Semiclassical tunneling and magnetic flux effects on the circle

HAL Id: hal-01152499

https://hal.archives-ouvertes.fr/hal-01152499v2

Submitted on 25 Aug 2015

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Semiclassical tunneling and magnetic flux effects on the circle

Virginie Bonnaillie-Noël, Frédéric Hérau, Nicolas Raymond

To cite this version:

Virginie Bonnaillie-Noël, Frédéric Hérau, Nicolas Raymond. Semiclassical tunneling and magnetic flux effects on the circle. Journal of Spectral Theory, European Mathematical Society, 2017, 7 (3), pp.771-796. �10.4171/JST/177�. �hal-01152499v2�

Semiclassical tunneling and magnetic flux effects on the circle

V. Bonnaillie-Noël^∗, F. Hérau^†and N. Raymond^‡ August 25, 2015

Abstract

This paper is devoted to semiclassical tunneling estimates induced on the circle by a double well electric potential in the case when a magnetic field is added. When the two electric wells are connected by two geodesics for the Agmon distance, we highlight an oscillating factor (related to the circulation of the magnetic field) in the splitting estimate of the first two eigenvalues.

Keywords. WKB expansion, magnetic Laplacian, Agmon estimates, tunnel effect.

MSC classification. 35P15, 35J10, 81Q10.

1 Introduction and motivations

1.1 Motivation

This paper is devoted to the spectral analysis of the self-adjoint realization of the electro- magnetic Laplacian (hDs+a(s))²+V(s) on L²(S¹) where the vector potential aand the electric potentialV are smooth functions on the circleS¹ and where we used the standard notation D= −i∂. In particular we are interested in estimating the spectral gap, in the semiclassical limit, between the first two eigenvalues when the electric potential admits a double symmetric well.

Assumption 1.1 In the parametrizationR∋s7→e^is ∈S¹, the functionV admits exactly two non degenerate minima at0andπwithV(0) =V(π) = 0and satisfiesV(π−s) =V(s).

It is well-known that, in dimension one, there is no magnetic field in the sense that the exterior derivative of the 1-form a(s) ds is zero. Nevertheless, since S¹ is not simply con- nected, we cannot gauge outathanks to an appropriate unitary transform: The circulation ofawill remain. This can be explained as follows. Let us define ϕ(s) =Rs

0 (a(σ)−ξ₀) dσ

∗Département de Mathématiques et Applications (DMA - UMR 8553), PSL, CNRS, ENS Paris, 45 rue d’Ulm, F-75230 Paris cedex 05, Francebonnaillie@math.cnrs.fr

†LMJL - UMR6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, F-44322 Nantes cedex 3, France,frederic.herau@univ-nantes.fr

‡IRMAR - UMR8625, Université Rennes 1, CNRS, Campus de Beaulieu, F-35042 Rennes cedex, France nicolas.raymond@univ-rennes1.fr

with ξ₀ = Rπ

−πa(σ) dσ so that ϕ is well-defined and smooth on S¹. Then let us consider the conjugate operator

L_h= e^iϕ/h

(hD_s+a(s))²+V(s) e^−iϕ/h

= (hDs+a(s)−ϕ^′(s))²+V(s)

= (hD_s+ξ₀)²+V(s).

The aim of this paper is to investigate the effect of the circulationξ₀ofaon the semiclassical spectral analysis.

1.2 Results

The analysis of this paper gives an asymptotic result of the splitting between the first two eigenvalues λ₁(h) and λ₂(h) of L_h, when the potential V has some symmetries.

Theorem 1.2 Let κ be the geometric constant defined by κ=

rV^′′(0)

2 . (1.1)

Then, as soon as h is small enough, there are only two eigenvalues of L_h in the interval I_h = (−∞,2κh) and they both satisfy

for j = 1,2, λ_j(h) =κh+o(h) as h→0.

Let us define the (positive) Agmon distances

S_u= Z

[0,π]

pV(σ) dσ, S_d= Z

[0,−π]

pV(σ) dσ, and S= min{S_u,S_d},

and the two constants

A_u= exp − Z

[0,^π₂]

∂_σ√ V −κ

√V dσ

!

, A_d= exp Z

[−^π₂,0]

∂_σ√ V +κ

√V dσ

! .

Then we have the spectral gap estimate

λ₂(h)−λ₁(h) = 2|w₀(h)|+h^3/2O(e^−S/h), (1.2) with

w₀(h) = 2h^1/2 rκ

π

Au

r V π

2

e^{iξ0π−h Su} +A_d r

V

−π 2

e^{−iξ0hπ−Sd}

. (1.3)

Remark 1.3 The constantsS_uandS_dcorrespond to integrations in the upper and respec- tively lower part of the circle for the Agmon distance. Then two situations may occur:

1. If the two Agmon distances S_u and S_d are different, only one term in the sum (1.3) definingw₀(h)is predominent andw₀(h) is not zero forhsmall enough. In this case, there exists a unique geodesic linking the two wells, corresponding either to the upper part of the circle, or to the lower part. Moreover, the circulation ξ₀ is not involved in the estimate of the tunneling effect: we get an estimate similar to what happens in the purely electric situation (see [3,10] and more generally [11,5,6]).

2. IfS_u=S_d, the situation is completely different: due to the circulation, the interaction term w₀(h) can vanish for some parameters h and the eigenvalues can be equal up to an error of order O(h^3/2e^−S/h). This corresponds to a crossing (up to the forementionned error) of these first two eigenvalues. Note that this does not mean that the eigenvaluesλ₁(h)andλ₂(h) effectively cross but the gap is inO(h^3/2e^−S/h).

When the potential V is even, we are in the second situation and we have A_u=A_d=A, S_u=S_d=S, V

−π 2

=V π 2

,

and we immediately deduce the following splitting estimate.

Theorem 1.4 Assume that V is even, then λ₂(h)−λ₁(h) = 8h^1/2A

r Vπ

2 rκ

π cos

ξ0π h

e^−S/h+h^3/2O e^−S/h

. Organization of the paper and strategy of the proofs

In order to prove Theorem1.2, we will follow the strategy developed by Helffer and Sjös- trand in [5,6] (see also the lecture notes by Helffer [4, Section 4]) for the pure electric case.

Thanks to a change of gauge, the investigation of the present paper can be reduced to the electric case only locally and not globally due to the circulationξ₀. In Section 2, we recall the WKB approximations of the first eigenfunction in the simple well case. In Section3, we explain how we can construct a2by2Hermitian matrix (the so-called “interaction matrix”) from the eigenfunctions of each well, which describes the splitting of first two eigenvalues ofL_h. This strategy is well-known (see for instance [2] for a short presentation and [3] for a complete description of the main terms) and is given here for completeness. The aim of the present paper is to highlight its oscillatory consequences on the interaction term in the non zero circulation case. To authors’ knowledge this strategy was never applied in this context and the understanding of this model might be a main step towards the estimate of the pure magnetic tunnel effect in higher dimension (see [7] and our recent contribution [1, Section 5.3]). Note here that the influence of the circulation on the first eigenvalue has also been analyzed in [4, Theorem 7.2.2.1] when V admits a unique and non degenerate minimum. This question was also tackled by Outassourt in [8] in a periodic framework.

Finally, in Section 4, we analyze the semiclassical behavior of the interaction matrix in terms of the WKB approximations.

2 Simple well cases

In this section we study simple well configurations. First, we consider the well s= 0. In the last part, we explain how we can transfer what was done for the well s= 0to the well s=π thanks to a unitary transform.

2.1 Local reduction to the pure electric situation

Let us introduce the Dirichlet realization attached to the well s = 0. For any ρ ∈ (0, π], we define

Br(ρ) :=B(0, ρ) = (−ρ, ρ).

Givenη >0, let us considerL_h,r the Dirichlet realization of(hD_s+ξ₀)²+V(s)on the space L²(B^r(π−η), ds). SinceB^r(π−η) is simply connected, we can perform a gauge transform so that the study ofL_h,r is reduced to the one of the operator

Lh,r = e^iξh0sL_h,re^−iξh0s =h²D²_s+V(s), (2.1) defined onDom(Lh,r) = H²(B^r(π−η))∩H¹₀(B^r(π−η)). Let us denote byλ(h)the ground state energy ofLh,randφ_h,rthe positive andL²-normalized eigenfunction ofLh,rassociated with the lowest eigenvalue λ(h). We have

Lh,rφ_h,r= h²D_s²+V

φ_h,r =λ(h)φ_h,r on B^r(π−η).

Then, by gauge tranform, the function defined on B^r(π−η) by

ϕ_h,r(s) = e^−iξ0hsφ_h,r(s), (2.2) is aL²-normalized eigenfunction of L_h,r associated withλ(h).

In the next section, we recall some results about the WKB analysis of the operator Lh,r. In Section 2.3 we recall Agmon estimates and in particular prove the exponential decay of eigenfunctions. In the following subsection, we establish uniform estimates of the difference between the eigenfunctions and the WKB quasimodes.

2.2 WKB approximations in a simple well

This section is devoted to recall the structure of the first WKB quasimode of Lh,r. Lemma 2.1 The asymptotic WKB series for the first quasimode of Lh,r is given by

ψ_h,r =χrΨ_h,r, with Ψ_h,r(s) =h^−1/4e^−Φr(s)h X

j>0

h^jaj(s), ∀s∈ B^r(π), (2.3) where

i) χr is a smooth cut-off function supported on B^r(π−η) with06χr61 andχr = 1 on B^r(π−2η),

ii) Φr is the standard Agmon distance to the well at s= 0:

Φr(s) = Z

[0,s]

pV(σ) dσ, ∀s∈ B^r(π), (2.4)

iii) a₀ is a solution of the associated transport equation Φ^′_r∂_sa₀+∂_s Φ^′_ra₀

=κa₀, (2.5)

withκ defined in (1.1). It can be given explicitly by a0(s) =κ

π 1/4

exp

− Z s

0

Φ^′′_r(σ)−κ 2Φ^′_r(σ) dσ

, ∀s∈ B^r(π).

The function ψ_h,r is a L²-normalized WKB quasimode in the sense that

e^Φr/h(Lh,r−µr(h))ψ_h,r =O(h^∞) in L²(B^r(π−2η)), (2.6) where µr(h) is the first quasi-eigenvalue given by the asymptotic series

µ_r(h) =κh+X

j>2

µ_r,jh^j. Moreover, we have

∂_sψ_h,r(s) =−h^−5/4Φ^′_r(s)e^−Φr(s)h a₀(s)(1 +O(h)), ∀s∈ B^r(π−2η).

Proof: The proof of the result is classical (see [3,10]) and we just recall the computation ofa₀, which is quite easy since we are in dimension one. For s∈ Br(π−η), we check that

V(s) =κ²s²+O(s³) and Φr(s) =κs²

2 +O(s³).

Solving the transport equation (2.5), we get a₀(s) =K₀exp

− Z s

0

Φ^′′_r(σ)−κ 2Φ^′_r(σ) dσ

, where K₀ is a normalization constant determined by

1 = Z

B^r(π−η)

ψ_h,r(s)²ds=K₀²h^−1/2 Z

R

e^−κs2/hds(1 +O(h)) =K₀² rπ

κ +O(h).

ThusK₀ = (κ/π)^1/4.

The explicit form of the quasimode will be used for the computation of the splitting between the first two eigenvalues of L_h in Section 4.

2.3 Agmon estimates and WKB approximation

Let us recall the following lemma (see [9] for a close version) which will be useful to prove localization estimates.

Lemma 2.2 Let H be a Hilbert space and P and Q be two unbounded and symmetric operators defined on a domain D ⊂ H. We assume that P(D) ⊂ D, Q(D) ⊂ D and [[P, Q], Q] = 0on D. Then, for u∈D, we have

RehP u, P Q²ui=kP Quk²− k[Q, P]uk².

This lemma will be applied with P the derivation and Qthe multiplication by a smooth function.

With the aim of proving that our Ansatz is a good approximation of the first eigen- function φ_h,r ofLh,r, we first establish some Agmon estimates.

Proposition 2.3 Let Φ be a Lipschitzian function such that

V(s)− |Φ^′(s)|² >0, ∀s∈ B^r(π−η), (2.7)

and let us assume that there exist M >0 and R >0 such that for allh∈(0,1),

V(s)− |Φ^′(s)|² >M h, ∀s∈ Br(π−η)∩∁Br(Rh^1/2), (2.8)

|Φ(s)|6M h, ∀s∈ B^r(Rh^1/2). (2.9) Then, for all C₀ ∈ (0, M), there exist positive constants c, C such that, for h ∈ (0,1), z∈[0, C₀h], u∈Dom(Lh,r),

chke^Φ/hukL²(B^r(π−η)) 6ke^Φ/h(Lh,r−z)ukL²(B^r(π−η))+ChkukL²(B^r(π−η)∩B^r(Rh^1/2)), (2.10) and

hD_s

e^Φ/hu²

L²(B^r(π−η))6 C

hke^Φ/h(Lh,r−z)uk²_L²_{(Br(π−η))}+Chkuk²_L2(B^r(π−η)∩B^r(Rh^1/2)). (2.11) Proof: We apply Lemma 2.2withP =hDs,Q= e^Φ/h and u∈Dom(Lh,r)to get

Re Z

B^r(π−η)

hD_su hD_s

e^2Φ/hu ds

!

= Z

B^r(π−η)|hD_s(e^Φ/hu)|²ds− Z

B^r(π−η)|Φ^′(s)|²e^2Φ/h|u|²ds.

Integrating by parts, adding the electric potential V, and recalling that Lh,r =h²D_s²+V, we find

Z

B^r(π−η)|hD_s(e^Φ/hu)|²ds+ Z

B^r(π−η)

(V(s)− |Φ^′(s)|²)e^2Φ/h|u|²ds

= Re Z

B^r(π−η)Lh,rue^2Φ/huds

!

6ke^Φ/hLh,rukke^Φ/huk. Using (2.7) and (2.8), we get

Z

B^r(π−η)|hD_s(e^Φ/hu)|²ds+M h Z

B^r(π−η)∩∁B^r(Rh^1/2)

e^2Φ/h|u|²ds6ke^Φ/hLh,rukke^Φ/huk. Thanks to (2.9),Φ/his uniformly bounded with respect to honBr(Rh^1/2) and we deduce

khD_s(e^Φ/hu)k²+M hke^Φ/huk² 6ke^Φ/hLh,rukke^Φ/huk+C_Rhkuk²_L²_{(Br(π−η)∩Br(Rh}^1/2₎₎. For|z|6C₀h, we get

khDs(e^Φ/hu)k²+ (M −C0)hke^Φ/huk²

6ke^Φ/h(Lh,r−z)ukke^Φ/huk+C_Rhkuk²_L2(B^r(π−η)∩B^r(Rh^1/2)). (2.12) SinceC₀ < M, this gives (2.10). Then we combine (2.12) with (2.10) to get (2.11).

Proposition 2.4 Let c₀ >0 such that

V(s)>c₀s² and Φr(s)>c₀s², ∀s∈ B^r(π−η). (2.13) Proposition 2.3applies in the following cases:

(a) for ε∈(0,1), the rough weight Φ_r,ε =√

1−εΦr withR >0 and M =c₀εR²,

(b) for N ∈ N∗ and h ∈ (0,1), the precised weight Φe_r,N,h = Φ_r −N hln max ^Φ_h^r, N , withR=q

N

c0 andM =Ninf_{Br(π−η)Φ}^V

r,

(c) for ε∈(0,1), N ∈N^∗ andh ∈(0,1), the intermediate weight Φb_r,N,h(s) = min

(

Φe_r,N,h(s),√

1−ε inf

t∈suppχ^′_r Φr(t) + Z

[s,t]

pV(σ) dσ

!)

, (2.14)

withR=q

N

c0 andM =Nmin

ε,inf_{Br(π−η)Φ}^V

r

, where we recall that χ^′_r is supported in Br(π−η)\ Br(π−2η).

Proof: Note that the existence of c0 > 0 is guarranted since the function V admits a unique and non degenerate minimum onB^r(π−η)at0. Using the definition (2.4) ofΦr, we have directly (2.9) forΦ_r and consequently for the other weights Φe_r,N,h and Φb_r,N,h which are smaller. Let us now prove (2.7) and (2.8) for each choice.

(a) We have V − |Φ^′_r,ε|² = εV. Combining this with the positivity of V or (2.13) gives (2.7) and (2.8).

(b) On{Φ_r < N h}, we have |Φe^′_r,N,h|² =|Φ^′_r|² =V. On{Φr >N h}, we get

Φe^′_r,N,h = Φ^′_r

1−N h Φ_r

, so that

V − |Φe^′_r,N,h|² =VN h Φ_r

2−N h Φ_r

>N hV

Φ_r >cN h>0, (2.15) since the function V /Φ_r is continuous and bounded from below by some c > 0 on B^r(π−η). This proves (2.7). According to (2.13), for all R >0andh∈(0,1), we have Φr>c₀R²hon B^r(π−η)∩∁B^r(Rh^1/2). In particular, for R>R₀=p

N/c₀, we get B^r(π−η)∩∁B^r(Rh^1/2)⊂ {Φr>N h}.

Recalling (2.15), this establishes (2.8).

(c) We notice that the infimum in the definition of Φb_r,N,h is a minimum. Thus, almost everywhere on B^r(π−η), we have either |Φb^′_r,N,h|=√

1−ε√

V, or |Φb^′_r,N,h| =|Φe^′_r,N,h|. Then we apply Proposition 2.4(a) and (b).

Remark 2.5 The weights introduced in Proposition 2.4 are essential to prove that the eigenfunctions of Lh,r are approximated by their WKB expansion in the space L²(e^Φr/hds) (as we will see in Proposition 2.7). The rough weight Φ_r,ε = √

1−εΦ_r would not be enough to get the main term of the tunneling estimate (1.2). The precised weight Φe_r,N,h is introduced to get an approximation of the eigenfunctions in the spaceL²(h^−Ne^Φr/hds)with a fixed and large N ∈N; the factor h^−N will be absorbed since the approximation is valid modulo O(h^∞). The intermediate weight Φb_r,N,h is only a slight modification ofΦe_r,N,h (see Lemma2.6) on ∁K where the weight Φe_r,N,h becomes bad.

We end this section with some properties, which will be used later, about the weightΦb_r,N,h defined in (2.14).

Lemma 2.6 Let K be a compact with K⊂ B^r(π−2η). We consider the weight defined in Proposition 2.4(c). For all N ∈N∗, there existsε₀ such that for all0< ε < ε₀, there exist h₀ >0 and R >0 such that, for allh∈(0, h₀), we have

(1) Φb_r,N,h6Φ_r on Br(π−η), (2) Φb_r,N,h=Φe_r,N,h on K, (3) Φb_r,N,h=√

1−εΦr on suppχ^′_r. Proof:

(1) The first inequality comes immediately from the definition ofΦb_r,N,h.

(2) By continuity and sinceK and the complementary ofB^r(π−2η) are disjoint compacts, there existsε₀ such that for all 0< ε < ε₀ and for all s∈K,

Φe_r,N,h(s)6Φr(s)6√

1−ε inf

t∈suppχ^′_r Φr(t) + Z

[s,t]

pV(σ) dσ

! . By definition ofΦb_r,N,h, we deduce that Φb_r,N,h=Φe_r,N,h on K.

(3) Let us now consider s∈suppχ^′_r. There exists h₀ >0 (depending on ε) such that for allh∈(0, h0), we have





inf_t∈suppχ^′r

Φr(t) +R

[s,t]

pV(σ) dσ

= Φr(s), Φe_r,N,h(s) = Φr(s) +O(hlnh) >√

1−εΦr(s).

ThusΦb_r,N,h =√

1−εΦr onsuppχ^′_r.

2.4 Weighted comparison between quasimodes and eigenfunctions We may now provide the approximation of φ_h,r by the WKB construction ψ_h,r defined in (2.3). Let us introduce the projection

Πrψ=hψ, φ_h,riφ_h,r.

Proposition 2.7 Let K be a compact set with K ⊂ Br(π −2η). We have both in the L^∞(K) and in the L²(K) sense

e^Φr/h(ψ_h,r−Π_rψ_h,r) =O(h^∞), (2.16) e^Φr/hD_s(ψ_h,r−Π_rψ_h,r) =O(h^∞). (2.17)

Proof: Let us apply Proposition 2.3withu=ψ_h,r−Πrψ_h,r andz=λ(h)and the weight Φ =Φb_r,N,h defined in Proposition 2.4(c). We get

chke^Φbr,N,h/huk²_L²_{(Br(π−η))}+hD_s

e^Φbr,N,h/hu²

L²(B^r(π−η))

6Ch⁻¹ke^Φbr,N,h/h(Lh,r−λ(h))ψ_h,rk²_L²_{(Br(π−η))}+Chkuk²_L2(B^r(π−η)∩B^r(Rh^1/2)). (2.18) Let us investigate the first term in the r.h.s. of (2.18). Using Lemma 2.1, we have, in the sense of differential operators,

e^Φbr,N,h/h(Lh,r−λ(h))ψ_h,r = e^Φbr,N,h/h(Lh,r−λ(h))χ_rΨ_h,r

= e^Φbr,N,h/hχr(Lh,r−λ(h))Ψ_h,r+ e^Φbr,N,h/h[Lh,r, χr]Ψ_h,r

= e^{(Φbr,N,h−Φr)/h}OL^∞(B^r(π−η))(h^∞) + e^{(Φbr,N,h−Φr)/h}OL^∞(suppχ^′_r)(1). (2.19) Using Lemma 2.6, there exists c₁ >0 such that

e^{(bΦr,N,h−Φr)/h}OL^∞(B^r(π−η))(h^∞) =OL^∞(B^r(π−η))(h^∞),

e^{(Φbr,N,h−Φr)/h}= e^{−(1−√1−ε)Φr/h}6e^−c1/h =O(h^∞) onsuppχ^′_r.

Putting these estimates in (2.19), we deduce that

Ch⁻¹ke^Φbr,N,h/h(Lh,r−λ(h))ψ_h,rk²_L²_{(Br(π−η))} =O(h^∞). (2.20)

Let us deal with the second term in the r.h.s. of (2.18). By definition, Π_rψ_h,r belongs to the kernel of Lh,r −λ(h) and, since the gap between the lowest eigenvalues of Lh,r is of orderh, the spectral theorem proves that there existsc >0 such that

chkukL²(B^r(π−η))=chkψ_h,r−Π_rψ_h,rkL²(B^r(π−η))

6k(Lh,r−λ(h))uk_L²_{(Br(π−η))}=k(Lh,r−λ(h))ψ_h,rk_L²_{(Br(π−η))}=O(h^∞), (2.21) where we have used (2.6) for the last estimate.

Consequently (2.18) becomes

chke^Φbr,N,h/huk²L²(B^r(π−η))+hD_s

e^Φbr,N,h/hu²

L²(B^r(π−η)) =O(h^∞). (2.22) By Sobolev embedding, we deduce that, as well as in L^∞(B^r(π−η))as in L²(B^r(π−η)),

he^Φbr,N,h/hu=O(h^∞).

To deduce (2.16), we first recall Lemma 2.6 (2), so that Φb_r,N,h = Φe_r,N,h on K. Then we have, in L^∞(K) and inL²(K),

he^Φer,N,h/hu=O(h^∞). (2.23)

Now the definition ofΦe_r,N,h (given in Proposition 2.4(b)) implies that inL^∞(K) we have

e^{(Φr−Φer,N,h)/h}=O(h^−N). (2.24)

By using (2.23), we get, inL^∞(K) and in L²(K),

e^Φr/hu=h⁻¹O(h^−N)O(h^∞) =O(h^∞). (2.25) This proves (2.16).

Now we deal with theL²(K) estimate in (2.17). Let us recall that Lemma 2.6(2) gives

Φb_r,N,h=Φe_r,N,h on K. (2.26)

We first write that

e^Φbr,N,h/hhD_su²

L²(K)

6hD_s

e^Φbr,N,h/hu

L²(K)+eΦ^′_r,N,h

e^Φbr,N,h/hu

L²(K). (2.27) Using that |Φe^′_r,N,h|² 6V which is bounded and (2.22), we deduce, by Lemma (2.6) (2),

e^Φer,N,h/hhD_su

L²(K) =e^Φbr,N,h/hhD_su²

L²(K)=O(h^∞). (2.28) Next using (2.24), we have the desiredL²(K) estimate in (2.17):

e^Φr/hhD_su

L²(K)=O(h^∞). (2.29)

As a complementary result and for further use, let us do a new commutation with hD_s. We have

khD_s(e^Φr/hu)kL²(K)6ke^Φr/hhD_sukL²(K)+kΦ^′_re^Φr/hukL²(K). Using (2.16) in L²(K), the fact that|Φ^′_r|² =V,V is bounded and (2.29), we infer

hD_s

e^Φr/hu

L²(K)=O(h^∞). (2.30)

We end up with the L^∞(K) estimate in (2.17). From (2.20) restricted to K, (2.26) and (2.24), we have

ke^Φr/h(Lh,r−λ(h))ψ_h,rk²_L²_(K)=O(h^∞). (2.31) SinceΠrψ_h,r is an eigenfunction, we get

e^Φr/h(Lh,r−λ(h))u²

L²(K)=e^Φr/h(Lh,r−λ(h)) (ψ_h,r−Πrψ_h,r)²

L²(K)=O(h^∞).

By definition ofLh,r, this provides

e^Φr/h h²D²_s+V(s)−λ(h) u²

L²(K)=O(h^∞).

Thanks to (2.16) in L²(K) and sinceλ(h) =O(h) andV is bounded, we infer

e^Φr/hh²D²_su

L²(K) =O(h^∞). (2.32)

We have

(h²D²_s)

e^Φr/hu

= e^Φr/h(h²D²_s)u+h

h²D²_s,e^Φr/hi

u, (2.33)

where h

h²D²_s,e^Φr/hi

u=−e^Φr/h 2ihΦ^′_rD_su+|Φ^′_r|²u+hΦ^′′_ru

. (2.34)

SinceΦ^′_r andΦ^′′_r are bounded functions, we can estimate each term in (2.34) thanks to the L²(K)estimate given in (2.16) and (2.17) and we get

h

h²D²_s,e^Φr/hi

u_L2(K)=O(h^∞). (2.35) From (2.32), (2.35) and (2.33), we get the following estimate

kh²D²_s(e^Φr/hu)kL²(K)=O(h^∞). (2.36) From Sobolev embedding, we deduce from (2.36) and (2.30) that

khD_s(e^Φr/hu)kL^∞(K)=O(h^∞). (2.37) Now doing again the commutation betweenhD_s and e^Φr/hgives

ke^Φr/hhD_sukL^∞(K)6khD_s(e^Φr/hu)kL^∞(K)+kΦ^′_re^Φr/hukL^∞(K). (2.38) Using then (2.37) for the term with the derivative, the fact thatΦ^′_r is bounded and (2.16) in the L^∞(K) sense, we get

ke^Φr/hhD_sukL^∞(K)=O(h^∞). (2.39) The proof of the L^∞(K) estimate in (2.17) is complete, and so is the proof of Proposi- tion2.7.

Remark 2.8 The estimate given by Proposition2.7is crucial and will be used in particular to get an estimate at the points ±π/2 in Section 4.

2.5 From one well to the other

In this section we explain how to transfer the informations for the well configurations= 0 to the one ofs=π. In the following we index by ℓthe quantities, operators, quasimodes, etc. related to the left-hand side well whose coordinate is s=π.

Let Bℓ(ρ) := B(π, ρ) = (π−ρ, π+ρ), for any ρ ∈ (0, π). The Dirichlet realization of (hD_s+ξ₀)²+V(s)on L²(Bℓ(π−η),ds)is denoted L_h,ℓ.

Let us consider the transformU defined by

U(f)(s) =f(π−s). (2.40)

For any ρ ∈ (0, π], the application U defines an anti-hermitian unitary transform from L²(B^r(ρ),ds)ontoL²(Bℓ(ρ),ds). According to Assumption1.1about the symmetry ofV, the two operators L_h,r and L_h,ℓ are unitary equivalent:

L_h,ℓ=UL_h,rU⁻¹. (2.41)

Thus they have the same spectrum and λ(h) is the first common eigenvalue. The eigen- functions ofL_h,ℓ are obviously deduced from those ofL_h,r thanks to the unitary transform U. We let φ_h,ℓ=U φ_h,r. Then the function φ_h,ℓ is a positiveL²-normalized eigenfunction

ofLh,ℓ (the Dirichlet realization of h²D²_s+V on L²(Bℓ(π−η),ds)) associated withλ(h).

Thus we have

Lh,ℓφ_h,ℓ= (h²D_s²+V)φ_h,ℓ=λ(h)φ_h,ℓ onBℓ(π−η).

The functionϕ_h,ℓ defined onBℓ(π−η) by

ϕ_h,ℓ=U ϕ_h,r, (2.42)

is an eigenfunction ofL_h,ℓ associated with λ(h) and satisfies

ϕ_h,ℓ(s) = e^iξ0hπe^−iξ0hsφ_h,ℓ(s), ∀s∈ Bℓ(π−η). (2.43)

3 Double wells and interaction matrix

3.1 Estimates of Agmon

In this section, we discuss the estimates of Agmon in the double well situation. These global estimates have a similar proof as in Proposition 2.3. From now on, Φ will denote the global Agmon distance

Φ(s) = min(Φr(s),Φ_ℓ(s)), with the Agmon distances defined as in (2.4) by

Φ_r(s) = Z

[0,s]

pV(σ) dσ, ∀s∈ Br(π) and Φ_ℓ(s) = Z

[π,s]

pV(σ) dσ, ∀s∈ Bℓ(π). (3.1)

The functionΦis Lipschitzian and satisfies the eikonal equation |Φ^′|² =V.

Proposition 3.1 Let us consider the ρ-neighborhood of the wells on S¹ identified with R/2πZ

Bb(ρ) =B^r(ρ)∪ Bℓ(ρ).

For all ε ∈ (0,1), C₀ > 0, there exist positive constants h₀, A, c, C such that, for all h∈(0, h₀), z∈[0, C₀h] andu∈ C^∞(S¹),

chke^{√1−εΦ/h}ukL²(S¹)6ke^{√1−ε)Φ/h}(L_h−z)ukL²(S¹)+Chkuk_L²₍Bb(Ah^1/2)), (3.2) and

(hD_s+ξ₀)

e^{√1−εΦ/h}u²

L²(S¹)

6 C

hke^{√1−εΦ/h}(L_h−z)uk²L²(S¹)+Chkuk²_L2(Bb(Ah^1/2)). (3.3) Proof: Forε∈(0,1), we let Φ_ε= √

1−εΦ. We apply Lemma 2.2 withP =hD_s+ξ₀, Q= e^Φε/h, and use that Φ is Lipschitzian. After an integration by parts, we obtain

Re Z

S¹

(hD_s+ξ₀)²ue^2Φε/huds= Z

S¹

(hD_s+ξ₀)

e^Φε/hu² ds− Z

S¹|Φ^′_ε|²e^2Φε/h|u|²ds.

Adding the electric potential V and recalling thatL_h = (hD_s+ξ₀)²+V, we get Z

S¹

(hD_s+ξ₀)

e^Φε/hu² ds+ Z

S¹

V − |Φ^′_ε|²

e^2Φε/h|u|²ds= Re Z

S¹

L_hue^2Φε/huds 6e^Φε/hL_hue^Φε/hu,

so that Z

S¹

(hD_s+ξ₀)

e^Φε/hu² ds+ Z

S¹

εVe^2Φε/h|u|²ds6e^Φε/hL_hu

e^Φε/hu. The rest of the proof is identical to the one of Proposition2.3, using again the non degen- eracy of the minima ofV at s= 0and s=π as in the proof of Proposition2.4. Then we get (3.3).

As a direct consequence of Proposition3.1 withu=ϕand z=λ, we get

Corollary 3.2 For all ε∈(0,1), there exist C >0 and h₀ >0 such that, for h ∈(0, h₀) andϕ an eigenfunction of L_h associated with λ=O(h),

ke^{√1−εΦ/h}ϕkL²(S¹)6CkϕkL²(S¹) and khD_s(e^{√1−εΦ/h}ϕ)kL²(S¹)6CkϕkL²(S¹). 3.2 Rough estimates on the spectrum

The main purpose of this article is to get an exponentially precise description of the lowest eigenvalues of L_h. For this we use the one well unitary equivalent operators L_h,r and L_h,ℓ defined respectively onB^r(π−η)andBℓ(π−η). Let us consider the quadratic approximation ofLh,r defined on Rby

h²D_s²+1

2V^′′(0)s².

From a direct and standard analysis, we know that its spectrum is discrete, made of the simple eigenvalues (2j+ 1)κh for j ∈ N. In particular, κh is a single eigenvalue in the interval I_h = (−∞,2κh). By quadratic approximation, we know that for any fixed η,L_h,r has only a single eigenvalue λ(h) inI_h satisfying

λ(h) =κh+O(h^3/2), (3.4)

since the eigenvalues are of type

(2j+ 1)κh+O(h^3/2), j>0. (3.5) In order to estimate the first two eigenvalues of the full operator L_h on S¹, which will appear to be very close toλ(h)and the only ones inI_h, we need to write the matrix of L_h on an appropriate invariant two dimensional subspace. For this we need to extend onS¹ the quasimodes built in the simple well cases.

Notation 3.3 We will use the following conventions and notation:

(i) We identify functions on S¹ and2π-periodic functions of the variables∈R. We also extend by0 onS¹\ B^r(π−η) the functions χr andϕ_h,r and by 0on S¹\ Bℓ(π−η)the functionsχ_ℓ and ϕ_h,ℓ.

(ii) We index by α and β the points r and ℓ, and identify r with 0 and ℓ with π on S¹. For convenience, we also denote by α¯ the complement of α in {r, ℓ}.

(iii) for a given function f, we say that a function is Oe(e^{−f /h}) if, for all ε >0, η >0, it isO(e^{(ε+γ(η)−f)/h}), where lim_η→0γ(η) = 0 (see [5,6, 2]).

Definition 3.4 We introduce two quasimodes f_h,r andf_h,ℓ defined on S¹ by

f_h,r =χrϕ_h,r and f_h,ℓ=χ_ℓϕ_h,ℓ, (3.6) with

χ_ℓ=U χr. (3.7)

We have in particular f_h,ℓ=U f_h,r. Since we want to compare the operatorsL_h andL_h,α, we first computeL_hf_h,α.

Lemma 3.5 Let us denote, for α∈ {ℓ,r},

r_h,α= (L_h−λ(h))f_h,α= (L_h,α−λ(h))χ_αϕ_h,α = [L_h,α, χ_α]ϕ_h,α. (3.8) For η sufficiently small, we have

(i) r_h,α(s) =Oe(e^−S/h),

(ii) hr_h,α, f_h,αi=Oe(e^−2S/h) and hr_h,α, f_h,βi=Oe(e^−S/h) for α6=β, (iii) hf_h,α, f_h,αi= 1 +Oe(e^−2S/h) andhf_h,α, f_h,βi=Oe(e^−S/h) for α6=β,

(iv) Let us introduce the finite dimensional vectorial space F=span{f_h,r, f_h,ℓ}. Then, for h small enough, dimF = 2.

Proof:

(i) Thanks to Corollary 3.2, we get inL^∞(S¹)and L²(S¹) sense that, for all ε >0, e^{√1−εΦα(s)/h}r_h,α(s) =O(1).

Since the support of [L_h,α, χ_α]is included inBα¯(2η), we get:

r_h,α(s) =Oe(e^−S/h). (3.9) (ii) is a consequence of (i) and the location of the support ofr_h,α.

(iii) We first recall, from Proposition 2.3and Proposition2.4(a), that

ϕ_h,α=Oe(e^−Φα/h), (3.10) in L²(Bα(π−η)) and H¹(Bα(π−η)). According to Agmon estimates, this gives in particular

hf_h,α, f_h,αi= 1 +Oe(e^−2S/h). (3.11) Forα6=β, using (3.10), the supports ofχ_α and χ_β and sinceΦ_α+ Φ_β >S, we get

hf_h,α, f_h,βi=Oe(e^−S/h).

(iv) The previous estimates imply that dimF = 2 for h small enough.

In the following series of lemmas, we show that the first two eigenvalues are exponentially close to λ(h) and are the only ones inI_h.

Lemma 3.6 Let us define G = range (1Ih(L_h)). Then dist(sp(L_h), λ(h)) = Oe(e^−S/h) and dim G>2.

Proof: This is a consequence of the spectral theorem. Indeed, using Lemma3.5, we get

∀u∈ F, k(L_h−λ(h))uk=Oe(e^−S/h)kuk. This achieves the proof sincedimF = 2.

Now we can prove the following.

Lemma 3.7 We have

(i) h(L_h−λ(h))u, ui>κhkuk², for all u∈ G^⊥, (ii) dimG = 2,

(iii) sp(L_h)∩I_h ⊂[λ(h)−Oe(e^−S/h), λ(h) +Oe(e^−S/h)].

Proof:

(i) We use again a localization formula and consider a partition of unity (χe_ℓ,χer) such that

e

χ²_ℓ +χe²_r = 1 on S¹,

where χe_ℓ =Uχer and χer is supported inB^r(3π/2), equal to 1 in B^r(π/2). Writing the

“IMS” formula, we deduce that, for u∈ F^⊥, h(L_h−λ(h))u, ui= X

α∈{ℓ,r}

h(L_h−λ(h))χe_αu,χe_αui+O(h²)kuk². LetΠ_α be the orthogonal projection onϕ_h,α, then

e

χ_αu−Π_αχe_αu∈ hϕ_h,αi^⊥. Withκ defined in (1.1), we get

h(L_h−λ(h))u, ui=X

α∈{ℓ,r}

h(L_h,α−λ(h))(χe_αu−Π_αχe_αu),(χe_αu−Π_αχe_αu)i+O(h²)kuk²

> X

α∈{ℓ,r}

2κhkχeαu−Παχeαuk²+O(h^3/2)kuk², (3.12) from (3.4) and (3.5).

Let us now check that there exists c >0(uniform in η) such that

kΠ_αχe_αuk=O(e^−c/h). (3.13) For this we introduce new cut-off functions χb_α such that χe_α ≺ χb_α ≺ χ_α, that is to say suppχe_α ⊂ {bχ_α ≡1} and suppχb_α ⊂ {χ_α ≡1}. Thanks to the condition on the support, we have

b

χ_αu⊥f_h,α. Since f_h,α=ϕ_h,α on the support of χe_α, we check that

kΠ_αχe_αuk=|heχ_αu, ϕ_h,αi|=|heχ_αu, f_h,αi|=|h(χe_α−χb_α)u, f_h,αi|

6k(χe_α−χb_α)f_h,αk kuk=O(e^−c/h)kuk, (3.14)