Virginie Bonnaillie-Noël

Frédéric Hérau
email: herau@univ-nantes.fr

Nicolas Raymond
email: nicolas.raymond@univ-angers.fr

PURELY MAGNETIC TUNNELING EFFECT IN TWO DIMENSIONS

The semiclassical magnetic Neumann Schrödinger operator on a smooth, bounded, and simply connected domain Ω of the Euclidean plane is considered. When Ω has a symmetry axis, the semiclassical splitting of the first two eigenvalues is analyzed. The first explicit tunneling formula in a pure magnetic field is established. The analysis is based on a pseudo-differential reduction to the boundary and the proof of the first known optimal purely magnetic Agmon estimates.

defined for ψ ∈ H 1 A (Ω) ⊂ L 2 (Ω), the set for which Q h (ψ) is finite. In this article, the magnetic field is B = ∇ × A = 1 and, by gauge invariance, we can choose A = (0, -x 1 ). The domain of L h is

Dom(L h ) = ψ ∈ H 1 A (Ω) : (-ih∇ -A) 2 ψ ∈ L 2 (Ω) , n • (-ih∇ -A)ψ = 0 on Γ = ∂Ω ,
where n is the outward pointing normal to the boundary. In this paper, L will denote the half-length of the boundary.

N. R. and F. H. are deeply grateful to the Mittag-Leffler Institute where part of the ideas of this article were discussed. N. R. also thanks Bernard Helffer, Pierig Keraval and Johannes Sjöstrand for many stimulating discussions.

1.1.2. From superconductivity to semiclassical analysis. The original motivation to study the spectrum of L h is the mathematical study of superconductivity. In particular, the asymptotic description of the third critical field (in the large magnetic field limit) is related to the groundstate energy of L h . For an overview of this vast subject, the reader is referred to the book [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]. Independently of superconductivity, the subject has acquired a life of its own (see the book [START_REF] Raymond | Bound states of the magnetic Schrödinger operator[END_REF]). Let us only point out some contributions directly related to the present framework. In [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF], the ground state energy is analyzed and the following asymptotic formula is established

λ 1 (h) = Θ 0 h -C 1 κ max h 3 2 + o(h 3 2 ) , (1.1)
where κ max is the maximum of the curvature of Γ, and Θ 0 ∈ (0, 1) and C 1 > 0 are related to the de Gennes operator (see [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF]Appendix A]). This operator is defined as follows. Consider, for all ξ ∈ R, L ξ the Neumann realization on R + of the operator D 2 t + (ξ -t) 2 . The eigenvalues of L ξ are simple and denoted by (µ n (ξ)) n 1 . It is known (see [START_REF] Dauge | Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators[END_REF]) that µ 1 has a unique and non-degenerate minimum at some ξ 0 > 0. We will denote by u ξ the positive L 2 -normalized ground state. Then,

Θ 0 = min ξ∈R µ 1 (ξ) , C 1 = u 2 ξ 0 (0) 6 . (1.2)
In relation with (1.1), Helffer and Morame also proved that the first eigenfunctions are somehow localized near the boundary points of maximal curvature (see [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF]Theorem 10.6] and the numerical simulation of the ground state when Ω is an ellipse, Figure 1). In contrast with [START_REF] Helffer | Magnetic bottles in connection with superconductivity[END_REF] where only the groundstate energy is considered, in [START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF], all the low lying eigenvalues are considered in the semiclassical limit when the curvature has a unique and non-degenerate minimum. Fournais and Helffer establish that, for all n 1, 4 ) ,

λ n (h) = Θ 0 h -C 1 κ max h 3 2 + (2n -1)C 1 Θ 1 4 0 3k 2 2 h 7 4 + o(h 7
with k 2 = -κ (s 0 ) where κ is the curvature as a function of the curvilinear coordinate and s 0 the point of maximal curvature.

1.1.3. Magnetic WKB constructions. In relation with (1.3), we may wonder how the corresponding eigenfunctions behave and if we can accurately describe them in the semiclassical limit. It has been an open question for many years to know if the eigenfunctions could be written in a WKB form. A positive and very explicit answer has been given in [START_REF] Bonnaillie-Noël | Magnetic WKB constructions[END_REF] (see also Section 2.4.2 where we recall the result). It turned out that the magnetic operator is deeply connected to an effective electric operator acting on the boundary. Letting the analysis there revealed the crucial role of the following effective eikonal equation v(s) -µ 1 (ξ 0 ) 2 ϕ (s) 2 = 0 .

v(s) = C 1 (κ max -κ(s)) 0 ,
(1.4)

1.1.4. An effective eikonal equation. The remarkable feature of the aforementionned WKB analysis is that the eikonal equation (1.4) is the same, up to a local change of gauge, as the one obtained when considering the following purely electric Hamiltonian acting on L 2 (R/(2LZ)),

L eff h = µ 1 (ξ 0 ) 2 h 1 2 D 2 s + V (s) , V (s) = 2v(s) µ 1 (ξ 0 ) .
Let us denote by (λ eff n (h)) n 1 the sequence of its eigenvalues. If v has exactly two symmetric non-degenerate minima at s r ∈ (-L, 0) and s ∈ (0, L), it is well-known that the low lying spectrum is made of exponentially close pairs of eigenvalues. In order to describe the corresponding tunneling formula, we consider

S = min (S u , S d ) , S u = [sr,s ] V (s) ds , S d = [s ,sr] V (s) ds , (1.5)
where [p, q] denotes the arc joining p and q in the "circle" R/(2LZ) counterclockwise. The indices u and d refer to the up and down parts of the "circle" (corresponding to the up and down parts of ∂Ω).

The tunneling formula is

λ eff 2 (h) -λ eff 1 (h) = 2|w(h)| + O(h 3 8 e -S/h 1/4 ) , (1.6)
where

w(h) = µ 1 (ξ 0 )h 1 8 π -1 2 g 1 2
A u V (0)e -Su/h 1/4 + A d V (L)e -S d /h 1/4 , (

with

A u = exp - [sr,0] (V 1
2 ) (s) + g V (s) ds ,

A d = exp - [s ,L] (V 1 2 ) (s) -g V (s) ds , g = (V (s r )/2) 1 2 = (V (s )/2) 1 2 .
(1.8)

Such a one dimensional result goes back to [START_REF] Harrell | Double wells[END_REF]. This formula may also be found in [START_REF] Bonnaillie-Noël | Semiclassical tunneling and magnetic flux effects on the circle[END_REF] up to a convenient rescaling. The reader might also want to consider the Bourbaki exposé [START_REF] Robert | Analyse semi-classique de l'effet tunnel[END_REF] based on the celebrated Helffer-Sjöstrand theory developped in [START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF][START_REF] Helffer | Puits multiples en limite semi-classique. II. Interaction moléculaire[END_REF][START_REF] Helffer | Multiple wells in the semiclassical limit. III. Interaction through nonresonant wells[END_REF][START_REF] Helffer | Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten[END_REF][START_REF] Helffer | Puits multiples en mécanique semi-classique. V. Étude des minipuits[END_REF][START_REF] Helffer | Puits multiples en mécanique semi-classique. VI. Cas des puits sous-variétés[END_REF][START_REF] Helffer | Effet tunnel pour l'équation de Schrödinger avec champ magnétique[END_REF] (see also the series of works by Simon [START_REF] Simon | Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions[END_REF][START_REF] Simon | Semiclassical analysis of low lying eigenvalues[END_REF][START_REF] Simon | Semiclassical analysis of low lying eigenvalues. III. Width of the ground state band in strongly coupled solids[END_REF][START_REF] Simon | Semiclassical analysis of low lying eigenvalues. IV. The flea on the elephant[END_REF]). In a periodic framework, flux effects are considered in [START_REF] Outassourt | Comportement semi-classique pour l'opérateur de Schrödinger à potentiel périodique[END_REF] (see also [START_REF] Bonnaillie-Noël | Semiclassical tunneling and magnetic flux effects on the circle[END_REF]).

1.1.5. Numerical simulations and conjecture. More than a decade ago, the first numerical simulations describing magnetic tunneling effects in two dimensions appeared (see for instance [START_REF] Bonnaillie-Noël | Computations of the first eigenpairs for the Schrödinger operator with magnetic field[END_REF] in the case of corner domains). For instance, in the case of the ellipse (see Figure 2), it was rather a surprise to be able to estimate an exponentially small effect and also to reveal the "oscillation" of λ 2 (h) -λ 1 (h), numerically. For more numerical simulations concerning smooth domains with symmetries, the reader may consult [START_REF] Bonnaillie-Noël | Magnetic WKB constructions[END_REF]Section 5.3.3] where "camels" (see Figure 3) and ellipses are considered. The case of varying (and vanishing) magnetic fields is also investigated.

Figure 3. Modulus and phase of the groundstate in a camel-like domain

Based on the WKB analysis in pure magnetic fields and the ideas à la Born-Oppenheimer developped in [START_REF] Bonnaillie-Noël | Magnetic WKB constructions[END_REF], we end up with the conjecture [2, Conjecture 1.4] of an explicit formula to describe a purely magnetic tunneling when Ω is an ellipse. This conjecture has been numerically checked (see Figure 4) and, to the authors' knowledge, is the first of its kind.

Let us recall this conjecture.

Conjecture 1.1. Assume that Ω is an ellipse. Then, there exists

α 0 ∈ R such that λ 2 (h) -λ 1 (h) = →0 h 13 8 A 2 5 2 C 3 4 1 √ π (k 2 µ 1 (ξ 0 )) 1 4 (κ max -κ min ) 1 2 × cos L γ 0 h - ξ 0 h 1 2 -α 0 e -S/h 1 4 + o(h 13
8 )e -S/h 1 4 ,

where . Ω is a smooth, open, bounded, connected, and simply-connected set of the plane. Moreover, it is assumed that Ω is symmetric and that the curvature has two non-degenerate maxima: i) Ω is symmetric with respect to the y-axis.

S = 2C 1 µ 1 (ξ 0 ) L 2 -L 2 κ max -κ(s) ds , A = exp   - [ L 2 ,L] ∂ s κ max -κ(s) -k 2 2 κ max -κ(s) ds   , γ 0 = |Ω| |Γ| = |Ω| 2L , k 2 = -κ L 2 .
ii) The curvature κ on the boundary Γ attains its maximum at exactly two points a 1 and a 2 which are not on the symmetry axis and belong to the same connected component of the boundary. We write

a 1 = (a 1,1 , a 1,2 ) ∈ Γ and a 2 = (a 2,1 , a 2,2 ) ∈ Γ , such that a 1,1 > 0 and a 2,1 < 0 . iii)
The second derivative of the curvature (w.r.t. arc-length) at a 1 and a 2 is negative.

We can now state the main theorem of this article, which gives, to the authors' knowledge, the first optimal purely magnetic tunneling estimate.

Theorem 1.3. Under Assumption 1.2, we have the tunneling formula

λ 2 (h) -λ 1 (h) = 2| w(h)| + o(h 13 8 e -S/h 1 4 ) , where w(h) = µ 1 (ξ 0 )h 13 8 π -1 2 g 1 2 A u V (0)e -Su/h 1/4 e iLf (h) + A d V (L)e -S d /h 1/4 e -iLf (h) , where V (s) = 2C 1 (κmax-κ(s)) µ 1 (ξ 0 ) and i. f (h) = γ 0 /h -ξ 0 /h 1/2 -α 0 ,
ii. α 0 is a constant involving the de Gennes operator and the geometry (see (2.15)), iii. A u , A d and g are defined in (1.8).

Remark 1.4. Let us make some remarks about Theorem 1.3. The proof actually allows to consider slightly more general situations. i) Theorem 1.3 implies Conjecture 1.1. In the case of the ellipse, we have s r = -s = -L 2 and κ(0) = κ(L) = κ min . Moreover, due the additionnal symmetry with respect to the horizontal axis, we have A u = A d (see (1.8)) and S u = S d (see (1.5)). This additionnal symmetry is thus responsible for the presence of the cosine in Conjecture 1.1.

ii) The assumption that Ω is bounded is not necessary to establish a tunneling result. Our strategy also applies to deal with camel-like domains (see Figure 3). In this simpler case, the "down" part in the tunneling formula has to be removed. Then, there is only one interaction term and no global flux effects.

In particular, no oscillation of λ 2 (h) -λ 1 (h) occurs. iii) The assumption that Ω is simply-connected is not necessary. The possible holes only contribute to change the value of γ 0 .

iv) The fact that we consider the first two eigenvalues, or only a domain with only one symmetry, is just for the simplicity of the presentation. The same strategy provides us with tunneling estimates in multiple well situations since our method reduces the analysis to one dimension electric tunneling (up to phase shifts). v) In [START_REF] Simon | Semiclassical analysis of low lying eigenvalues. IV. The flea on the elephant[END_REF], Simon described the "flea on the elephant effect". This effect occurs when the electric potential is slightly perturbed and/or when the symmetry is broken. In this case, the first two eigenfunctions end up living in separate wells. In our case, such a phenomenon could be described as well (if we perturb the geometry of the boundary). In the special case of the ellipse, the oscillating effect is due to the existence of two minimal geodesics connecting the two curvature wells: If we slightly perturb the boundary (by keeping the symmetry) in such a way that S u = S d , this kills one of the minimal geodesics and the beautiful oscillating effect disappears.

Remark 1.5. The investigation will reveal the microlocal nature of the tunelling estimate given in Theorem 1.3. It contrasts with the electric tunneling à la Helffer-Sjöstrand, and even with recent contributions about purely geometric tunneling [START_REF] Helffer | Tunneling for the Robin Laplacian in smooth planar domains[END_REF] and [START_REF] Kachmar | Tunnel effect in a shrinking shell enlacing a magnetic field[END_REF] where microlocal analysis is absent.

1.3. Organization and strategy. In Section 2, we explain how the spectral analysis of L h can be reduced to the one of an operator L h,δ on a tubular neighborhood of the boundary, see Proposition 2.2. Then, L h,δ is written in the classical tubular coordinates (s, t) ∈ R/(2LZ) × (0, δ) and rescaled in the transverse variable t = τ , with = h 1 2 . The spectral analysis is then reduced to the one of N , see Proposition 2.7.

In Section 3, we consider a "one well problem" by removing the left maximum and gluing an infinite strip. Then, the resulting operator N ,r can be interpreted as a pseudo-differential operator with operator valued symbol the principal symbol of which being the de Gennes operator. Such operators and their spectrum have been extensively studied by Martinez via Grushin reductions. A concise presentation can be found in [START_REF] Martinez | A general effective Hamiltonian method[END_REF]. More details and extensions may also be found in the Ph. D. thesis of Keraval [START_REF]Formules de Weyl par réduction de dimension[END_REF]. To some extent, our presentation will be similar to [START_REF] Martinez | Développements asymptotiques et effet tunnel dans l'approximation de Born-Oppenheimer[END_REF] where tunneling estimates are provided in the case of partially semiclassical electric operators. In order to construct a parametrix of N ,r 1 , one will need a convenient symbol class, see Notation 3.2. For that purpose, we will use a microlocal cutoff function and construct a parametrix for the "microlocalized" operator Op W p (near ξ 0 ), see Theorem 3.5.

In Section 4, we use the parametrix to show that tangential elliptic estimates for N ϕ ,r may be deduced from the one of an effective pseudo-differential operator acting on the boundary, see Theorem 4.2. , where ϕ is an appropriate subsolution of the effective eikonal equation.

In Section 5, we establish Theorem 5.1. It is devoted to remove the frequency cutoff function introduced in Section 3.1 up to using the transverse Agmon estimates, and the behavior at infinity of the de Gennes function µ 1 .

In Section 6, we explain how to deduce optimal tangential Agmon estimates from Theorem 5.1 (see Corollary 6.1). We also establish slightly rougher tangential estimates for the "double well operator" N from the one well estimates, see Proposition 6.2.

Section 7 is devoted to the proof of Theorem 1.3. We construct an approximate basis from the WKB Ansätze attached to each curvature well and compute the spectrum of the interaction matrix thanks to the accurate WKB approximation of the ground state in each simple well.

2.

A reduction to a tubular neighborhood of the boundary 2.1. Normal Agmon estimates and spectral consequence. The following proposition is well-known (see [START_REF] Fournais | Accurate eigenvalue asymptotics for the magnetic Neumann Laplacian[END_REF]Theorem 4.1]). It comes from the fact that the magnetic Laplacian on Ω with Dirichlet boundary condition is bounded from below by h since

∀ψ ∈ C ∞ 0 (Ω) , Ω |(-ih∇ -A)ψ| 2 dx h Ω |ψ| 2 dx . Proposition 2.1. Let M > 0.
There exist C, h 0 , α > 0 such that, for all h ∈ (0, h 0 ), and all eigenpairs (λ, ψ) of

L h with λ Θ 0 h + M h 3 2 , Ω e 2αdist(x,Γ)/h 1 2 |ψ| 2 dx C ψ 2 ,
and

Ω e 2αdist(x,Γ)/h 1 2 |(-ih∇ -A)ψ| 2 dx Ch ψ 2 .
This proposition tells us that the first eigenfunctions of L h are exponentially localized in a neighbrohood of size h 1 2 of Γ. This invites us to define the new operator L h,δ . Consider the (possibly h-dependent) δ-neighborhood of the boundary

Ω δ = {x ∈ Ω : dist(x, Γ) < δ} .
(The dependence of δ w.r.t. h will be precised later.) Then, consider L h,δ the self-adjoint realization of (-ih∇ -A) 2 with the following boundary conditions n • (-ih∇ -A)ψ = 0 , on Γ , and ψ = 0 , on {x ∈ Ω : dist(x, Γ) = δ} , where δ < δ 0 with δ 0 small enough to ensure the smoothness of the boundary of

Ω δ . The quadratic form Q h,δ associated with L h,δ is defined for all ψ ∈ V δ , Q h,δ (ψ) = Ω δ |(-ih∇ -A)ψ| 2 dx , with V δ = {ψ ∈ H 1 (Ω δ ) : ψ(x) = 0 , on {x ∈ Ω : dist(x, Γ) = δ}} .
The operator L h,δ still has a compact resolvent and we can consider the nondecreasing sequence of its eigenvalues (λ n (h, δ)) n 1 repeated according to their multiplicity.

Proposition 2.2. Let n 1. There exist C, h 0 , α > 0 such that, for all h ∈ (0, h 0 ) and δ ∈ (0, δ 0 ),

λ n (h) λ n (h, δ) λ n (h) + Ce -αδ/h 1 2 .
Proof. The first inequality follows from the fact that Ω δ ⊂ Ω, the Dirichlet condition and the min-max principle. The second inequality follows from the Agmon estimates. Indeed, consider an orthonomal family of eigenfunctions (ψ j ) 1 j n associated with (λ j (h)) 1 j n and let

E n (h, δ) = span 1 j n χ δ ψ j .
Here χ δ is defined by

χ δ (x) = χ dist(x,Γ) δ where χ is a smooth function such that χ(x) = 1 for x ∈ [0, 1/2) and χ(x) = 0 for x 1. Thus, E n (h, δ) ⊂ V δ . Consider ψ ∈ E n (h, δ) and write ψ = χ δ ψ = χ δ n j=1 β j ψ j .
We have

Q h,δ (χ δ ψ) = Ω |χ δ (-ih∇ -A)ψ -ihψ∇χ δ | 2 dx (-ih∇ -A)ψ 2 + 2h (-ih∇ -A)ψ L 2 (Ω\Ω δ/2 ) ψ∇χ δ + h 2 ψ∇χ δ 2 .
Then, since the (ψ j ) 1 j n are orthogonal eigenfunctions, we get

(-ih∇ -A)ψ 2 λ n (h) ψ 2 .
From Proposition 2.1, we have

ψ∇χ δ Cδ -1 e -αδ/2h 1 2 ψ , (-ih∇ -A)ψ L 2 (Ω\Ω δ/2 ) Ch 1 2 e -αδ/2h 1 2 ψ .

It follows that

Q h,δ (χ δ ψ) λ n (h) + C(h 3 2 δ -1 + h 2 δ -2 )e -αδ/h 1 2 ψ 2 ,
and then

Q h,δ (χ δ ψ) λ n (h) + C(h + h 3 2 δ -1 + h 2 δ -2 )e -αδ/h 1 2 χ δ ψ 2 .
2.2. Tubular coordinates and truncated operator. We will use the canonical tubular coordinates (s, t) where s is the arc-length and t the distance to the boundary. We recall some elementary properties of these coordinates. Let

(-L, L] s → M (s) ∈ Γ (2.1)
be a parametrization of Γ. The unit tangent vector of Γ at the point M (s) of the boundary is given by T (s) := M (s). We define the curvature κ(s) by the following identity

T (s) = -κ(s) n(s),
where n(s) is the unit vector, normal to the boundary, pointing outward at the point M (s). We choose the orientation of the parametrization M to be counterclockwise, so det(T (s), n(s)) = 1, ∀s ∈ (-L, L]. We introduce the change of coordinates

Φ : R/((2L)Z) × (0, δ) (s, t) → x = M (s) -t n(s) ∈ Ω δ .
(2.

2)

The determinant of the Jacobian of Φ is given by

m(s, t) = 1 -tκ(s). (2.3)
Thanks to this change of coordinates, L h,δ is unitarily equivalent to M h,δ the self-adjoint realization on L 2 (Γ × (0, δ), mdsdt), of the differential operator

-h 2 m -1 ∂ t m∂ t + m -1 -ih∂ s + γ 0 -t + κ 2 t 2 m -1 -ih∂ s + γ 0 -t + κ 2 t 2 , where m(s, t) = 1 -tκ(s) , γ 0 = |Ω| |Γ| ,
with the boundary conditions

∂ t ψ(s, 0) = 0 , ψ(s, δ) = 0 .
This fact can be found in [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]Appendix F]. The first eigenfunctions of M h,δ also satisfy Agmon estimates (with respect to t).

Proposition 2.3. Let M > 0. There exist C, h 0 , α > 0 such that, for all h ∈ (0, h 0 ), and all eigenpair (λ,

ψ) of M h,δ with λ Θ 0 h + M h 3 2 , Ω e 2αt/h 1 2 |ψ| 2 dsdt C ψ 2 ,
and

Ω e 2αt/h 1 2 (-ih∂ s + γ 0 -t - κ 2 t 2 )ψ 2 + |h∂ t ψ| 2 dsdt Ch ψ 2 .
These estimates invite us to consider an operator on the space domain Γ × (0, +∞) instead of Γ × (0, δ). For this we insert cutoff functions in the preceding operator. Let c be a smooth real function equal to 1 on [0, 1] and 0 for t 2. Then, we let m(s, t) = 1 -tc(δ -1 t)κ(s) . Instead of M h,δ , we consider M h,δ the self-adjoint realization on the Hilbert space L 2 (Γ × (0, +∞), mdsdt), of the differential operator with associated eigenvalues λ n (h, δ).

-

h 2 m -1 ∂ t m∂ t + m -1 -ih∂ s + γ 0 -t + c(δ -1 t) κ 2 t 2 m -1 -ih∂ s + γ 0 -t + c(δ -1 t) κ 2 t 2 ,
with Neumann boundary condition on t = 0. Note here that the additional truncation in front of κ is introduced in order to make this term bounded (and later a lower order term) when t is large.

Using the same truncation trick as in the proof of Proposition 2.2, similar Agmon type estimates for M h,δ , and the min-max principle, we get the following.

Proposition 2.4. Let n 1. There exist C, h 0 , α > 0 such that, for all h ∈ (0, h 0 ) and δ ∈ (0, δ 0 ),

λ n (h, δ) λ n (h, δ) λ n (h, δ) + Ce -αδ/h 1 2 .
Remark 2.5. Actually, at this stage, we have not proved that the low-lying spectrum of M h,δ is discrete. This will be a consequence of the forthcoming analysis.

From now on we fix

δ = h 1 4 -η h 1 4
, for some fixed 0 < η < 1/4. Note that this assumption is sufficient to ensure that remainder terms appearing in the latter proposition are indeed controlled by the main term which is of order e -S/h 1 4 for some constant S (see the main statement in Theorem 1.3).

The rescaled operator.

The exponential localization at the scale h 1 2 near t = 0 suggests to consider the partial rescaling

(s, t) = (σ, τ ) , with = h 1 2 .
We also let

a (σ, τ ) = 1 -τ κ(σ)c µ (τ ) , c µ (τ ) = c(µτ ) for µ = 1 2 +2η
, where we recall that η is positive and small, and c is the cutoff function introduced in the preceding section.

Remark 2.6. The notation µ will be convenient later when expanding the operator in powers of , with coefficients depending on µ.

Note that

a = 1 + O( 1 2 -2η
) . Upon dividing M h,δ by h, we get the new operator N acting on the space L 2 (Γ × R + , a dsdt) = L 2 (Γ × R + , dsdt), as the differential operator

N = -a -1 ∂ τ a ∂ τ + a -1 -i ∂ σ + -1 γ 0 -τ + c µ κ 2 τ 2 a -1 -i ∂ σ + -1 γ 0 -τ + c µ κ 2 τ 2
with Neumann condition on τ = 0. Note that

Dom (N ) = u ∈ L 2 (Γ × R + ) : -∂ 2 τ u ∈ L 2 (Γ × R + ) , -i ∂ σ + -1 γ 0 -τ 2 u ∈ L 2 (Γ × R + ) , ∂ τ u(•, 0) = 0 .
We denote by (ν n ( )) n 1 its eigenvalues. Using then Propositions 2.2 and 2.4, we get Proposition 2.7. Let n 1. There exist K > S, C, h 0 > 0 such that, for all h ∈ (0, h 0 ),

λ n (h) -Ce -K/h 1 4 2 ν n ( ) λ n (h) + Ce -K/h 1 4 .
This means that, in order to estimate the expected splitting between eigenvalues λ 2 (h) -λ 1 (h) of the original operator, we can consider the corresponding splitting for the reduced and rescaled operator N . The rest of the article is devoted to this problem. 2.4. One well operators.

2.4.1. Definitions. Let us consider the "one well operator" (attached to the right well). It is geometrically defined by surgery by removing a small neighborhood of the left curvature maximum, and gluing an infinite strip, see Figure 6. For this we choose first the curvilinear origin at the intersection of the upper part of Γ and the vertical axis, and we identify Γ with [s -2L, s ]. Note that in these coordinates, we have s r < 0 < s . We consider then the following right well differential operator

N ,r,γ 0 := -a -1 ∂ τ a ∂ τ + a -1 -i ∂ σ + -1 γ 0 -τ + c µ κ r 2 τ 2 a -1 -i ∂ σ + -1 γ 0 -τ + c µ κ r 2 τ 2 , (2.4)
acting on L 2 (R × R + , a dσdτ ) where κ r is an appropriate extension of κ defined as follows:

κ r = κ , on I r,η := (s -2L + η, s -η) , s r s s = -L s = L s = 0 s -η s + η Figure 6
. One well domain attached to the right well and κ r = 0 on (-∞, s -2L) ∪ (s + ∞). This extension may be chosen so that κ r has a unique and non-degenerate maximum at s r < 0.

Since the space domain is now simply connected, N ,r,γ 0 is unitarily equivalent to the flux-free operator N ,r := N ,r,0 since e iσγ 0 / 2 N ,r,γ 0 e -iσγ 0 / 2 = N ,r,0 . Note that the domain is the same as the one of the operator with constant magnetic field on

R 2 + = R × R + : Dom (N ,r ) = u ∈ L 2 (R 2 + ) : -∂ 2 τ u ∈ L 2 (R 2 + ) , -i ∂ σ -τ 2 u ∈ L 2 (R 2 + ) , ∂ τ u(•, 0) = 0 .
Let us now consider u ,r a groundstate of the flux-free operator N ,r,0 with well at s r < 0. The bottom of the spectrum is indeed discrete. Let us briefly explain this. Firstly, the essential spectrum is [Θ 0 , +∞), since, at infinity with respect to σ, the operator coincides with the Neumann magnetic Laplacian on the half-plane, whose spectrum is [Θ 0 , +∞). Secondly, one knows (see Theorem 2.8) that the spectrum below Θ 0 is not empty.

The function φ ,r (σ, τ ) = e -iγ 0 σ/ 2 u ,r (σ, τ ) (2.5) is then a groundstate for N ,r,γ 0 .

In order to define an operator adapted to the left well, we use the symmetry of Γ. More precisely, we consider the symmetry operator

U f (σ, τ ) = f (-σ, τ ) ,

and define

N , ,γ 0 = U -1 N ,r,γ 0 U . Note that this operator also corresponds to the following construction. Identifying Γ with [s r , s r + 2L], we can define on R the extended curvature κ (•) := κ r (-•) and note that it is equal to κ on (s r + η, s r + 2L -η) and 0 on (-∞, s r ) ∪ (s r + 2L, +∞).

In this way, κ has a unique and non-degenerate maximum at s > 0. The operator N , ,γ 0 acting on L 2 (R × R + , a dσdτ ) with well at s has then the same expression as the one of N ,r,γ 0 . A natural groundstate for N , ,γ 0 is then

φ , := U φ ,r
and, letting u , = U u ,r , we have φ , (σ, τ ) = e -iγ 0 σ/ 2 u , (σ, τ ) .

(2.6)

In the following, we will focus on the right well and find a WKB approximation of u ,r .

V (s) = 2C 1 µ 1 (ξ 0 ) (κ max -κ r (s)) .
In what follows, we consider formal series in the sense of [3, Notation 1.13], where the neighbourhood on which the approximation (at any given order) occurs can be taken arbitrarily large but bounded.

Theorem 2.8. Let us consider the following Agmon distance to the right well s r :

Φ(σ) = [sr,σ] V (σ)dσ . (2.7)
There exist formal series (b n ( ) n 0 and (δ n ( )

) n 0 such that b n ( ) ∼ j 0 b n,j j 2 , δ n ( ) ∼ j 0 δ n,j j 2 ,
and

(N ,r -δ n ( )) Ψ ,r,n = O( ∞ )e -Φ(σ)/ 1 2 , with Ψ ,r,n ∼ →0 -1 8 b n ( )e -Φ(σ)/ 1 2 e iσξ 0 / . (2.8)
Moreover,

δ n,0 = Θ 0 , δ n,1 = 0 , δ n,2 = -C 1 κ max , δ n,3 = (2n -1)C 1 Θ 1 4 0 3k 2 2 ,
and

b n,0 (σ, τ ) = f n,0 (σ)u ξ 0 (τ ) , b n,1 (σ, τ ) = iΦ (σ)f n,0 (σ)(∂ ξ u ξ ) ξ 0 (τ ) + f n,1 (σ)u ξ 0 (τ ) , (2.9
) where f n,0 solves the effective transport equation

µ 1 (ξ 0 ) 2 (Φ ∂ σ + ∂ σ Φ )f n,0 + F (σ)f n,0 = (2n -1)C 1 Θ 1 4 0 3k 2 2 f n,0 , (2.10)
f n,1 is a solution of a similar transport equation, and where F is a smooth function such that F (s r ) = 0 and Re F = 0.

Remark 2.9. Let us consider (2.10). We may write f n,0 in the form f n,0 (σ) = e iα n,0 (σ) fn,0 (σ) for some real-valued function α n,0 , and where fn,0 solves the real classical transport equation

µ 1 (ξ 0 ) 2 (Φ ∂ σ + ∂ σ Φ ) fn,0 = (2n -1)C 1 Θ 1 4 0 3k 2 2 fn,0 . (2.11)
Note that f1,0 (0) be can chosen to be positive and we will assume that it is the case. Following e.g. [4, Section 2.2] we also choose the normalization Ψ ,r,n = 1 (see also Section 6.2). This gives

f 2 1,0 (0) = g π 1/2 A u , (2.12)
where g and A u are defined in (1.8).

Let us now consider the phase shifts. The α n,0 are chosen so that

µ 1 (ξ 0 )iΦ α n,0 + F = 0 , or, equivalently, Φ α n,0 = iF µ 1 (ξ 0 )
.

Since F (s r ) = 0 and Φ vanishes linearly at s r , we can write

α n,0 (σ) = iF (σ) µ 1 (ξ 0 )Φ (σ) , (2.13)
where the right-hand-side can be seen as a real-valued (Re F = 0) smooth function defined at s r (by using the natural continuous extension). This determines the phase shift α n,0 up to an additive constant. At this stage, we fixed the normalization of the WKB Ansatz. Later on, in Section 6.2, we will take profit of this appropriate normalization, which determines the functions f1,0 and α 1,0 . This normalization of f1,0 is the one that we used in [4, Section 2.2] when considering the tunneling effect for purely electric Schrödinger operators on the circle. This will be suitable to recognize, in our final computation, the interaction term for an electric Hamiltonian. The equation (2.11) is indeed the same as the one we obtain when performing a WKB construction for the semiclasssical electric Hamiltonian

µ 1 (ξ 0 ) 2 D 2 σ + v(σ) , v = C 1 (κ max -κ r ) . (2.14)
We define

α 0 = α 1,0 (0) -α 1,0 (-L) L , (2.15)
which is the phase shift appearing in Theorem 1.3

A Grushin problem

In this section, we focus on the one well operator. Let us consider a smooth nonnegative function, with bounded derivative, σ → ϕ(σ) and consider the conjugate operator

N ϕ ,r = e ϕ/ 1 2 N ,r e -ϕ/ 1 2 ,
still acting on Dom (N ,r ). Explicitly,

N ϕ ,r = -a -1 ∂ τ a ∂ τ + a -1 -i ∂ σ -τ + i 1 2 ϕ + c µ κ r 2 τ 2 a -1 -i ∂ σ -τ + i 1 2 ϕ + c µ κ r 2 τ 2 .
In order to lighten the notation, we write κ and N ϕ instead of κ r and N ϕ ,r . In all what follows we shall use the following notation in order to compare operators and deal with remainders:

Notation 3.1. For formal operators A, B, C, . . . in L 2 (R) we say that A = O(B, C) if there is a constant c 0 such that for all u in S(R) Au c( Bu + Cu + . . .) .
This definition naturally extends to L 2 (R × R + ) and similar pivot spaces when taking test function satisfying in addition the good boundary conditions.

3.1.

A pseudo-differential operator with operator-valued symbol. We notice that N ϕ can be written as an -pseudo-differential operator with an operatorvalued symbol n (σ, ξ) having an expansion in powers of

1 2 : N ϕ = Op W n ,
with Op W n acting on S(R σ , S(R +,τ )) through the usual quantization formula (see [START_REF]Formules de Weyl par réduction de dimension[END_REF]Definition 2.1.7])

Op W n u(σ) = 1 (2π ) R 2 e i(σ-σ)•ξ n σ + σ 2
, ξ u(σ)dσdξ,

with here n = n 0 + 1 2 n 1 + n 2 + 3 2 n 3 + 2 r
, and where after a computation using the usual symbolic rules, we get

n 0 = -∂ 2 τ + (ξ -τ ) 2 , n 1 = 2i(ξ -τ )ϕ , n 2 = -ϕ 2 + κc µ ∂ τ + c µ κ(ξ -τ )τ 2 + 2κτ c µ (ξ -τ ) 2 + κτ c µ (τ ) , Re n 3 = 0 , r = O(τ 4 , (ξ -τ ) 2 τ 2 , (ξ -τ )τ, τ 2 ∂ τ ) . (3.1)
In the last expression, the notation O is defined in Notation 3.1. The expansion was performed with respect to , with µ considered a parameter (see Remark 2.6). It will be explained later how to deal with the remainder r . It involves in particular powers of τ which can be controlled via the normal localization estimates, and thus are not really problematic. Note the in (3.1), µ is considered as a parameter although it may depend on . Now the frequency variable ξ is a priori unbounded, and in the next step of the analysis, we therefore "truncate" our operator in ξ to get a bounded symbol. Let us consider a smooth, bounded, and increasing odd function

χ such that χ(ξ) = ξ for ξ ∈ [-ξ 0 2 , ξ 0 2 ].
We let η ± = ± lim ξ→±∞ χ(ξ) and assume that η -∈ (0, ξ 0 ). We let, for all ξ ∈ R, χ 1 (ξ) = ξ 0 + χ(ξ -ξ 0 ) . Then, the function ξ → µ 1 (χ 1 (ξ)) is bounded and still has a unique minimum at ξ 0 , which is non-degenerate and not attained at infinity. Note that, by construction, we have, for all ξ ∈ [ ξ 0 2 , 3ξ 0 2 ], µ 1 (χ 1 (ξ)) = µ 1 (ξ). Since µ 1 (ξ) < 1 for all ξ > 0 and η -∈ (0, ξ 0 ), we also notice that

µ 1 • χ 1 (R) ⊂ [Θ 0 , 1) . (3.2)
We will consider

Op W p , with p (s, ξ) = n (s, χ 1 (ξ)) , (3.3)
and notice in particular that the principal operator symbol of Op W p is

p 0 (s, ξ) = -∂ 2 τ + (χ 1 (ξ) -τ ) 2 .
For a recent panorama of pseudo-differential operators with operator symbols, we refer to [20, Chapitre 2] (see also [START_REF] Gérard | A mathematical approach to the effective Hamiltonian in perturbed periodic problems[END_REF]Appendix B]). The introduction of the function χ 1 is inspired by [20, Section 6.3].

3.2.

The Grushin problem for the principal operator symbol. Let us first consider the principal symbol p 0 (whose domain is independent of ξ). Let z ∈ C such that Re z ∈ (Θ 0 -ε, Θ 0 + ε), with ε > 0 such that Θ 0 + ε < 1. Consider the matrix operator:

P 0,z (ξ) := p 0 -z •v ξ •, v ξ 0 ∈ S(R 2 s,ξ , L (Dom p 0 × C, L 2 (R + ) × C)) , acting on Dom (p 0 ) × C and valued in L 2 (R + ) × C. Here v ξ = u χ 1 (ξ)
. We also denote by Π ξ , or simply Π the orthogonal projection on Cv ξ .

Notation 3.2. The notation

P ∈ S(R 2 , L (Dom p 0 × C, L 2 (R + ) × C)) means that -P = P (x, ξ
) is a family of closed operators whose domain does not depend on (x, ξ), and whose graph norms are equivalent uniformly in (x, ξ), -for all α ∈ N 2 , there exists C α > 0 such that ∂ α s,ξ P • C α • P , uniformly with respect to (x, ξ), and where • P is the graph norm of P . This class can be thought as a generalization of the standard class of scalar symbols

S(1) = {p ∈ C ∞ (R 2 , C) , ∀α ∈ N 2 , ∃C α > 0 , ∂ α s,ξ p C α }
to operator-valued symbols. Note however that, contrary to the scalar case, this is not an algebra. More details can be found in [20, Section 6.3].

Lemma 3.3. For all ξ ∈ R, P 0,z (ξ) is bijective and

Q 0,z (ξ) := P -1 0,z (ξ) = (p 0 -z) -1 Π ⊥ •v ξ •, v ξ z -µ 1 (ξχ 1 (ξ)) , and Q 0,z ∈ S(R 2 s,ξ , L (L 2 (R + ) × C, Dom p 0 × C)) .
Here Π ⊥ denotes the orthogonal projection on v ξ ⊥ .

Proof. Let (v, β) ∈ L 2 (R + ) × C and let us look for (u, α) ∈ Dom (p 0 ) × C such that P 0,z (ξ)(u, α) T = (v, β) T . In other words,

(p 0 -z)u = v -αv ξ , u, v ξ = β , or (p 0 -z)Π ⊥ u = v -αv ξ -β(p 0 -z)v ξ = v -αv ξ -β(µ 1 (χ 1 (ξ)) -z)v ξ , (3.4)
with u, v ξ = β. The operator p 0 -z stabilizes (Cv ξ ) ⊥ and induces an operator. Moreover, there exists c > 0 such that for all u ∈ Dom (p 0 ) ∩ (Cv ξ ) ⊥ and all z ∈ C such that Re z ∈ (Θ 0 -ε, Θ 0 + ε),

Re (p 0 -z)u, u = (p 0 -Re z)u, u (µ 2 (χ 1 (ξ)) -Re z) u 2 c u 2 ,
where we used the self-adjointness of p 0 , the min-max principle and the fact that min µ 2 > 1 (see [7, Proposition 3.2.2 & Remark 3.2.6]) and Θ 0 + ε < 1. Thus, the operator (p 0 -z) |(Cv ξ ) ⊥ is injective with closed range and, by considering the adjoint, it is bijective. We also notice that

(p 0 -z) -1 Π ⊥ (µ 2 (χ 1 (ξ)) -Re z) -1 c -1 .
The equation (3.4) has a solution if and only if the r.h.s. belongs to

(Cv ξ ) ⊥ , that is α = v, v ξ -β(µ 1 (χ 1 (ξ)) -z) .
This unique solution is given by

Π ⊥ u = (p 0 -z) -1 Π ⊥ (v -αv ξ -β(µ 1 (χ 1 (ξ)) -z)v ξ ) = (p 0 -z) -1 Π ⊥ v .
Therefore, u = βv ξ + (p 0 -z) -1 Π ⊥ v.

Pseudo-differential dimensional reduction and subprincipal terms.

Let us now consider the full symbol

P z (s, ξ) := p -z •v ξ •, v ξ 0 ∈ S(R 2 s,ξ , L (Dom p 0 × C, L 2 (R + ) × C)) ,
and notice that we can write

P z = P 0,z + 1 2 P 1 + P 2 + 3 2 P 3 P [3] z + 2 R ,
where for j 1 , P j = p j 0 0 0 , R = r 0 0 0 and from (3.1) and using the fact that χ 1 (ξ) is now bounded, we can write

p 0 = -∂ 2 τ + (χ 1 (ξ) -τ ) 2 , p 1 = 2i(χ 1 (ξ) -τ )ϕ , p 2 = -ϕ 2 + κc µ ∂ τ + c µ κ(χ 1 (ξ) -τ )τ 2 + 2κτ c µ (χ 1 (ξ) -τ ) 2 + κτ c µ (τ ) , Re p 3 = 0 , r = O(τ 4 , τ 2 ∂ τ ) .
(3.5) Remark 3.4. Note that in the last expansion at order 3 w.r.t. 1 2 , we do not need the exact expression of p 3 and will use later that it is purely imaginary. The structure of the last Taylor expansion is rather subtle. Indeed we do not care about the cutoff in variable τ induced by c µ , but we have to keep in mind that up to loosing powers of , the involved operators are indeed in S(1). This property allows to do all the computations with test functions in Dom(p 0 ) × C and gives a meaning to the composition of operators done in the next theorem. In particular, this expansion is uniform in the parameter µ. Let us notice that the powers of τ and ∂ τ in r will be compensated later by the normal decay.

The following theorem gives then an approximated parametrix of operator Op W P z , that is, in our context, an inverse up to a remainder of order 2 . Theorem 3.5. Consider the operator symbol

Q [3] z = Q 0,z + 1 2 Q 1,z + Q 2,z + 3 2 Q 3,z where Q 0,z is given in Lemma 3.3 and Q 1,z = -Q 0,z P 1 Q 0,z , Q 2,z = -Q 0,z P 2 Q 0,z -Q 1,z P 1 Q 0,z , Q 3,z = -Q 0,z P 3 Q 0,z -Q 1,z P 2 Q 0,z -Q 2,z P 1 Q 0,z -C z , (3.6
)

with 2iC z = ({Q 0,z , P 1 } + {Q 1,z , P 0,z }) Q 0,z ,
where we used the classical notation for the Poisson bracket

{Q, P} = ∂ ξ Q • ∂ s P -∂ s Q • ∂ ξ P .
Then, we have Op W (Q [3] z )Op W (P z ) = Id + 2 O( τ 6 ) .

(3.7)

Moreover, we have the following explicit description. Letting

Q [3] z = q z q + z q - z q ± z , we write q ± z = q ± 0,z + 1 2 q ± 1,z + q ± 2,z + 3 2 q ± 3,z , with q ± 0,z = z -µ 1 (χ 1 (ξ)) , q ± 1,z = -iϕ (s)µ 1 (χ 1 (•)) (ξ) , q ± 2,z = κ(σ)C 1 (ξ, µ) + C 2 (ξ, z)ϕ 2 , (3.8)
where

C 1 (ξ, µ) = c µ ∂ τ + c µ (χ 1 (ξ) -τ )τ 2 + 2τ c µ (χ 1 (ξ) -τ ) 2 v ξ , v ξ -τ c µ (τ )∂ τ v ξ , v ξ , C 2 (ξ, z) = 1 -4 (p 0 -z) -1 Π ⊥ (χ 1 (ξ) -τ )v ξ , (χ 1 (ξ) -τ )v ξ .
and when z is real we have Re q ± 3,z = 0 . Moreover, q - z , q + z , and q ± z are uniformly (with respect to µ) bounded symbols. Remark 3.6. From [6, Prop. A.2] (see also the definition of C 1 in (1.2)), we have

C 1 (ξ 0 , 0) = C 1 ,
and, from the exponential decay of v ξ and its derivative (in the τ variable) and the confinement in τ induced by the truncation c µ , we have, uniformly in ξ,

C 1 (ξ 0 , µ) = C 1 + O( ∞ ) , τ c µ (τ )∂ τ v ξ , v ξ = O( ∞ ) .
From [6, Prop. A.3], we have

C 2 (ξ 0 , Θ 0 ) = µ (ξ 0 ) 2 .
Remark 3.7. Let us recall here that the bijectivity of Op W (p ) -z is related to the one of Op W (q ± z ). In this case, we have, modulo some remainders,

(Op W (p ) -z) -1 Op W q z -Op W q - z [Op W q ± z ] -1 Op W q + z .
Proof. The proof is constructive. In order to see where the expressions (3.6) are coming from, let us consider the product Op W (Q [3] z )Op W (P [3] z ) , and its the expansion in half-powers of . The symbols Q j,z are chosen so that (3.7) holds. Let us explain how these choices are made.

Terms of order 0 . The terms of order 1 give Q 0,z P 0,z = Id . Now, one wants to cancel the other terms.

Terms of order 1 2 . Cancelling the terms of order 1 2 , we find

Q 1,z P 0,z + Q 0,z P 1 = 0 , (3.9)
or, equivalently,

Q 1,z = -Q 0,z P 1,z Q 0,z . Explicitly, Q 1,z = - q 0,z p 1 q 0,z q 0,z p 1 q + 0 q - 0 p 1 q 0,z q - 0 p 1 q + 0 . Note that q ± 1,z = -p 1 v ξ , v ξ , p 1 = 2iϕ (χ 1 (ξ) -τ )
. By the Feynman-Hellmann theorem,

q ± 1,z = -2iϕ (χ 1 (ξ) -τ )v ξ , v ξ = -iϕ (s)µ 1 (χ 1 (•)) (ξ)
. Terms of order 1 . Let us cancel the terms of order :

Q 1,z P 1 + 1 2i {Q 0,z , P 0,z } + Q 0,z P 2 + Q 2,z P 0,z = 0 .
Since the principal symbol does not depend on s, the Poisson bracket is zero, and thus

Q 1,z P 1 + Q 0,z P 2 + Q 2,z P 0,z = 0 . It follows that Q 2,z = -Q 1,z P 1 Q 0,z -Q 0,z P 2 Q 0,z . We have Q 0,z P 2 Q 0,z = q 0,z p 2 q 0,z q 0,z p 2 q + 0 q - 0 p 2 q 0,z p 2 v ξ , v ξ ,
and from the expression of Q 1,z above Q 1,z P 1 Q 0,z = -q 0 p 1 q 0 p 1 q 0 q 0 p 1 q 0 p 1 q + 0 q - 0 p 1 q 0 p 1 q 0 q - 0 p 1 q 0 p 1 q + 0 .

In particular, we have

q ± 2,z = q - 0 p 1 q 0 p 1 q + 0 -p 2 v ξ , v ξ = p 1 (p 0 -z) -1 Π ⊥ p 1 v ξ , v ξ -p 2 v ξ , v ξ .
With (3.1) and (3.3), we have

p 1 (p 0 -z) -1 Π ⊥ p 1 v ξ , v ξ = -4ϕ 2 (p 0 -z) -1 Π ⊥ (χ 1 (ξ) -τ )v ξ , (χ 1 (ξ) -τ )v ξ , p 2 v ξ , v ξ = -ϕ 2 + κC 1 (ξ, µ) .
Terms of order 3 2 . In the same way, we determine Q 3,z by solving

Q 0,z P 3 + Q 1,z P 2 + Q 2,z P 1 + Q 3,z P 0 + 1 2i ({Q 0,z , P 1 } + {Q 1,z , P 0,z }) = 0 .
which gives

Q 3,z = -Q 0,z P 3 Q 0,z + Q 1,z P 2 Q 0,z + Q 2,z P 1 Q 0,z -C z (3.10)
which is the last equality in (3.6).

We show now that when z is real, Re (q ± 3 ) is purely imaginary. For this we notice that the first term in parenthesis in (3.10) gives rise to a purely imaginary term in the right bottom of its matrix expression. Then, we show that C z is actually skew-self-adjoint. First, since P 0,z does not depend on s,

2iC z = ∂ ξ Q 0,z ∂ s P 1 Q 0,z -∂ s Q 1,z ∂ ξ P 0,z Q 0,z .
Then, recalling that P 0,z Q 0,z = Id and (3.9) and taking the derivatives of these formulas with respect to ξ and s, respectively, we get

2iC z = ∂ ξ Q 0,z ∂ s P 1 Q 0,z + ∂ s Q 1,z P 0,z ∂ ξ Q 0,z = -∂ ξ Q 0,z P 0,z ∂ s Q 1,z + ∂ s Q 1,z P 0,z ∂ ξ Q 0,z = (P 0,z ∂ ξ Q 0,z ) * (∂ s Q 1,z ) * + ∂ s Q 1,z P 0,z ∂ ξ Q 0,z = (∂ s Q 1,z P 0,z ∂ ξ Q 0,z ) * + ∂ s Q 1,z P 0,z ∂ ξ Q 0,z ,
where we used that P 0,z , Q 0,z are self-adjoint and P 1 , Q 1,z are skew-self-adjoint.

Remainders and order 2 . Therefore, with the definition of Q

[3]
z , and composition of pseudo-differential operators, the operator symbol of Op W (Q

[3] z )Op W (P [3]
z ) coincides with Id modulo terms of orders at least O( 2 ). By the Calderón-Vaillancourt theorem, this remainder is a bounded operator, but the bound depends on the parameter µ. To avoid this problem, we observe that, by Taylor expansion, the remainder is of order 2 in the worse topology of L 2 ( τ 6 dτ ds). This power 6 comes from the product of the terms of order 3 2 . In the same way, we see that Op W (Q [3] z ) Op W (P z ) -Op W (P [3] z ) is again of order 2 for the topology L 2 ( τ 6 dτ ds). Using that

2 (p 0 -z) -1 c µ τ 2 ∂ τ = 2 O( τ 2 ), (3.11)
we can get rid of the derivatives in the remainder term involving τ 2 ∂ τ .

The fact that q - z , q + z , and q ± z are bounded comes from their explicit expressions and the fact that v ξ is exponentially decaying uniformly in ξ with respect to τ .

Tangential coercivity estimates

We will use Theorem 3.5 for z ∈ C such that

z = Θ 0 -C 1 κ max + O( 2 ) ,
and assume that ϕ is an appropriate sub-solution of the eikonal equation in the following sense Assumption 4.1. Let ϕ 0 be a Lipschitzian function such that, for all M > 0 there exist C, R > 0 such that

(i) for all σ ∈ R, v(σ) - µ 1 (ξ 0 ) 2 ϕ (σ) 2 0, (ii) for all σ such that |σ -s r | R 1 2 , v(σ) - µ 1 (ξ 0 ) 2 ϕ (σ) 2 M .
Note that ϕ = 0 is such a subsolution (much more useful solutions will be introduced later) and that for all σ such that |σ -

s r | R 1 2 , v(σ) - µ 1 (ξ 0 ) 2 ϕ (σ) 2 C .
Theorem 4.2. Let K > 0. Under Assumption 4.1, there exist 0 , c, R 0 > 0 such that, for all R > R 0 , there exists C R > 0 such that the following holds. For all ∈ (0, 0 ) and all z ∈ C such that |z -

Θ 0 + C 1 κ max | K 2
, and for all ψ ∈ Dom (Op W p ),

cR 2 2 ψ (Op W p -z)ψ + C R 2 χ 0 ( -1 2 R -1 (σ -s r ))ψ + 2 τ 6 ψ where χ 0 ∈ C ∞ 0 (R) is 1 in a neighborhood of 0. Remark 4.3. The domain of Op W p is Dom (Op W p ) = L 2 (R σ , B 2 N (R +,τ )) , with B 2 N (R + ) = {u ∈ H 2 (R + ) : τ 2 u ∈ L 2 (R + ) , u (0)
= 0}. In Theorem 4.2, we use the convention that if τ 6 ψ does not belong to L 2 (R 2 + ), we have τ 6 ψ = +∞ in which case the inequality is true. The same kind of convention will be used in Section 5. Anyway, in the proofs, ψ can be assumed to belong to the Schwartz class S(R 2 + ). 4.1. From the effective operator... Proposition 4.4. Let K > 0. There exist h 0 , C > 0 such that, for all z ∈ C such that |z -

Θ 0 + C 1 κ max | K 2 , R v(σ) - µ 1 (ξ 0 ) 2 ϕ 2 (σ) |ψ| 2 dσ -C 2 ψ 2 -Re Op W q ± z ψ, ψ .
In particular, for some c > 0 and all R > 0, there exists C R > 0 such that

cR 2 2 ψ Op W q ± z ψ + C R 2 χ 0 ( -1 2 R -1 (σ -s r ))ψ .
Proof. Using the assumption on z and (3.8), we have 2) , and also

-Re q ± z = µ 1 (χ 1 (ξ)) -Θ 0 + -κ(σ)C 1 (ξ, µ) + C 1 κ max -C 2 (ξ, Θ 0 )ϕ 2 + O(
-Re q ± z = µ 1 (χ 1 (ξ))-Θ 0 + -κ(σ)C 1 (ξ, 0) + C 1 (ξ 0 , 0)κ max -C 2 (ξ, Θ 0 )ϕ 2 +O( 2 ) . We write -Re q ± z v(σ) -C 2 (ξ 0 , Θ 0 )ϕ 2 (σ) + r , (4.1) where
r = µ 1 (χ 1 (ξ)) -Θ 0 + s , with |s | C min(1, |ξ -ξ 0 |) . Since µ 1 (χ 1 (ξ)) -Θ 0 c min (ξ -ξ 0 ) 2 , 1 , we get, from the Young inequality, r -C 2 . (4.2)
Using (4.1), (4.2), and the standard Fefferman-Phong inequality, the result follows.

4.2. ... to the bidimensional operator. We can now establish Theorem 4.2.

Let us recall the relation between Op W p and Op W q ± z . We have by Theorem 3.5

Op W q z Op W q + z Op W q - z Op W q ± z Op W p -z B * B 0 = Id + O L 2 (R×R + , τ 6 dσdτ )→L 2 (R×R + ) ( 2 ) ,
where B = Op W ( •, v ξ ) . In particular,

Op W q z (Op W p -z) + Op W q + z B = Id + O L 2 (R×R + , τ 6 dσdτ )→L 2 (R + ) ( 2 ) Op W q - z (Op W p -z) + Op W q ± z B = O L 2 (R×R + , τ 6 dσdτ )→L 2 (R×R + ) ( 2 ) . (4.3) Thus, ψ C (Op W p -z)ψ + C Bψ + C 2 τ 6 ψ , and
Op W q ± z (Bψ) C (Op W p -z)ψ + C 2 τ 6 ψ . From Proposition 4.4, we deduce cR 2 2 Bψ C (Op W p -z)ψ + C R 2 χ 0 ( -1 2 R -1 (σ -s r ))
Bψ , and then, choosing R large enough,

cR 2 2 ψ C (Op W p -z)ψ + C R 2 χ 0 ( -1 2 R -1 (σ -s r ))Bψ + C 2 τ 6 ψ .
Moreover, by rescaling and using the fact that the symbol of B only depends on ξ, we have

[B, χ 0 ( -1 2 R -1 (σ -s r ))] = O( 1 2
), we get

cR 2 2 ψ C (Op W p -z)ψ + 2 Bχ 0 ( -1 2 R -1 (σ-s r ))ψ +C 5 2 ψ +C 2 τ 6 ψ ,
and the conclusion follows.

Removing the frequency cutoff

Let us now replace in Theorem 4.2 the "truncated" operator Op W p (defined in Section 3.1) by the operator without frequency cutoff N ϕ . This can be done up to convenient additional remainders.

Theorem 5.1. Under Assumption 4.1, there exist c, 0 > 0 such that for all ∈ (0, 0 ) and all ψ ∈ Dom (N ϕ ),

c 2 ψ τ 6 (N ϕ -z)ψ + 2 χ 0 ( -1 2 R -1 (σ -s r ))ψ , and c 2 2 D 2 σ ψ τ 6 (N ϕ -z)ψ + 2 χ 0 ( -1 2 R -1 (σ -s r ))ψ .
5.1. Preliminary lemmas. Let us consider a smooth function χ 2 = χ 2 (ξ) equal to 1 away from a compact and whose support avoids ξ 0 .

Lemma 5.2. There exist C, 0 > 0 such that for all ∈ (0, 0 ) and all ψ ∈ Dom (N ϕ ),

Op W χ 2 ψ + ( D σ -τ )Op W χ 2 ψ + D τ Op W χ 2 ψ C (N ϕ -z)Op W χ 2 ψ .
Proof. We write

Re (N ϕ -z)Op W χ 2 ψ, Op W χ 2 ψ (1 + o(1)) D 2 τ + ( D σ -τ ) 2 Op W χ 2 ψ, Op W χ 2 ψ -Re z Op W χ 2 ψ 2 .
(5.1) Thus, by using the support of χ 2 and the properties of µ 1 ,

Re (N ϕ -z)Op W χ 2 ψ, Op W χ 2 ψ ((1 + o(1))c 1 -Re z) Op W χ 2 ψ 2 , c 1 > Θ 0 .
Using again (5.1) and the Cauchy-Schwarz inequality, the conclusion follows.

Actually, we have also an "H 2 -control" with respect to the "magnetic derivatives".

Lemma 5.3. There exist C, 0 > 0 such that for all ∈ (0, 0 ) and all ψ ∈ Dom (N ϕ ),

( D σ -τ ) 2 Op W χ 2 ψ + D 2 τ Op W χ 2 ψ C (N ϕ -z)Op W χ 2 ψ .
Proof. This is obtained through standard elliptic estimates by controlling first the magnetic tangential derivative.

Lemma 5.4. Let N ∈ N. There exist C, 0 > 0 such that for all ∈ (0, 0 ) and all ψ ∈ Dom (N ϕ ),

Op W χ 2 ψ + D τ Op W χ 2 ψ + ( D σ -τ )Op W χ 2 ψ + D 2 τ Op W χ 2 ψ + ( D s -τ ) 2 Op W χ 2 ψ C (N ϕ -z)ψ + O( N ) ψ .
Proof. From Lemmas 5.2 and 5.3, we have

Op W χ 2 ψ + ( D σ -τ )Op W χ 2 ψ + D τ Op W χ 2 ψ + ( D σ -τ ) 2 Op W χ 2 ψ + D 2 τ Op W χ 2 ψ C (N ϕ -z)Op W χ 2 ψ . (5.2)
Let us deal with the r.h.s. and notice that

(N ϕ -z)Op W χ 2 ψ (N ϕ -z)ψ + [N ϕ , Op W χ 2 ]ψ . (5.3)
Let us consider the commutator. One of the terms is

[a -1 D τ a D τ , Op W χ 2 ]ψ = [a -1 ∂ τ a , Op W χ 2 ]D τ ψ O( ∞ ) D τ ψ + Op W χ 2 D τ ψ , (5.4)
where χ 2 has a support slightly larger than the one of χ 2 , and where we used classical results of composition of pseudo-differential operators. The other term is

[a -1 ( D σ -τ + κτ 2 /2)a -1 ( D s -τ + c µ κτ 2 /2), Op W χ 2 ]ψ O( ∞ )( ( D σ -τ )ψ + ψ ) + C Op W χ 2 ( D σ -τ )ψ + Op W χ 2 ψ .
(5.5) Using (5.2), (5.3), (5.4), and (5.5), an induction argument (on the size of the support of χ 2 ) provides us with

Op W χ 2 ψ + D τ Op W χ 2 ψ + ( D σ -τ )Op W χ 2 ψ + D 2 τ Op W χ 2 ψ + ( D σ -τ ) 2 Op W χ 2 ψ C (N ϕ -z)ψ +O( N )( ψ + ( D σ -τ )ψ + D τ ψ ) .
Noticing that

( D σ -τ )ψ + D τ ψ C N ϕ ψ + C ψ C (N ϕ -z)ψ + C ψ ,
the conclusion follows. We would like to get a control D σ instead of D σ -τ . In particular, one should control τ with the normal Agmon estimates. Proposition 5.6. Let N ∈ N. There exist C, 0 > 0 such that for all ∈ (0, 0 ) and all ψ ∈ Dom (N ϕ ),

Op W χ 2 ψ + D τ Op W χ 2 ψ + D s Op W χ 2 ψ + D 2 τ Op W χ 2 ψ + ( D s ) 2 Op W χ 2 ψ + τ D s Op W χ 2 ψ C (N ϕ -z)ψ + τ (N ϕ -z)ψ + τ 2 (N ϕ -z)ψ +O( N ) ψ .
Proof. Let us apply Lemma 5.4 to τ ψ (recall Remark 5.5). We get

τ Op W χ 2 ψ C (N ϕ -z)τ ψ + O( N ) τ ψ .
Replacing ψ by Op W χ 2 ψ, commuting N ϕ with τ and using Lemma 5.4, we get

τ Op W χ 2 ψ C τ (N ϕ -z)ψ + O( N )( ψ + τ ψ ) .
With Lemma 5.4, we get

( D s )Op W χ 2 ψ C (N ϕ -z)ψ + C τ (N ϕ -z)ψ + O( N )( ψ + τ ψ ) .
In the same spirit, we get

( D s ) 2 Op W χ 2 ψ C (N ϕ -z)ψ + C τ (N ϕ -z)ψ + C τ 2 (N ϕ -z)ψ + O( N )( (1 + τ + τ 2 )ψ ) , τ D s Op W χ 2 ψ C (N ϕ -z)ψ + C τ (N ϕ -z)ψ + C τ 2 (N ϕ -z)ψ + O( N )( (1 + τ + τ 2 )ψ ) .
Due to the Dirichlet condition, we have, for k 1,

τ k ψ C (N ϕ -z)τ k ψ . (5.6)
Computing commutators and controlling them by N ϕ ψ , the result follows upon noticing that N ϕ ψ (N ϕ -z)ψ + |z| ψ .

5.2.

Proof of Theorem 5.1. With the triangle inequality,

(N ϕ -z)ψ (P -z)ψ -(N ϕ -P )ψ . (5.7)
Using Proposition 5.6 with χ 2 such that 1 -χ 2 is supported in {χ 1 (ξ) = ξ} to control the terms ( D s ) 2 and τ D σ , we get

(N ϕ -P )ψ C (N ϕ -z)ψ + C τ (N ϕ -z)ψ + C τ 2 (N ϕ -z)ψ + O( N ) ψ . (5.8)
Combining (5.7) and (5.8) with Theorem 4.2, provides us with

cR 2 2 ψ τ 2 (N ϕ -z)ψ +C R 2 χ 0 ( -1 2 R -1 (σ-s r ))ψ + 2 τ 6 ψ +C N ψ .
(5.9) By using again (5.6), we get

τ 6 ψ C (N ϕ -z)τ 6 ψ C τ 6 (N ϕ -z)ψ + C [N ϕ , τ 6 ]ψ .
Computing explicitly the commutator, we get, by induction,

τ 6 ψ C τ 6 (N ϕ -z)ψ + C N ϕ ψ C τ 6 (N ϕ -z)ψ + C (N ϕ -z)ψ + C|z| ψ .
With (5.9) and choosing R large enough (to absorb the C|z| term), we deduce the first estimate in Theorem 5.1. Combining this estimate with Proposition 5.6, the conclusion follows.

5.3.

A slight improvement. The considerations in the previous section give the following improvement of Theorem 5.1.

Corollary 5.7. Under Assumption 4.1, there exist c, 0 > 0 such that for all ∈ (0, 0 ) and all ψ ∈ Dom (N ϕ ),

c 2 τ ψ τ 6 (N ϕ -z)ψ + 2 χ 0 ( -1 2 R -1 (σ -s r ))ψ , (5.10)
and

c 2 τ 2 D 2 σ ψ τ 6 (N ϕ -z)ψ + 2 χ 0 ( -1 2 R -1 (σ -s r ))ψ .
(5.11)

Proof. We recall (5.6), and we use it with k = 1. Estimating a commutator, this shows that

τ ψ C τ (N ϕ -z)ψ + C ∂ τ ψ + C ψ .
We also notice that

∂ τ ψ 2 C N ϕ ψ ψ .
(5.12)

Then, since z is bounded,

∂ τ ψ + τ ψ C τ (N ϕ -z)ψ + C ψ . (5.13)
Applying Theorem 5.1, we get (5.10).

To get (5.11), we apply Proposition 5.6 with ψ replaced by τ ψ. Then, we estimate the commutators by using (5.13), (5.12) (with ψ replaced by τ k ψ, k = 1, 2) and (5.6) (with k = 2), and we use (5.10).

6. Optimal tangential Agmon estimates 6.1. Agmon estimates. Let us discuss here some important consequences of our elliptic estimates. An immediate corollary of Theorem 5.1 is the following. Corollary 6.1. Under Assumption 4.1 and with the notation introduced in Section 2.4.1, for all K > 0, there exist C, 0 > 0 such that for all ∈ (0, 0 ) and all λ eigenvalue of N ,r such that |λ -(Θ 0 -C 1 κ max )| K 2 and all associated eigenfunction Ψ ∈ Dom (N ,r ),

R 2 + e 2ϕ/ 1 2 |Ψ| 2 dsdτ C Ψ 2 .
Proof. We apply Theorem 5.1 with z = λ and ψ = e ϕ/ Let us now explain how to get tangential Agmon estimates for the two wells operator N from the estimates on the one well operators (acting on L 2 (R × R + )).

By using that u is an eigenfunction and by choosing η small enough (and adapting χ r accordingly), we get the following estimate,

c 2 χ r e ϕ/ 1 2 u e ε/2 1 2 τ 6 [N , χ r ]u L 2 ([-L,L)×R + ) + C 2 u L 2 ([-L,L)×R + ) .
Thanks to the normal Agmon estimates, we get

χ r e ϕ/ 1 2 u Ce ε/ 1 2 u L 2 ([-L,L)×R + ) .
By considering the left well, we get by symmetry

χ e ϕ/ 1 2 u Ce ε/ 1 2 u L 2 ([-L,L)×R + ) .
Since the supports of χ r and χ overlap [-L, L), the conclusion follows.

6.2. WKB approximation in the right well. Let us now discuss a crucial application of Theorem 5.1. We work here on the real line. Let us apply the theorem to

ψ = e ϕ/ 1 2 (ψ ,r -Π r ψ ,r ) , (6.1)
where ψ ,r (σ, τ ) = χ η,r Ψ ,τ (σ, τ ), χ η,r is a cut-off function supported in I η,r and such that χ η = 1 on I 2η,r , -Ψ ,r is the WKB solution introduced in (2.8) (with n = 1) and scaled so that ψ ,r = 1 (see Remark 2.9), -Π r is the orthogonal projection on the first eigenspace, spanned by u ,r , of the operator N ,r .

Theorem 5.1 and Corollary 5.7 yield the following WKB approximations (see, for instance, [10, Prop. 5.1] in the context of the Robin Laplacian for a similar estimate).

Proposition 6.3. We have

ψ ,r -Π r ψ ,r L 2 (R 2 + ) = O( ∞ ) , (6.2)
and we can assume that ψ ,r , u ,r = 1 + O( ∞ ) up to the multiplication of Ψ ,r by a complex number of modulus 1. Moreover, let K ⊂ I 2η,r be a compact set. The following estimate

τ e Φr/ √ (Ψ ,r -u ,r ) = O( ∞ ) , (6.3
)

holds in C 1 (K; L 2 (R + )).
Remark 6.4. Note that the choice ψ ,r , u ,r = 1 + O( ∞ ), in addition to the normalization of ψ ,r , completely determines the quasimode, especially the value of α 1,0 which was only defined up to an additive constant in (2.13).

Proof. To get (6.2), we remember that the expansion of the first eigenvalue is given by δ 1 ( ) in Theorem 2.8, we use the spectral gap of the one well case (see for instance (1.3)) and we apply the spectral theorem.

To get (6.3), we use our normalization of the WKB Ansatz, and we choose ϕ(s) = Φr,η,N, (s) = min Φr,N, (s),

√ 1 -θ inf σ∈I 2η,r \Iη,r Φ r (σ) + [s,σ] V (σ) dσ , (6.4) where Φr,N,ε (s) = Φ r (s) -N √ ln max Φ r √ , N . (6.5)
Here N ∈ N, 0 < θ < 1.

Interaction matrix and tunneling effect

We now have all the elements in hand to prove Theorem 1.3. Let us consider the common "single well" groundstate energy µ sw 1 ( ) of the operators N ,r and N , (it depends on η). It results from the Agmon estimates in Corollary 6.1 and Proposition 6.2, and the min-max principle that

µ sw 1 ( ) -Õ(e -S/ √ ) ν 1 ( ) ν 2 ( ) µ sw 1 ( ) + Õ(e -S/ √ ) , (7.1)
where Õ(e -S/ √

) means O(e -(S-ε)/

√

) for all ε > 0.

7.1. WKB quasimodes and approximated basis. In this section, we recall the main lines of the strategy to reduce the asymptotic study of the spectral gap ν 2 ( )-ν 1 ( ) to the study of the two by two interaction matrix. Once the tangential exponential decay of the eigenfunctions is established (see Proposition 6.2), the derivation of the interaction matrix can be done as if we were in dimension one.

In this section, one will precisely refer to the estimates obtained in [4, Section 3] where the strategy has been described in great detail for an electric Hamiltonian in dimension one (see also the Bourbaki exposé [25, Section 2] describing the Helffer-Sjöstrand results in [START_REF] Helffer | Multiple wells in the semiclassical limit[END_REF]).

To construct the interaction matrix, we will use the ground states of the one well problems and use them to provide an approximate basis of the space

E = 2 i=1 Ker(N -ν i ( )) .
We will truncate them, project them on E and orthonormalize them. We define, for α ∈ { , r}, f ,α = χ η,α φ ,α ,

where the φ ,α are essentially the functions φ ,α (see (2.5) and (2.6)) seen on the circle identified with [-L, L), and precisely defined by

φ ,r (σ, τ ) = e -iγ 0 σ/ 2 u ,r (σ, τ ) if -L σ s -η 2 e -iγ 0 (σ-2L)/ 2 u ,r (σ -2L, τ ) if s + η 2 < σ < L , (7.3) φ , (σ, τ ) = e -iγ 0 (σ+2L)/ 2 u , (σ + 2L, τ ) if -L σ s r -η/2 e -iγ 0 σ/ 2 u , (σ, τ ) if s r + η/2 < σ < L . (7.4)
Note here that, due to the flux term γ 0 σ/ 2 , there is no natural extension by periodicity. In the following, we work on Γ identified with [-L, L).

Thanks to Proposition 6.2, the set {f , , f ,r } is quasi-orthonormal in the sense that

f ,α 2 = 1 + Õ(e -2S/ √ ) and f ,α , f ,β = Õ(e -S/ √ ) for α = β .
Furthermore, the function r ,α = (N -µ sw ( ))f ,α , α ∈ { , r}, satisfies, r ,α = Õ(e -S/ √

) .

These estimates, in dimension one, are proved, for instance, in [4, Lemma 3.5].

7.1.2. Projection. Since we want to describe the first two eigenvalues of N , it is convenient to build a basis of E from the quasimodes f ,r and f , . Thus, we consider the new quasimodes, for α ∈ { , r}, g ,α = Πf ,α ,

where Π is the orthogonal projection on E. The following estimate holds, for α ∈ { , r},

g ,α -f ,α + ∂ s (g ,α -f ,α ) = Õ(e -S/ √ ) ,
and its proof is the same as the one of [4, Lemma 3.8].

7.1.3. Orthonormalization. Starting from the basis {g , , g ,r }, we obtain by the Gram-Schmidt algorithm the orthonormal basis {g , , g ,r }. In other words, we have g = gG -1 2 where G is the Gram-Schmidt matrix ( g ,α , g ,β ) α,β∈{r, } . Let M be the matrix of N in the basis {g , , g ,r }. We have Spec(M) = {ν 1 ( ), ν 2 ( )} and, by solving the equation det(M -λId) = 0, we deduce, as in dimension one (see [START_REF] Bonnaillie-Noël | Semiclassical tunneling and magnetic flux effects on the circle[END_REF]Proposition 3.11]), that ν 2 ( ) -ν 1 ( ) = 2|w ,r | + Õ(e -2S/ √ ) , w ,r = r , , f ,r .

(7.6) 7.2. Computing the interaction. We may now estimate the interaction term w ,r . In contrast with Section 7.1, we provide here more details since the proof deviates from the usual computation of the interaction in dimension one. Firstly, the tangential derivative D σ is replaced by the magnetic derivative

D = D σ + -1 γ 0 -τ + c µ κ 2 τ 2 .
(7.7) Lemma 7.1 gives an explicit formula for w ,r involving this magnetic derivative. Secondly, the separation of variables, responsible for the final reduction to an interaction in dimension one, is explained in Section 7.2.2.

7.2.1. An explicit formula for w ,r . The aim of this section is to prove the following. Proof. We have w ,r = (N -µ sw 1 ( ))f , , f ,r = [N , χ η, ]φ , , χ η,r φ ,r . For shortness we use in the following the notation χ r for χ η,r and similarly on the left-side. We recall that χ r and χ do not depend on τ . Thus w ,r = [a -1 D a -1 D , χ ]φ , , χ r φ ,r .

(7.9)

For shortness, we let φ ,α = φ α .

In the following we let S L = (-L, L) × (0, +∞). Writing the commutator, integrating by parts, and using the Leibniz formula, we get where we have used χ χ r = 0 and χ χ r = χ . Note also that χ is supported in (-L, 0). We let φα (σ, τ ) = e iγ(σ,τ )/ φ α (σ, τ ) on S L , where γ satisfies ∂ σ γ(σ, τ ) = τ -γ 0 / -c µ κ τ ). We also recall that φ , and φ ,r are explicitly described in (7.3) and (7.4). We find that where Ψ , (0, τ ) = U Ψ ,r (0, τ ) (there is no phase shift since we are at σ = 0). Using again that Φ r (0) + Φ (0) = S u and the explicit expression of the first term of Ψ ,r , we get By using that a = 1 + o(1) and the exponential decay of b 1, with respect to τ , we find that where we used the explicit expression of the first term of b 1, given in (2.9) and Remark 2.9. Let us now replace b 1, by its first two terms b 1,0 + 1 2 b 1,1 (modulo ) given in (2.9). We recall the following two formulas (see, for instance, [3, Lemma 5.8]): With the first formula, we get

2 +∞ 0 (ξ 0 -τ )U b 1, b 1, dτ =2

1 . Introduction 1 . 1 .

 111 A long-term investigation. 1.1.1. The magnetic Laplacian with Neumann boundary condition. Consider Ω a smooth, open, and simply-connected set of the plane. This article is devoted to the spectral analysis of the magnetic Laplacian L h defined as the self-adjoint operator associated with the quadratic form Q h (ψ) = Ω |(-ih∇ -A)ψ| 2 dx .

Figure 1 .

 1 Figure 1. Modulus of the ground state when Ω is an ellipse.

Figure 2 .

 2 Figure 2. λ 2 (h) -λ 1 (h) as a function of 1/h in the case of the ellipse

Figure 4 . 1 . 2 .a 2 ΩFigure 5 .

 41225 Figure 4. λ 2 (h) -λ 1 (h) as a function of 1/h; numerical simulation (blue) vs our conjecture (dashed)

1 1 2N

 1 and actually of the conjugated operator N ϕ ,r = e ϕ/ ,r e -ϕ/ 1 2

Remark 5 . 5 .

 55 The estimates in Lemmas 5.2, 5.3, and 5.4 are also true for ψ satisfying the Dirichlet condition (instead of the Neumann condition).

7. 1 . 1 .

 11 Truncation. Let χ η,r (respectively χ η, ) be a cut-off function satisfying χ η,r = 1 in {|s -s | 2η} (respectively χ η, = 1 in {|s -s r | 2η}) and χ η,r = 0 in {|s -s | η} (respectively χ η, = 0 in {|s -s r | η}).

Lemma 7 . 1 .

 71 We havew ,r = i +∞ 0 a -1 φ D φ r + D φ φ r (0, τ ) -a -1 φ D φ r + D φ φ r (-L, τ )dτ . (7.8)

w

  ,r = S L D (a -1 D (χ φ ))χ r φ r -χ χ r φ r D (a -1 D φ ) dσdτ = S L a -1 D (χ φ )D (χ r φ r ) -D φ D (χ χ r φ r ) dσdτ = S L a -1 [-i χ φ + χ D φ ] D (χ r φ r ) -D φ χ D (χ r φ r ) + i χ χ r φ r dσdτ = -i S L a -1 χ φ D (χ r φ r ) + D φ χ r φ r dσdτ = i S L a -1 χ χ r φ D φ r + D φ φ r dσdτ = i S L a -1 χ φ D φ r + D φ φ r dσdτ ,

2 2 .a - 1 φa - 1 φ 7 . 2 . 2 .a - 1 φ

 2117221 Using this change of function, we get w ,r = -Sr a -1 D σ χ φ D σ φr + D σ φ φr dσdτ ,where S r = (-L, 0) × R + . Then, by integration by parts,w ,r = w ,r + i +∞ 0 D σ φr + D σ φ φr (0, τ ) -a -1 φ D σ φr + D σ φ φr (-L, τ )dτ , (7.10) with w ,r = Sr χ D σ a -1 φ D σ φr + (D σ φ ) φr dσdτ . Note that w ,r = Sr χ -φ ( D σ a -1 D σ ) φr + ( D σ a -1 D σ ) φ φr dσdτ ,and, coming back to φ α ,w ,r = Sr χ -φ D a -1 D φ r + D a -1 D φ φ r dσdτ .Using the fact that the φ α are eigenfunctions associated with the same eigenvalue, we get w ,r = 0. From (7.10), we deduce thatw ,r = i +∞ 0 D φ r + D φ φ r (0, τ ) -a -1 φ D φ r + D φ φ r (-L, τ )dτ .End of the proof of Theorem 1.3. Let us explain how to analyze the asymptotic behavior of the first term, related to the upper part of Γ, in (7.8): D φ r + D φ φ r (0, τ )dτ . Note that a = 1 + o(1), | c µ τ 2 | = o( -2η

a - 1 ua - 1 Ψ 2 τ 2 Ψ 2 τ 2 Ψ

 112222 , ( D σ -τ + c µ κ 2 τ 2 )u ,r + ( D σ -τ + c µ κ 2 τ 2 )u , u ,r (0, τ )dτ .Then, we use the uniform approximation given in Proposition 6.3, the explicit expression of the WKB Ansatz in Theorem 2.8, and the fact that Φ r (0)+Φ (0) = S u to get , D σ -τ + c µ κ ,r + D σ -τ + c µ κ , Ψ ,r dτ + O( ∞ )e -Su/ 1/2 ,

1 4 e 1 2 1 2

 411 Su/ 1/2 w u ,r = +∞ 0 a -1 U b 1, D σ + ξ 0 -τ + i Φ r (0) + c µ κ 2 τ 2 b 1, dτ + +∞ 0 a -1 D σ + ξ 0 -τ + i Φ (0) + c µ κ 2 τ 2 U b 1, b 1, dτ + O( ∞ ) .

1 4 eξ 0 -τ -i 1 2 0 ξ 0 -τ + i 1 2 0 (ξ 0 1 2

 401001001 Su/ 1/2 w u ,r = +∞ 0 Φ r (0) U b 1, b 1, (0, τ )dτ + +∞ Φ (0) U b 1, b 1, (0, τ )dτ + O( ) =2 +∞ -τ )U b 1, b 1, (0, τ )dτ -2i Φ r (0) f 2 1,0 (0)e -2iα 1,0(0)+ O( ) ,

2 +∞ 0 (ξ 0

 200 -τ )u 2 ξ 0 (τ )dτ = 0 , 1 + -τ )u ξ 0 (τ )(∂ ξ u) ξ 0 (τ )dτ = µ 1 (ξ 0 ) 2 .

  -τ ) U b 1,0 b 1,1 + U b 1,1 b 1,0 dτ + O( ) = -4i f 2 1,0 (0)Φ r (0)e -2iα 1,0 (0) +∞ 0 (ξ 0 -τ )u ξ 0 (τ )v ξ 0 (τ )dτ + O( ) .

  2.4.2. WKB construction. The following fundamental theorem has been established in [3, Theorem 5.6 & Section 5.3.2]. Let us recall that

Let us recall the two Agmon distances

From them, we can construct a weight to estimate the decay of the eigenfunctions of N away from s r and s . Let us consider some periodic versions of Φ r and Φ . We let

The functions Φr and Φ are defined on [-L, L) but not on (s -η, s + η] and (s r -η, s r + η], respectively. Thus, we consider smooth extensions of Φr and Φ such that Φr > Φ near s and Φ > Φr near s r . Thus, Φr and Φ can be seen as 2L-periodic functions.

For the following we shall identify functions on Γ with 2L-periodic functions and mainly consider [-L, L) as interval of integration. The preceding construction of weights on [-L, L) is adapted to this point of view and will allow to give estimates on the two-well case in this setting. Proposition 6.2. Set θ ∈ (0, 1) and consider the 2L-periodic function defined on [-L, L) by

Let ε > 0 and assume that η is small enough. There exist C, 0 > 0 such that for all ∈ (0, 0 ) and all λ eigenvalue of N such that |λ -

and all associated eigenfunction u ∈ Dom (N ),

Proof. Consider an eigenfunction u as in the assumptions. Let χ r be a smooth cutoff function equal to 1 near s r and being 0 near s . Away from the support of χ r , ϕ can be modified and extended to R × R + so that Assumption 4.1 is satisfied. We can then consider ψ = χ r e iσγ 0 / 2 e ϕ/ 1 2 u as a function on R and apply to it Theorem 5.1 with z = λ. We get then

With the second formula, we deduce that

We recognize here the interaction term associated with the effective Hamiltonian (2.14) (see also (1.7)). Using (2.12), and differentiating at 0 formula (2.7), the estimate (7.11) becomes

In the same way, we can deal with the integral corresponding to the down part of the boundary in (7.8):

and we find that

V (-L)A d e -2iα 1,0 (-L) e i(-2Lγ 0 / 2 +2Lξ 0 / ) +O( )

where there is an additional phase shift coming from the ±2L-translation in σ appearing in the second and first expression in (7.3) and (7.4), respectively. Now recalling that, from (7.8),

, and multiplying by e iLf (h) in order to have a more symmetric formula (only the modulus of this quantity is relevant), we get the expression w(h) given in the statement of Theorem 1.3. Then, Theorem 1.3 follows from (7.6) and Proposition 2.7.