})$ which has geometric multiplicity one. Let us also define

[TABLE]

where

[TABLE]

According to Assumption 4 and Remark 1.7, we recall that there is precisely one connected component $\widehat{E}(\mathbf{m})\neq E(\mathbf{m})$ of $\{V<\boldsymbol{\sigma}(\mathbf{m})\}$ such that $\overline{E(\mathbf{m})}\cap\overline{\widehat{E}(\mathbf{m})}\neq\emptyset$ (see examples in Figure 1.3). Moreover, it holds ${\bf j}(\mathbf{m})=\partial\widehat{E}(\mathbf{m})\cap\partial E(\mathbf{m})$ and the global minimum $\hat{\mathbf{m}}$ of $V|_{\widehat{E}(\mathbf{m})}$ satisfies $\boldsymbol{\sigma}(\hat{\mathbf{m}})>\boldsymbol{\sigma}(\mathbf{m})$ and $V(\hat{\mathbf{m}})<V(\mathbf{m})$ (see in this connection [21], where the notation $\widehat{E}(\mathbf{m})$ is introduced for an arbitrary Morse function).

According to the geometry of the Morse function $V$ around $\partial E(\mathbf{m})$ and to Lemma 1.8, we have then the following result.

Lemma 4.1**.**

Assume that Assumption 4 is satisfied and let $\mathbf{m}\in\mathcal{U}^{(0)}\setminus\{\underline{\mathbf{m}}\}$, $\mathbf{s}\in{\bf j}(\mathbf{m})$, and $\xi(\mathbf{s})$ be some unitary eigenvector of the matrix ${\rm Hess\,}V(\mathbf{s})+B^{*}(\mathbf{s})$ associated with its unique negative eigenvalue (see Lemma 1.8). Then, there exists a neigborhood $\mathcal{O}$ of $\mathbf{s}$ such that:

[TABLE]

It follows that there exist $\rho_{0},\delta_{0}>0$ sufficiently small such that for all $\rho\in(0,\rho_{0}]$ and $\delta\in(0,\delta_{0}]$, the set $E_{\mathbf{m},3\rho,3\delta}$ defined in (4.2) has exactly two connected components, $E_{\mathbf{m},3\rho_{0},3\delta_{0}}^{+}$ and $E^{-}_{\mathbf{m},3\rho,3\delta}$, containing respectively $\mathbf{m}$ and $\hat{\mathbf{m}}$.

Proof.

For shortness, we denote $\xi=\xi(\mathbf{s})$. By a continuity argument, note that to prove the first part of Lemma 4.1, it is sufficient to prove that the linear hyperplane $\xi^{\perp}$ does not meet the cone $\{X\in{\mathbb{R}}^{n}\,;\ \langle{\rm Hess\,}V(\mathbf{s})X,X\rangle\leq 0\}$ outside the origin. The second part of the lemma then simply follows from the observation that the set $\mathcal{C}_{\mathbf{s},\rho,\delta}$ defined in (4.1) is thus an arbitrary small neighborhood of $\mathbf{s}$ when $\rho,\delta>0$ tend to [math].

When $d\geq 3$, it is then enough to show that for any column vector $X\in{\mathbb{R}}^{d}\setminus\{0\}$ such that $\langle{\rm Hess\,}V(\mathbf{s})X,X\rangle=0$, it holds ${\rm Span\,}X\oplus\xi^{\perp}={\mathbb{R}}^{d}$, i.e. $\langle X,\xi\rangle\neq 0$. Indeed, when $d\geq 3$, any linear hyperplane meets $\{X\in{\mathbb{R}}^{n}\,;\ \langle{\rm Hess\,}V(\mathbf{s})X,X\rangle>0\}$ and then meets $\{X\in{\mathbb{R}}^{d}\setminus\{0\}\,;\ \langle{\rm Hess\,}V(\mathbf{s})X,X\rangle=0\}$ if and only if it meets $\{X\in{\mathbb{R}}^{d}\setminus\{0\}\,;\ \langle{\rm Hess\,}V(\mathbf{s})X,X\rangle\leq 0\}$. Let us then consider $X\in{\mathbb{R}}^{d}\setminus\{0\}$ such that $\langle{\rm Hess\,}V(\mathbf{s})X,X\rangle=0$ and let us prove that $\langle X,\xi\rangle\neq 0$. To show this, let us work in orthonormal coordinates of ${\mathbb{R}}^{d}$ where ${\rm Hess\,}V(\mathbf{s})$ is diagonal, i.e. where ${\rm Hess\,}V(\mathbf{s})={\rm Diag}(-\mu_{1},\mu_{2},\dots,\mu_{d})$. It then follows from $\langle{\rm Hess\,}V(\mathbf{s})X,X\rangle=0$ and from the third item of Lemma 1.8 that

[TABLE]

It holds in particular $X_{1}\neq 0$ and thus, by multiplying the two above relations,

[TABLE]

the last inequality resulting from the Cauchy-Schwarz inequality. The relation $\langle X,\xi\rangle\neq 0$ follows.

When $d=2$, the situation is slightly different since for any hyperplane $H$, either $H\setminus\{0\}\subset\{X\in{\mathbb{R}}^{2}\setminus\{0\}\,;\ \langle{\rm Hess\,}V(\mathbf{s})X,X\rangle\leq 0\}$ or $H\setminus\{0\}\subset\{X\in{\mathbb{R}}^{2}\setminus\{0\}\,;\ \langle{\rm Hess\,}V(\mathbf{s})X,X\rangle>0\}$. Take again orthonormal coordinates where ${\rm Hess\,}V(\mathbf{s})={\rm Diag}(-\mu_{1},\mu_{2})$. We have then only to prove that the vector $\xi^{\prime}:=(-\xi_{2},\xi_{1})^{*}$, which spans $\xi^{\perp}$, satisfies

[TABLE]

This is obviously satisfied since equivalent to

[TABLE]

which holds true thanks to iii) of Lemma 1.8. This concludes the proof of Lemma 4.1.

Let us now define, for every $h\in(0,1]$ and for every $\rho_{0},\delta_{0}>0$ small enough, the function $\kappa_{\mathbf{m},h}$ on the sublevel set $E_{-}(\mathbf{m})\cap\{V<\boldsymbol{\sigma}(\mathbf{m})+3\delta_{0}\}$ (see (4.3)) as follows:

On the disjoint open sets $E^{+}_{\mathbf{m},3\rho_{0},3\delta_{0}}$ and $E^{-}_{\mathbf{m},3\rho_{0},3\delta_{0}}$ introduced in Lemma 4.1,

[TABLE] 2. 2.

For every $\mathbf{s}\in{\bf j}(\mathbf{m})$ and $x\in\mathcal{C}_{\mathbf{s},3\rho_{0},3\delta_{0}}\cap\{V<\boldsymbol{\sigma}(\mathbf{m})+3\delta_{0}\}$ (see (4.1)),

[TABLE]

where the orientation of $\xi(\mathbf{s})$ is chosen in such a way that there exists a neighborhood $\mathcal{O}$ of $\mathbf{s}$ such that $E(\mathbf{m})\cap\mathcal{O}$ is included in the half-space $\{\xi(\mathbf{s})\cdot(x-\mathbf{s})>0\}$ (see Lemma 4.1 and Figures 4.1 and 4.2), $\chi\in C^{\infty}(\mathbb{R};[0,1])$ is even and satisfies $\chi\equiv 1$ on $[-1,1]$, $\chi(\eta)=0$ for $|\eta|\geq 2$, and

[TABLE]

Note in particular that

[TABLE]

Note also that for every $\rho_{0},\delta_{0}>0$ small enough, thanks to the definitions (4.4) and (4.5), and since the sets $E^{+}_{\mathbf{m},3\rho_{0},3\delta_{0}}$, $E^{-}_{\mathbf{m},3\rho_{0},3\delta_{0}}$, and $\mathcal{C}_{\mathbf{s},3\rho_{0},3\delta_{0}}$’s, $\mathbf{s}\in{\bf j}(\mathbf{m})$, are two by two disjoint (see Lemma 4.1), $\kappa_{\mathbf{m},h}$ is well defined and is ${\mathcal{C}}^{\infty}$ on $E_{-}(\mathbf{m})\cap\{V<\boldsymbol{\sigma}(\mathbf{m})+3\delta_{0}\}$.

Consider now a smooth function $\theta_{\mathbf{m}}$ such that

[TABLE]

The function $\theta_{\mathbf{m}}\kappa_{\mathbf{m},h}$ then belongs to ${\mathcal{C}}_{c}^{\infty}({\mathbb{R}}^{d};[-1,1])$ and

[TABLE]

Definition 4.2**.**

For any $\mathbf{m}\in{\mathcal{U}}^{(0)}$ let us define the function $\psi_{\mathbf{m},h}$ by

[TABLE]

when $\mathbf{m}\neq\underline{\mathbf{m}}$ and, when $\mathbf{m}=\underline{\mathbf{m}}$, $\psi_{\mathbf{m},h}(x):=1$. We then define, for any $\mathbf{m}\in{\mathcal{U}}^{(0)}$, the quasimode $\varphi_{\mathbf{m},h}$ by

[TABLE]

Note that, for every $h\in(0,1]$, it holds $L_{V,b,\nu}\varphi_{\underline{\mathbf{m}},h}=0$ and for every $\mathbf{m}\in{\mathcal{U}}^{(0)}\setminus\{\underline{\mathbf{m}}\}$, the quasimodes $\psi_{\mathbf{m},h}$ and $\varphi_{\mathbf{m},h}$ belong to ${\mathcal{C}}_{c}^{\infty}(\mathbb{R}^{d};{\mathbb{R}}^{+})$ with supports included in $E_{-}(\mathbf{m})\cap\{V<\boldsymbol{\sigma}(\mathbf{m})+2\delta_{0}\}$. We have more precisely the following lemma resulting from the previous construction.

Lemma 4.3**.**

Assume that Assumption 4 is satisfied. For every $\mathbf{m}\in{\mathcal{U}}^{(0)}$ and every small $\epsilon>0$ fixed, there exist $\rho_{0},\delta_{0}>0$ small enough such that for every $h\in(0,1]$, one has:

• i)

It holds

[TABLE]

• ii)

When $\mathbf{m}\neq\underline{\mathbf{m}}$, there exists a neighborhood $\mathcal{O}_{\rho_{0},\delta_{0}}$ of $\overline{E(\mathbf{m})}$ such that:

[TABLE]

In particular, it holds

[TABLE]

• iii)

When $\mathbf{m}\neq\underline{\mathbf{m}}$, it holds:

[TABLE]

Let moreover $\mathbf{m}^{\prime}$ belong to ${\mathcal{U}}^{(0)}$ with $\mathbf{m}\neq\mathbf{m}^{\prime}$. The following then hold true for every $\rho_{0},\delta_{0}>0$ small enough and every $h\in(0,1]$:

• iv)

if $\boldsymbol{\sigma}(\mathbf{m})=\boldsymbol{\sigma}(\mathbf{m}^{\prime})$, then $\mathop{\rm supp}\nolimits(\psi_{\mathbf{m},h})\cap\mathop{\rm supp}\nolimits(\psi_{\mathbf{m}^{\prime},h})=\emptyset$,

• v)

if $\boldsymbol{\sigma}(\mathbf{m})>\boldsymbol{\sigma}(\mathbf{m}^{\prime})$, then

• –

either $\mathop{\rm supp}\nolimits(\psi_{\mathbf{m},h})\cap\mathop{\rm supp}\nolimits(\psi_{\mathbf{m}^{\prime},h})=\emptyset$,

• –

or $\psi_{\mathbf{m},h}=2$ on $\mathop{\rm supp}\nolimits(\psi_{\mathbf{m}^{\prime},h})$ and $V(\mathbf{m}^{\prime})>V(\mathbf{m})$.

*Proof. *The first part of Lemma 4.3 follows from Assumption 4 and from the construction of the quasimodes $\varphi_{\mathbf{m},h}$ defined in Definition 4.2 for $\mathbf{m}\in\mathcal{U}^{(0)}$, see indeed (4.4), (4.5), and (4.7). Let us then prove the second part of Lemma 4.3.

When $\boldsymbol{\sigma}(\mathbf{m})=\boldsymbol{\sigma}(\mathbf{m}^{\prime})$ and $\mathbf{m}\neq\mathbf{m}^{\prime}$, note first that $\mathbf{m}$ and $\mathbf{m}^{\prime}$ differ from $\underline{\mathbf{m}}$ since $\boldsymbol{\sigma}(\mathbf{m})=+\infty$ if and only if $\mathbf{m}=\underline{\mathbf{m}}$. When moreover $\mathbf{m}^{\prime}\notin E_{-}(\mathbf{m})$, it holds $E_{-}(\mathbf{m})\neq E_{-}(\mathbf{m}^{\prime})$ and hence $E_{-}(\mathbf{m})\cap E_{-}(\mathbf{m}^{\prime})=\emptyset$, implying $\mathop{\rm supp}\nolimits(\psi_{\mathbf{m},h})\cap\mathop{\rm supp}\nolimits(\psi_{\mathbf{m}^{\prime},h})=\emptyset$. In the case when $\mathbf{m}^{\prime}\in E_{-}(\mathbf{m})$, the statement of Lemma 4.3 follows from ii) of Assumption 4 and of Remark 1.7, which indeed imply that $\overline{E(\mathbf{m})}\cap\overline{E(\mathbf{m}^{\prime})}=\emptyset$ (see the first item of Lemma 4.3).

When $\boldsymbol{\sigma}(\mathbf{m})>\boldsymbol{\sigma}(\mathbf{m}^{\prime})$ and $\mathbf{m}^{\prime}\notin E(\mathbf{m})$, it holds $\overline{E(\mathbf{m})}\cap\overline{E(\mathbf{m}^{\prime})}=\emptyset$, and again, according to the first item of Lemma 4.3, it holds $\mathop{\rm supp}\nolimits(\psi_{\mathbf{m},h})\cap\mathop{\rm supp}\nolimits(\psi_{\mathbf{m}^{\prime},h})=\emptyset$ for every $\rho_{0},\delta_{0}>0$ small enough. Lastly, when $\boldsymbol{\sigma}(\mathbf{m})>\boldsymbol{\sigma}(\mathbf{m}^{\prime})$ and $\mathbf{m}^{\prime}\in E(\mathbf{m})$, it holds $\overline{E(\mathbf{m}^{\prime})}\subset E_{-}(\mathbf{m}^{\prime})\subset E(\mathbf{m})$ and then, according to the second item of Lemma 4.3, $\psi_{\mathbf{m},h}=2$ on $\mathop{\rm supp}\nolimits(\psi_{\mathbf{m}^{\prime},h})$ for every $\rho_{0},\delta_{0}>0$ small enough. Besides, the relation $V(\mathbf{m}^{\prime})>V(\mathbf{m})$ follows from $\mathbf{m}^{\prime}\in E(\mathbf{m})$ and from the first item of Assumption 4. $\square$

4.2. Quasimodal estimates

We write in the sequel $a\eqsim b$ and $a\lesssim b$ to mean, in the limit $h\to 0,$ equality/inequality up to a multiplicative factor $1+\mathcal{O}(h)$. Moreover, we define for shortness, for any critical point $\mathbf{u}$ of $V$:

[TABLE]

Proposition 4.4**.**

Assume that Assumption 4 is satisfied and consider the families \big{(}\psi_{\mathbf{m},h},\mathbf{m}\in{\mathcal{U}}^{(0)}\big{)} and \big{(}\varphi_{\mathbf{m},h},\mathbf{m}\in{\mathcal{U}}^{(0)}\big{)} of Definition 4.2. Then, for every $\mathbf{m}\in\mathcal{U}^{(0)}\setminus\{\underline{\mathbf{m}}\}$ and $\rho_{0},\delta_{0}>0$ small enough, it holds in the limit $h\to 0$:

[TABLE]

Moreover, there exists $C>0$ such that for every $\mathbf{m},\mathbf{m}^{\prime}\in{\mathcal{U}}^{(0)}$, it holds in the limit $h\to 0$:

[TABLE]

*Proof. *To prove the relation (4.8), write, according to Definition 4.2,

[TABLE]

where $Z_{h}$ is the normalizing constant defined by (1.10). Hence, according to Lemma 4.3 and standard tail estimates and Laplace asymptotics, we get, in the limit $h\to 0$,

[TABLE]

as well as

[TABLE]

The estimate (4.8) then follows easily.

Let us now prove the relation (4.9). According to Definition 4.2, note first that $\langle\varphi_{\mathbf{m},h},\varphi_{\mathbf{m},h}\rangle=1$ for every $\mathbf{m}\in{\mathcal{U}}^{(0)}$. Moreover, when $\mathbf{m},\mathbf{m}^{\prime}\in{\mathcal{U}}^{(0)}$ and $\mathbf{m}\neq\mathbf{m}^{\prime}$, it follows from Lemma 4.3 that, up to switching $\mathbf{m}$ and $\mathbf{m}^{\prime}$, we are in one of the two following cases:

• –

either $\mathop{\rm supp}\nolimits(\varphi_{\mathbf{m},h})\cap\mathop{\rm supp}\nolimits(\varphi_{\mathbf{m}^{\prime},h})=\emptyset$, and then

[TABLE]

• –

or $\psi_{\mathbf{m},h}=2$ on $\mathop{\rm supp}\nolimits(\psi_{\mathbf{m}^{\prime},h})$ and $V(\mathbf{m}^{\prime})>V(\mathbf{m})$, and then, using the preceding estimates,

[TABLE]

where $C=\frac{V(\mathbf{m}^{\prime})-V(\mathbf{m})}{2}>0$.

This leads to (4.9). $\square$

Proposition 4.5**.**

For every $\mathbf{m}\in\underline{\mathcal{U}}^{(0)}$ and $\rho_{0},\delta_{0}>0$ small enough, it holds in the limit $h\to 0$:

[TABLE]

and then

[TABLE]

*Proof. *Note first that thanks to (1.3), one has $\operatorname{div}(b_{h}m_{h})=0$ and hence:

[TABLE]

Using this relation together with (1.4), (4.4)–(4.7), Definition 4.2, and Lemma 4.3, we get, in the limit $h\to 0$,

[TABLE]

where for short we denote $\xi=\xi(\mathbf{s})$ and $\mu=\mu(\mathbf{s})$. From the second item in Lemma 1.8 and the Taylor expansion of $V+|\mu|\langle\xi,\cdot-\mathbf{s}\rangle^{2}$ around $\mathbf{s}\in{\bf j}(\mathbf{m})$,

[TABLE]

it is clear that for $\rho_{0}$ and $\delta_{0}$ small enough, $V+|\mu|\langle\xi,\cdot-\mathbf{s}\rangle^{2}$ uniquely attains its minimal value in $\mathcal{C}_{\mathbf{s},3\rho_{0},3\delta_{0}}$ at $\mathbf{s}$ since:

[TABLE]

Moreover, using again the second item in Lemma 1.8 and a standard Laplace method, it holds in the limit $h\to 0$, for every $\mathbf{s}\in{\bf j}(\mathbf{m})$,

[TABLE]

where we also used (4.6) at the last line. The statement of Proposition 4.5 then follows from (4.12) and (4.13), using also $Z_{h}\eqsim(2\pi h)^{\frac{d}{2}}D_{\underline{\mathbf{m}}}^{-1}\,e^{-\frac{V(\underline{\mathbf{m}})}{h}}$. $\square$

Proposition 4.6**.**

Let $\mathbf{m}\in\underline{\mathcal{U}}^{(0)}$. For $\rho_{0}$ and $\delta_{0}$ sufficiently small, it holds in the limit $h\to 0$:

[TABLE]

and

[TABLE]

*Proof. *Let $\mathbf{s}\in{\bf j}(\mathbf{m})$ and denote for short $\xi=\xi(\mathbf{s})$ and $\mu=\mu(\mathbf{s})$. We first recall the Taylor expansion of $V+|\mu|\langle\xi,\cdot-\mathbf{s}\rangle^{2}$ around $\mathbf{s}$,

[TABLE]

which implies, according to the second item of Lemma 1.8, that for $\rho_{0}$ and $\delta_{0}$ small enough:

• –

\nabla\big{(}\,V+|\mu|\langle\xi,\cdot-\mathbf{s}\rangle^{2}\,\big{)}(\mathbf{s})=0,

• –

$V+|\mu|\langle\xi,\cdot-\mathbf{s}\rangle^{2}$ uniquely attains its minimal value in $\mathcal{C}_{\mathbf{s},3\rho_{0},3\delta_{0}}$ at $\mathbf{s}$.

Note now that according to (1.4), it holds

[TABLE]

with on $\mathcal{C}_{\mathbf{s},3\rho_{0},3\delta_{0}}$, for every $\mathbf{s}\in{\bf j}(\mathbf{m})$, according to (4.5),

[TABLE]

where we recall that $b_{h}=b+h\nu$. It then follows from (4.4)–(4.7) that in the limit $h\to 0$,

[TABLE]

for some real constant $c\in(0,\delta_{0})$. Moreover, using $b(\mathbf{s})=0$ and the first item of Lemma 1.8, the Taylor expansion of $\nabla V+b$ around $\mathbf{s}$ satisfies

[TABLE]

It then follows from Proposition 4.5, standard tail estimates, and Laplace asymptotics, that in the limit $h\to 0$,

[TABLE]

which proves (4.14).

To prove (4.15), we observe that since $L_{V,b,\nu}^{*}=L_{V,-b,-\nu}$, the same computation as above shows that in the limit $h\to 0$,

[TABLE]

However, contrary to the preceding case, one has here only

[TABLE]

which implies, in the limit $h\to 0$,

[TABLE]

which is exactly (4.15). $\square$

4.3. Proof of Theorem 1.9

Throughout this section, we denote for shortness

[TABLE]

and we label the local minima $\mathbf{m}_{1},\ldots,\mathbf{m}_{n_{0}}$ of $V$ in so that $(S(\mathbf{m}_{j}))_{j\in\{1,\dots,n_{0}\}}$ is non-increasing (see (1.19)):

[TABLE]

For all $j\in\{1,\ldots,n_{0}\}$, we will also denote for shortness

[TABLE]

From Proposition 4.5, one knows that for all $j\in\{2,\ldots,n_{0}\}$, one has

[TABLE]

Moreover, since $(S_{j})_{j\in\{1,\dots,n_{0}\}}$ is non-increasing, we deduce from this estimate that there exists $h_{0}>0$ and $C>0$ such that for all $h\in(0,h_{0}]$ and all $i,j\in\{1,\ldots,n_{0}\}$, one has

[TABLE]

The two following lemmata are straightforward consequence of the previous analysis.

Lemma 4.7**.**

For every $j,k\in\{1,\ldots,n_{0}\}$ and $h\in(0,1]$, one has

[TABLE]

*Proof. *When $j=k$, the statement if obvious. When $j\neq k$, then it follows from Lemma 4.3 that we are in one of the three following cases:

• –

either $\mathop{\rm supp}\nolimits(\varphi_{j})\cap\mathop{\rm supp}\nolimits(\varphi_{k})=\emptyset$ and the conclusion is obvious,

• –

either there exists $c_{h}>0$ such that $\varphi_{j}=c_{h}$ on $\mathop{\rm supp}\nolimits(\varphi_{k})$ and

[TABLE]

• –

or there exists $c_{h}>0$ such that $\varphi_{k}=c_{h}$ on $\mathop{\rm supp}\nolimits(\varphi_{j})$ and

[TABLE]

$\square$

Lemma 4.8**.**

For $\rho_{0},\delta_{0}$ sufficiently small and every $j\in\{1,\ldots,n_{0}\}$, it holds in the limit $h\to 0$,

[TABLE]

and

[TABLE]

*Proof. *This is a simple rewriting of Proposition 4.6, using the fact that for every $\mathbf{m}\in\mathcal{U}^{(0)}$ and $h\in(0,1]$, $\varphi_{\mathbf{m},h}=\frac{\psi_{\mathbf{m},h}}{\|\psi_{\mathbf{m},h}\|}$. $\square$

We now introduce, for every $h>0$ small enough, the spectral projector $\Pi_{h}$ associated with the $n_{0}$ smallest eigenvalues of $L_{V}$ as described in Theorem 1.3. Let then $\epsilon_{0}$ be given by Theorem 1.3. According to Theorem 1.3, for every $h>0$ small enough, $\Pi_{h}$ satisfies

[TABLE]

and in particular:

[TABLE]

Lemma 4.9**.**

For all $j\in\{1,\ldots,n_{0}\}$, we have, in the limit $h\to 0$,

[TABLE]

and

[TABLE]

*Proof. *Thanks to the resolvent identity, one has

[TABLE]

Moreover, it follows from Theorem 1.3 and from (1.14) that for any $z\in\partial D(0,\frac{\epsilon_{0}}{2})$,

[TABLE]

Combined with (4.18), this proves (4.22). On the other hand, one has similarly

[TABLE]

and $\|(z-L_{V}^{*})^{-1}\|_{L^{2}(m_{h})\rightarrow L^{2}(m_{h})}={\mathcal{O}}(1)$. Then, (4.23) follows immediately from (4.19). $\square$

Proposition 4.10**.**

For every $j\in\{1,\dots,n_{0}\}$ and $h>0$ small enough, let us define $v_{j}:=\Pi_{h}\varphi_{j}$. Then, there exists $c>0$ such that for all $j,k\in\{1,\ldots,n_{0}\}$, one has in the limit $h\to 0$,

[TABLE]

and

[TABLE]

In particular, it follows from (4.24) that for every $h>0$ small enough, the family $(v_{1},\dots,v_{n_{0}})$ is a basis of ${\rm Ran}\,\Pi_{h}$.

*Proof. *Since, for some $c>0$, every $j\in\{1,\dots,n_{0}\}$, and every $h>0$ small enough, it holds $\tilde{\lambda}_{j}(h)={\mathcal{O}}(e^{-\frac{c}{h}})$, the first identity follows directly from (4.9), (4.22), and from the relation

[TABLE]

To prove the second estimate, observe that

[TABLE]

Moreover, thanks to Lemma 4.8, (4.21), and Lemma 4.9, one has

[TABLE]

and

[TABLE]

Gathering these two estimates and using Lemma 4.7, we obtain (4.25). $\square$

We now orthonormalize the basis $(v_{1},\dots,v_{n_{0}})$ of ${\rm Ran}\,\Pi_{h}$ by a Gram-Schmidt procedure: for all $j\in\{1,\dots,n_{0}\}$, let us define by induction

[TABLE]

Lemma 4.11**.**

There exists $c>0$ such that for all $j\in\{1,\dots,n_{0}\}$, one has in the limit $h\to 0$:

[TABLE]

with $\alpha_{jk}={\mathcal{O}}(e^{-\frac{c}{h}})$. In particular, it holds:

[TABLE]

*Proof. *One proceeds by induction on $j$. For $j=1$, one has $\tilde{e}_{1}=v_{1}=\varphi_{1}=1$ and there is nothing to prove. Suppose now that the above formula is true for all $\tilde{e}_{l}$ with $1\leq l\leq j<n_{0}$. Then $\tilde{e}_{j+1}=v_{j+1}-r_{j+1}$ with

[TABLE]

Since by induction, $\|\tilde{e}_{k}\|=1+{\mathcal{O}}(e^{-\frac{c}{h}})$ for all $k\in\{1,\dots,j\}$, it follows that

[TABLE]

Moreover, for all $k\in\{1,\dots,j\}$, one also has by induction

[TABLE]

with $\beta_{k,l}={\mathcal{O}}(1)$ for any $l\in\{1,\dots,k\}$ (and actually $\beta_{k,l}={\mathcal{O}}(e^{-\frac{c}{h}})$ when $l<k$), which implies

[TABLE]

Since, thanks to Proposition 4.10, it holds $\langle v_{j+1},v_{l}\rangle={\mathcal{O}}(e^{-\frac{c}{h}})$ for all $l,m\leq k<j+1$, then

[TABLE]

where $\gamma_{j,m}={\mathcal{O}}(e^{-\frac{c}{h}})$ for all $m\in\{1,\dots,j\}$. This proves the first part of the lemma. The second one is obvious. $\square$

Proposition 4.12**.**

For all $j,k\in\{1,\ldots,n_{0}\}$, one has in the limit $h\to 0$:

[TABLE]

*Proof. *Thanks to Lemma 4.11, one has for all $j,k\in\{1,\ldots,n_{0}\}$,

[TABLE]

where, for all $p,q$, it holds $\alpha^{\prime}_{p,q}=\alpha_{j,p}\alpha_{k,q}={\mathcal{O}}(e^{-\frac{c}{h}})$. Combined with Proposition 4.10, this implies

[TABLE]

On the other hand, thanks to Proposition 4.10 and (4.17), one has in the limit $h\to 0$, for all $1\leq p<j$ and $1\leq q<k$,

[TABLE]

Combined with (4.27) and using the fact that $\alpha^{\prime}_{p,q}={\mathcal{O}}(e^{-\frac{c}{h}})={\mathcal{O}}(\sqrt{h})$, this shows that

[TABLE]

Eventually, since $e_{k}=(1+{\mathcal{O}}(e^{-\frac{c}{h}}))\tilde{e}_{k}$ according to Lemma 4.11, we obtain

[TABLE]

which completes the proof. $\square$

We are now in position to prove Theorem 1.9. We recall that $(e_{1},\ldots,e_{n_{0}})$ is an orthonormal basis of ${\rm Ran}\,\Pi_{h}$ and that $L_{V}|_{{\rm Ran}\,\Pi_{h}}:{\rm Ran}\,\Pi_{h}\rightarrow{\rm Ran}\,\Pi_{h}$ has exactly $n_{0}$ eigenvalues $\lambda_{1},\dots,\lambda_{n_{0}}$, with $\lambda_{j}=0$ iff $j=1$, counted with algebraic multiplicity. Let us denote $\hat{e}_{j}=e_{n_{0}+1-j}$ and let $\mathcal{M}$ denote the matrix of $L_{V}$ in the basis $(\hat{e}_{1},\ldots,\hat{e}_{n_{0}})$. Since this basis is orthonormal, it holds

[TABLE]

Moreover, since

[TABLE]

then $\mathcal{M}$ has the form

[TABLE]

On the other hand, denoting $\hat{\lambda}_{j}(h):=\tilde{\lambda}_{n_{0}+1-j}(h)$ for $j\in\{1,\dots,n_{0}-1\}$, one deduces from Proposition 4.12 that for every $j,k\in\{1,\ldots n_{0}-1\}$, it holds in the limit $h\to 0$,

[TABLE]

that is

[TABLE]

For all $j\in\{1,\ldots,n_{0}-1\}$, let us now define

[TABLE]

where $\zeta(\mathbf{m})$, $\mathbf{m}\in{\mathcal{U}}^{(0)}\setminus\{\underline{\mathbf{m}}\}$, is defined in (1.21), and the last estimate follows from (4.16). Since the sequence $(S_{j})_{j\in\{2,\dots,n_{0}\}}$ is non-increasing, there exists a partition $J_{1}\sqcup\ldots\sqcup J_{p}$ of $\{1,\ldots,n_{0}-1\}$ such that for all $k\in\{1,\ldots,p\}$, there exists $\iota(k)\in\{1,\ldots,n_{0}-1\}$ such that

[TABLE]

Hence, we deduce from (4.28) that

[TABLE]

with

[TABLE]

and

[TABLE]

where, for every $k\in\{1,\dots,p\}$, $r_{k}={\rm card}(J_{k})$. Factorizing by $e^{-\frac{\hat{S}_{\iota(1)}}{h}}$, we get

[TABLE]

with

[TABLE]

Denoting $\tau_{1}=1$ and, for $k\in\{2,\dots,p\}$, $\tau_{k}=e^{\frac{\hat{S}_{\iota(k-1)}-\hat{S}_{\iota(k)}}{2h}}$, we observe that, thanks to (4.29), $\tau_{k}$ is exponentially small when $h\rightarrow 0$. Moreover, with this notation, one has

[TABLE]

This shows that $e^{-\frac{\hat{S}_{\iota(1)}}{h}}\mathcal{M}^{\prime}$ is a graded matrix in the sense of Definition A.1. Hence, we can apply Theorem A.4 and we get that in the limit $h\to 0$,

[TABLE]

where for every $k\in\{1,\dots,p\}$, $\varepsilon_{k}=\prod_{l=1}^{k}\tau_{l}$ and $M_{k}={\rm\,diag\,}(\nu_{j},\,j\in J_{k})$. Moreover, still according to Theorem A.4, $\mathcal{M}^{\prime}$ admits in the limit $h\to 0$, for every $k\in\{1,\dots,p\}$ and every eigenvalue $\lambda$ of $M_{k}$ with multiplicity $r^{\prime}_{k}$, exactly $r^{\prime}_{k}$ eigenvalues counted with multiplicity of order e^{-\frac{\hat{S}_{\iota(1)}}{h}}\varepsilon_{k}^{2}\big{(}\lambda+{\mathcal{O}}(\sqrt{h})\big{)}.

Going back to the initial parameters, one has, for every $k\in\{1,\dots,p\}$,

[TABLE]

Hence, the eigenvalues of $\mathcal{M}^{\prime}$ satisfy:

[TABLE]

which is exactly the announced result.

4.4. Proof of Theorem 1.11

As in the preceding subsection, we denote for shortness

[TABLE]

and we label the local minima $\mathbf{m}_{1},\ldots,\mathbf{m}_{n_{0}}$ of $V$ so that $(S(\mathbf{m}_{j}))_{j\in\{1,\dots,n_{0}\}}$ is non-increasing (see (1.19)):

[TABLE]

Let moreover $\mathbf{m}^{*}\in{\mathcal{U}}^{(0)}\setminus\{\underline{\mathbf{m}}\}$ be such that

[TABLE]

where the prefactors $\zeta(\mathbf{m})$, $\mathbf{m}\in{\mathcal{U}}^{(0)}\setminus\{\underline{\mathbf{m}}\}$, are defined in (1.21), and let us define, for any $h>0$,

[TABLE]

According to the unitary equivalence (see (1.14))

[TABLE]

and to the localization of the spectrum of $P_{\phi}$ stated in Proposition 1.1 and in Theorem 1.3, it holds for every $h>0$ small enough, taking $\epsilon_{0}$ as in the statement of Theorem 1.3,

[TABLE]

where, as in the preceding subsection,

[TABLE]

Moreover, it follows from Proposition 1.1 that $\sigma(P_{\phi})\subset\Gamma_{\Lambda_{0}}\subset\tilde{\Gamma}_{\Lambda_{0}}$ with $\tilde{\Gamma}_{\Lambda_{0}}=\{z\in{\mathbb{C}},\;|\operatorname{Im}(z)|\leq\Lambda_{0}(\operatorname{Re}(z)+1)\}$. Hence, for every $t>0$, the operator $e^{-tL_{V}}(I-\Pi_{h})$ can be written as the complex integral

[TABLE]

where

[TABLE]

and

[TABLE]

From the resolvent estimates proven in Theorem 1.3, it holds $(z-L_{V})^{-1}={\mathcal{O}}(1)$ uniformly on $\Gamma_{0}$, and then, for every $t>0$,

[TABLE]

Using in addition the resolvent estimates proven in Proposition 1.1, it holds $\|(z-L_{V})^{-1}\|\leq\frac{1}{\operatorname{Re}z}\leq\frac{2}{\epsilon_{0}}$ on $\Gamma_{\pm}$, and then

[TABLE]

It follows that for every $t>0$, it holds

[TABLE]

Moreover, $e^{-tL_{V}}\,({\rm Id}-\Pi_{h})={\mathcal{O}}(1)$ since $\Pi_{h}={\mathcal{O}}(1)$ (see (4.21)) and $e^{-tL_{V}}={\mathcal{O}}(1)$ (by maximal accretivity of $L_{V}$). Hence, there exists $C>0$ such that for every $t\geq 0$ and $h>0$ small enough, it holds

[TABLE]

Thus, according to (4.31), it just remains to show that

[TABLE]

To this end, let us first recall from Proposition 1.1 that the spectral projector $\Pi_{\{0\}}$ associated with the eigenvalue [math] of $L_{V}$ has rank $1$ and is actually the orthogonal projector $\Pi_{0}$ on ${\rm Span}\{1\}$ according to the relations

[TABLE]

It follows that

[TABLE]

Since moreover $\Pi_{h}-\Pi_{\{0\}}={\mathcal{O}}(1)$ (thanks to the resolvent estimate of Theorem 1.3), it suffices to show that

[TABLE]

Using the notation of the preceding subsection, this means proving that the matrix $\mathcal{M}^{\prime}$ of $L_{V}$ in the orthonormal basis $(\hat{e}_{1},\ldots,\hat{e}_{n_{0}-1})$ of ${\rm Ran}(\Pi_{h}-\Pi_{0})$ satisfies

[TABLE]

Let us now consider a subset $\mathcal{V}^{(0)}$ (in general non unique) of ${\mathcal{U}}^{(0)}\setminus\{0\}$ such that

[TABLE]

Then, for any $K>0$ and for every $h>0$ small enough, the closed disks of the complex plane

[TABLE]

are included in $\{\operatorname{Re}z>0\}$ and two by two disjoint. Moreover, according to Theorem 1.9, $K>0$ can be chosen large enough so that when $h>0$ is small enough, the $n_{0}-1$ non zero small eigenvalues of $L_{V}$ are included in

[TABLE]

In particular, for every $t\geq 0$ and for every $h>0$ small enough, it holds

[TABLE]

Using now the specific form of $\mathcal{M}^{\prime}$ exhibited in the preceding section and Theorem A.4, it holds for every $\mathbf{m}\in{\mathcal{V}}^{(0)}$, in the limit $h\to 0$,

[TABLE]

Indeed, the resolvent estimate of Theorem A.4 implies

[TABLE]

The relation (4.32) follows easily, which concludes the first part of Theorem 1.11.

Finally, let us assume that the element $\mathbf{m}^{*}$ satisfying (4.30) is unique. In this case, $\mathbf{m}^{*}$ necessarily belongs to ${\mathcal{V}}^{(0)}$ and the associated eigenvalue $\lambda(\mathbf{m}^{*},h)$ (see (1.20)) is then real and simple for every $h>0$ small enough. In particular, it holds

[TABLE]

where $\Pi_{\{\lambda(\mathbf{m}^{*},h)\}}$ is the spectral projector (whose rank is one)

[TABLE]

Moreover, the resolvent estimate (4.33) shows that $\Pi_{\{\lambda(\mathbf{m}^{*},h)\}}={\mathcal{O}}(1)$. Since in addition, it holds in this case (see (1.20))

[TABLE]

for every $K>0$ and for every $h>0$ small enough, we obtain that in the limit $h\to 0$,

[TABLE]

and thus the relation (4.32) remains valid if ones replaces $\lambda(h)-C\sqrt{h}$ there by $\lambda(\mathbf{m}^{*},h)$. This concludes the proof of Theorem 1.11.

Appendix A Some results in linear algebra

The aim of this appendix is to give some handy tools of linear algebra adapted to the setting of non-reversible metastable problems considered in this paper. Let us start with some notations.

Given any matrix $M\in{\mathcal{M}}_{d}({\mathbb{C}})$ and $\lambda\in\sigma(M)$, we denote by $m(\lambda)$ the multiplicity of $\lambda$, $m(\lambda)={\rm\,dim\,}{\rm\,Ker\,}(M-\lambda)^{d}$. We recall that for every $r>0$ small enough,

[TABLE]

where

[TABLE]

We denote by ${\mathscr{D}}_{0}(E)$ the set of complex matrices on a vector space $E$ which are diagonalizable and invertible.

Given two subsets $A,B\subset{\mathbb{C}}$, we say that $A\subset B+{\mathcal{O}}(h)$ if there exists $C>0$ such that $A\subset B+B(0,Ch)$.

Definition A.1**.**

Let ${\mathscr{E}}=(E_{j})_{j=1,\ldots,p}$ be a sequence of finite dimensional vector spaces $E_{j}$ of dimension $r_{j}>0$, let $E=\oplus_{j=1,\ldots,p}E_{j}$ and let $\tau=(\tau_{2},\ldots,\tau_{p})\in({\mathbb{R}}_{+}^{*})^{p-1}$. Suppose that $(h,\tau)\mapsto{\mathcal{M}}_{h}(\tau)$ is a map from $(0,1]\times({\mathbb{R}}_{+}^{*})^{p-1}$ into the set of complex matrices on $E$.

We say that ${\mathcal{M}}_{h}(\tau)$ is an $({\mathscr{E}},\tau,h)$-graded matrix if there exists ${\mathcal{M}}^{\prime}\in{\mathscr{D}}_{0}(E)$ independent of $(h,\tau)$ such that ${\mathcal{M}}_{h}(\tau)=\Omega(\tau)({\mathcal{M}}^{\prime}+{\mathcal{O}}(h))\Omega(\tau)$ with $\Omega(\tau)$ and ${\mathcal{M}}^{\prime}$ such that

• –

${\mathcal{M}}^{\prime}={\rm\,diag\,}(M_{j},\,j=1,\ldots,p)$* with $M_{j}\in{\mathscr{D}}_{0}(E_{j})$,*

• –

$\Omega(\tau)={\rm\,diag\,}(\varepsilon_{j}(\tau)I_{r_{j}},\,j=1,\ldots,p)$* with $\varepsilon_{1}(\tau)=1$ and $\varepsilon_{j}(\tau)=(\prod_{k=2}^{j}\tau_{k})$ for all $j\geq 2$.*

Throughout, we denote by ${\mathscr{G}}({\mathscr{E}},\tau,h)$ the set of $({\mathscr{E}},\tau,h)$-graded matrices.

Lemma A.2**.**

Suppose that ${\mathcal{M}}_{h}(\tau)$ is a family of $({\mathscr{E}},\tau,h)$-graded matrices and that $p\geq 2$. Then, one has

[TABLE]

where

• –

$J(h)=M_{1}+{\mathcal{O}}(h)$* with $M_{1}\in{\mathscr{D}}_{0}(E_{1})$,*

• –

${\mathcal{N}}_{h}(\tau^{\prime})\in{\mathscr{G}}({\mathscr{E}}^{\prime},\tau^{\prime},h)$* with $\tau^{\prime}=(\tau_{3},\ldots,\tau_{p})$ and ${\mathscr{E}}^{\prime}=(E_{j})_{j=2,\ldots,p}$,*

• –

$B^{\pm}_{h}(\tau^{\prime})\in{\mathscr{M}}(E_{1},\oplus_{j=2}^{p}E_{j})$* satisfies*

[TABLE]

with $b^{\pm}_{j}(h):E_{1}\rightarrow E_{j}$ independent of $\tau$ and $b_{j}(h)={\mathcal{O}}(h)$.

Moreover, the matrix ${\mathcal{N}}_{h}(\tau^{\prime})-B^{-}_{h}(\tau^{\prime})J(h)^{-1}B^{+}_{h}(\tau^{\prime})^{*}$ belongs to ${\mathscr{G}}({\mathscr{E}}^{\prime},\tau^{\prime},h)$.

*Proof. *Assume that ${\mathcal{M}}_{h}(\tau)=\Omega(\tau)({\mathcal{M}}^{\prime}+{\mathcal{O}}(h))\Omega(\tau)$ with $\Omega(\tau)$ and ${\mathcal{M}}^{\prime}$ as in Definition A.1. First observe that

[TABLE]

with

[TABLE]

On the other hand, we can write

[TABLE]

where $J(h)=M_{1}+{\mathcal{O}}(h)$ with $M_{1}\in{\mathscr{D}}_{0}(E_{1})$, $B^{\prime\pm}(h)={\mathcal{O}}(h)$, and ${\mathcal{N}}^{\prime}(h)={\mathcal{N}}^{\prime}_{0}+{\mathcal{O}}(h)$ with ${\mathcal{N}}^{\prime}_{0}={\rm\,diag\,}(M_{j},\,j=2,\ldots,p)$. Therefore,

[TABLE]

has exactly the form (A.2) with $B^{\pm}_{h}(\tau^{\prime})=\Omega^{\prime}(\tau^{\prime})B^{\prime\pm}(h)$ and ${\mathcal{N}}_{h}(\tau^{\prime})=\Omega^{\prime}(\tau^{\prime}){\mathcal{N}}^{\prime}(h)\Omega^{\prime}(\tau^{\prime})$. By construction, ${\mathcal{N}}_{h}(\tau^{\prime})$ belongs to ${\mathscr{G}}({\mathscr{E}}^{\prime},\tau^{\prime},h)$ and $B^{\pm}_{h}(\tau^{\prime})$ has the required form. $\square$

Lemma A.3**.**

Let $M$ be a complex diagonalizable matrix. Then there exists $C>0$ such that

[TABLE]

*Proof. *Let $P$ be an invertible matrix such that $PMP^{-1}=D$ is diagonal. Then

[TABLE]

$\square$

The following theorem gives precise informations on the spectrum of graded matrices as introduced above. The proof is based on standard arguments, namely on the Schur complement method and complex analysis. The use of these two tools permits to work by induction and to decompose the base vector space in order to isolate eigenspaces corresponding to eigenvalues of the same order and to see the remainder of the matrix as a perturbation. Similar arguments were used in [21] in a self-adjoint framework. We believe that this result could be useful in other contexts where the computation of clouds of eigenvalues cannot be carried out by standard self-adjoint arguments.

Theorem A.4**.**

Suppose that ${\mathcal{M}}_{h}(\tau)$ is $({\mathscr{E}},\tau,h)$-graded. Then, there exists $\tilde{\tau}_{0},h_{0}>0$ such that for all $0<\tau_{j}\leq\tilde{\tau}_{0}$ and all $h\in(0,h_{0}]$, one has

[TABLE]

*Moreover, for any eigenvalue $\lambda$ of $M_{j}$ with multiplicity $m_{j}(\lambda)$, there exists $K>0$ such that, denoting $D_{j}:=\{z\in{\mathbb{C}},\;|z-\varepsilon_{j}(\tau)^{2}\lambda_{j}|<K\varepsilon_{j}(\tau)^{2}h\}$, one has *

[TABLE]

where $n(D_{j};{\mathcal{M}}_{h}(\tau))$ is defined by (A.1). Moreover, there exists $C>0$ such that

[TABLE]

for all $z\in{\mathbb{C}}\setminus\cup_{j=1}^{p}\cup_{\lambda\in\sigma(M_{j})}B(\varepsilon_{j}(\tau)^{2}\lambda,\varepsilon_{j}(\tau)^{2}Kh)$.

*Proof. *We prove the theorem by induction on $p$. Throughout the proof the notation ${\mathcal{O}}(\cdot)$ is uniform with respect to the parameters $h$ and $\tau$. For $p=1$, one has ${\mathcal{M}}_{h}(\tau)=M_{1}+{\mathcal{O}}(h)$ with $M_{1}\in{\mathcal{M}}_{r_{1}}({\mathbb{R}})$ independent of $h$, diagonalizable and invertible. Let us denote $\lambda_{j}^{1}$, $j=1,\ldots,n_{1}$ its eigenvalues and $m_{j}=m(\lambda_{j}^{1})$ the corresponding multiplicities. The function $z\mapsto({\mathcal{M}}_{h}-z)^{-1}$ is meromorphic on ${\mathbb{C}}$ with poles in $\sigma({\mathcal{M}}_{h})$. Moreover, Lemma A.3 and the identity

[TABLE]

show that for any $C>0$ large enough, $({\mathcal{M}}_{h}-z)$ is invertible on ${\mathbb{C}}\setminus\cup_{j=1}^{n_{1}}D(\lambda_{j}^{1},Ch)$ with $\|(M_{1}-z)^{-1}\|={\mathcal{O}}(\frac{1}{Ch})$ and

[TABLE]

Hence, for every $C>0$ large enough, the associated spectral projector writes

[TABLE]

This implies that for $C>0$ large enough,

[TABLE]

which is exactly (A.3). As a consequence

[TABLE]

is maximal and hence $\sigma({\mathcal{M}}_{h})\subset\cup_{j=1}^{n_{1}}D(\lambda_{j}^{1},Ch)$. Eventually, (A.4) shows that for any $z\in{\mathbb{C}}\setminus\cup_{j=1}^{n_{1}}D(\lambda_{j}^{1},Ch)$, one has

[TABLE]

for some constant $C^{\prime}>0$. Using Lemma A.3 we get

[TABLE]

for all $z\in{\mathbb{C}}\setminus\cup_{j=1}^{n_{1}}D(\lambda_{j}^{1},2Ch)$. This completes the initialization step.

Suppose now that $p\geq 2$ and let ${\mathcal{M}}_{h}(\tau)\in{\mathscr{G}}({\mathscr{E}},\tau,h)$. We have

[TABLE]

with $J(h),B^{\pm}_{h}(\tau^{\prime})$ and ${\mathcal{N}}_{h}(\tau^{\prime})$ as in Lemma A.2. In order to lighten the notation, we will drop the variables $\tau,\tau^{\prime}$ in the proof below. For $\lambda\in{\mathbb{C}}$, let

[TABLE]

This is an holomorphic function, and since it is non trivial, its inverse is well defined excepted for a finite number of values of $\lambda$ which are exactly the spectral values of ${\mathcal{M}}_{h}$.

We first study the part of the spectrum of ${\mathcal{M}}_{h}$ which is of largest modulus. Let $\lambda_{n}^{1}$, $n=1,\ldots,n_{1}$, denote the eigenvalues of the matrix $M_{1}$. Since $J(h)=M_{1}+{\mathcal{O}}(h)$ and $M_{1}\in{\mathscr{D}}_{0}(E_{1})$, then the initialization step shows that there exists $C>0$ such that $\sigma(J(h))\subset\cup_{n=1}^{n_{1}}D(\lambda_{n}^{1},Ch)$. Moreover, since $M_{1}$ is invertible, there exists $c_{1},d_{1}>0$ and $h_{0}>0$ such that for all $n=1,\ldots,n_{1}$, one has $\lambda_{n}^{1}\in K(c_{1},d_{1})$ where $K(c_{1},d_{1})=\{z\in{\mathbb{C}},\;c_{1}\leq|z|\leq d_{1}\}$. Let $n\in\{1,\ldots,n_{1}\}$ be fixed and consider $D_{n}=D_{n}(h)=\{z\in{\mathbb{C}},\;|z-\lambda^{1}_{n}|\leq Mh\}$ for some $M>C>0$ and $\tilde{D}_{n}=\{z\in{\mathbb{C}},\;|z-\lambda^{1}_{n}|\leq 2Mh\}$. Observe that for $h>0$ small enough, the disks $\tilde{D}_{n}$ are disjoint. By definition, one has ${\mathcal{N}}_{h}(\tau^{\prime})={\mathcal{O}}(1)$ and since $|\lambda|\geq c_{1}-{\mathcal{O}}(h)\geq c_{1}/2$, this implies that for $\tau_{2}>0$ small enough with respect to $c_{1}$ and $\lambda\in\tilde{D}_{n}$, the matrix $\tau_{2}^{2}{\mathcal{N}}_{h}(\tau^{\prime})-\lambda$ is invertible, and $(\tau_{2}^{2}{\mathcal{N}}_{h}(\tau^{\prime})-\lambda)^{-1}={\mathcal{O}}(1)$. Moreover, it follows from the initialization step that for $\lambda\in\tilde{D}_{n}\setminus D_{n}$, $J(h)-\lambda$ is invertible and

[TABLE]

Combined with the fact that $B^{\pm}_{h}={\mathcal{O}}(h)$, this implies that for $h>0$ small enough and $\lambda\in\tilde{D}_{n}\setminus D_{n}$, $J(h)-\lambda-\tau_{2}^{2}B_{h}^{+,*}(\tau_{2}^{2}{\mathcal{N}}_{h}-\lambda)^{-1}B^{-}_{h}$ is invertible with

[TABLE]

Hence, the standard Schur complement procedure shows that for $\lambda\in\tilde{D}_{n}\setminus D_{n}$, ${\mathcal{P}}(\lambda)$ is invertible with inverse ${\mathcal{E}}(\lambda)$ given by

[TABLE]

with

[TABLE]

and

[TABLE]

Let us now consider the spectral projector $\Pi_{D_{n}}({\mathcal{M}}_{h})$. Then,

[TABLE]

where we defined

[TABLE]

On the other hand, an elementary computation shows that

[TABLE]

with

[TABLE]

where the last equality follows from (A.6). It follows that for $h>0$ small enough, the rank of $E_{n}$ is bounded from below by the multiplicity $m(\lambda_{n}^{1})$ of $\lambda_{n}^{1}$ and hence

[TABLE]

for all $n=1,\ldots,n_{1}$.

Let us now study the part of the spectrum of order smaller than $\tau_{2}^{2}$. Thanks to the last part of Lemma A.2, the matrix ${\mathcal{Z}}_{h}(\tau^{\prime}):={\mathcal{N}}_{h}-B_{h}^{-}J(h)^{-1}B_{h}^{+,*}$ is classical $({\mathscr{E}}^{\prime},\tau^{\prime})$-graded. Hence, it follows from the induction hypothesis that uniformly with respect to $h$, one has

[TABLE]

with $\tilde{\varepsilon}_{j}=\tau_{2}^{-1}\varepsilon_{j}=\prod_{l=3}^{j}\tau_{l}$ for $j\geq 3$ and $\tilde{\varepsilon}_{2}=1$. One also knows that for all $j=2,\ldots,p$ and all $\lambda\in\sigma(M_{j})$, one has

[TABLE]

where $D_{j}=D(\lambda\tilde{\varepsilon}_{j}^{2},Kh\tilde{\varepsilon}_{j}^{2})$ for some $K>0$. Moreover, one has for all $z\notin\cup_{j=2}^{p}\cup_{\lambda\in\sigma(M_{j})}D(\lambda\tilde{\varepsilon}_{j}^{2},Kh\tilde{\varepsilon}_{j}^{2})$ the resolvent estimate

[TABLE]

For $j=2,\ldots,p$, let $\lambda_{1}^{j},\ldots,\lambda_{n_{j}}^{j}$ denote the eigenvalues of the matrix $M_{j}\in{\mathcal{D}}_{0}$. As above, there exists $c_{j},d_{j}>0$ such that $\lambda_{n}^{j}\in K(c_{j},d_{j})$ for all $n=1,\ldots,n_{j}$. Suppose now that $j\in\{2,\ldots,p\}$ and $n\in\{1,\ldots,n_{j}\}$ are fixed and consider, for $M>K$,

[TABLE]

Since $M_{1}$ is invertible, $J(h)-\lambda$ is invertible and $(J(h)-\lambda)^{-1}=\mathcal{O}(1)$ for $\lambda$ in $D^{\prime}_{j,n}$ and $h,\tau_{2}$ small enough. Moreover, for any $\lambda\in\partial D^{\prime}_{j,n}$, it holds, noting $\lambda^{\prime}=\tau_{2}^{-2}\lambda$,

[TABLE]

Hence, according to the relations (A.9), (A.10), and to $\varepsilon_{j}=\tau_{2}\tilde{\varepsilon}_{j}$, it holds

[TABLE]

The latter operator is then invertible around $\partial D^{\prime}_{j,n}$ for $h,\tau_{2}$ small enough, and the Schur complement formula then permits to write the inverse of ${\mathcal{P}}(\lambda)$ as

[TABLE]

with

[TABLE]

and

[TABLE]

As above, let us consider the corresponding projector $\Pi_{D^{\prime}_{j,n}}({\mathcal{M}}_{h})$. From $\lambda=\tau_{2}^{2}\lambda^{\prime}$, we get

[TABLE]

with $\hat{D}^{\prime}_{j,n}=\{z^{\prime}\in{\mathbb{C}},\;|z^{\prime}-\tilde{\varepsilon}_{j}^{2}\lambda^{j}_{n}|\leq Mh\tilde{\varepsilon}_{j}^{2}\}$. It follows moreover from (A.11) that for every $\lambda^{\prime}\in\partial\hat{D}^{\prime}_{j,n}$ and $h,\tau_{2}$ small enough,

[TABLE]

and the same argument as above shows that ${\rm\,rank\,}(\Pi_{D^{\prime}_{j,n}}({\mathcal{M}}_{h}))\geq{\rm\,rank\,}(E^{\prime}_{n})$ with

[TABLE]

By the induction hypothesis, this shows that for $h$ small enough, the rank of $E^{\prime}_{n}$ is exactly the multiplicity of $\lambda_{n}^{j}$ and hence

[TABLE]

for all $j=2,\ldots,p$ and $n=1,\ldots,n_{j}$. Combined with (A.8), this shows that for all $j=1,\ldots,p$ and $n=1,\ldots,n_{j}$, one has

[TABLE]

with $D_{j,n}=\varepsilon^{2}_{j}D(\lambda_{n}^{j},Mh)$. Since $\sum_{j,n}m(\lambda_{n}^{j})$ is equal to the total dimension of the space, this implies that

[TABLE]

which proves the localization of the spectrum and (A.3).

It remains to prove the resolvent estimate. Suppose that $\lambda\in{\mathbb{C}}$ is such that $\lambda\notin\cup_{j=1}^{p}\cup_{\mu\in\sigma(M_{j})}D(\varepsilon^{2}_{j}(\tau)\mu,\varepsilon^{2}_{j}(\tau)Kh)$. We suppose first that $|\lambda|\geq c_{0}$ for $c_{0}>0$ such that $|\lambda_{n}^{1}|\geq 2c_{0}$ for all $n=1,\ldots,n_{1}$. Then ${\mathcal{P}}(\lambda)={\mathcal{M}}_{h}(\tau)-\lambda$ is invertible with inverse ${\mathcal{E}}(\lambda)$ given by (A.7). Using (A.6) it is clear that $E(\lambda)={\mathcal{O}}(h^{-1})={\mathcal{O}}({\rm\,dist\,}(\lambda,\sigma({\mathcal{M}}_{h}(\tau))^{-1})$. On the other hand, since $(\tau_{2}^{2}{\mathcal{N}}_{h}-\lambda)^{-1}={\mathcal{O}}(1)$ and $B_{h}^{\pm}={\mathcal{O}}(h)$ we have also $E_{0}(\lambda)={\mathcal{O}}(1)$ and then ${\mathcal{E}}(\lambda)={\mathcal{O}}({\rm\,dist\,}(\lambda,\sigma({\mathcal{M}}_{h}(\tau)))^{-1})$.

Suppose now that $|\lambda|\leq c_{0}$. Then ${\mathcal{P}}(\lambda)={\mathcal{M}}_{h}(\tau)-\lambda$ is invertible with inverse ${\mathcal{E}}(\lambda)$ given by (A.12). Setting $\lambda^{\prime}=\tau_{2}^{-2}\lambda$ one deduces from (A.13) and from (A.9),(A.10) that

[TABLE]

This completes the proof. $\square$