4 Computation of the interaction

Semiclassical tunneling and magnetic flux effects on the circle

V. Bonnaillie-Noël^∗, F. Hérau^†and N. Raymond^‡ May 6, 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 potentialaand 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 parametrizationR3s7→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, France[email protected]

†LMJL - UMR6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, F-44322 Nantes cedex 3, France,[email protected]

‡IRMAR - UMR8625, Université Rennes 1, CNRS, Campus de Beaulieu, F-35042 Rennes cedex, France [email protected]

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

(hDs+a(s))²+V(s) e^−iϕ/h

= (hD_s+a(s)−ϕ⁰(s))²+V(s)

= (hDs+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 eigenvaluesλ₁(h) and λ₂(h) of L_h, when the potentialV has some symmetries.

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

rV⁰⁰(0)

2 . (1.1)

Then, as soon as h is small enough, there are only two eigenvalues of L_h in the interval Ih = (−∞,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

Au= exp − Z

[0,^π₂]

∂σ

√ V −κ

√V dσ

!

, A_d= exp − Z

[0,−^π

2]

∂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), with

w₀(h) = 2h^1/2 rκ

π

A_u r

V π 2

e^iξ0π−Suh −A_d r

V

−π 2

e

−iξ0π−Sd h

. (1.2)

Remark 1.3 The constantsSuandSdcorrespond 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 Su and Sd are different, only one term in the sum (1.2) 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 [3,9] and more generally [10,5,6]).

2. IfSu=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 potentialV is even, we are in the second situation and we have Au=Ad=A, Su=Sd=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 V

π 2

r κ 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. 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 wells= 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 wells = 0. For any ρ ∈ (0, π], we define

B_r(ρ) :=B(0, ρ) = (−ρ, ρ).

Let us considerL_h,r the Dirichlet realization of (hDs+ξ0)²+V(s) on 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

L_h,r = e^iξh0sL_h,re

−iξ0s

h =h²D²_s+V(s), (2.1)

defined onDom(L_h,r) = H²(B_r(π−η))∩H¹₀(B_r(π−η)). Let us denote byλ(h)the ground state energy ofL_h,randφ_h,rthe positive andL²-normalized eigenfunction ofL_h,r associated with the lowest eigenvalue λ(h). We have

L_h,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^−iξh0sφ_h,r(s), (2.2) is a L²-normalized eigenfunction ofL_h,r associated withλ(h).

In the next section, we recall some results about the WKB analysis of the operator L_h,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 L_h,r. Lemma 2.1 The asymptotic WKB series for the first quasimode of L_h,r is given by

ψh,r =χrΨh,r, with Ψh,r(s) =h^−1/4e^−Φr(s)h X

j>0

h^jaj(s), ∀s∈ B_r(π), (2.3) where

i) χ_r is a smooth cut-off function supported on B_r(π−η) with 06χ_r61 andχ_r = 1 on B_r(π−2η),

ii) Φ_r 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 Φ⁰_r∂_sa₀+∂_s Φ⁰_ra₀

=κa₀, (2.5)

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

π 1/4

exp − Z s

0

∂_σ√ V −κ 2√

V dσ

!

, ∀s∈ B_r(π).

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

e^Φr/h(L_h,r−µ_r(h))ψ_h,r =O(h^∞) in L²(B_r(π−2η)), (2.6) where µ_r(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^−5/4p

V(s)e^−Φrh(s)a0(s)(1 +O(h)), ∀s∈ B_r(π−2η).

Proof: The proof of the result is classical (see [3,9]) and we just recall the computation ofa₀, which is quite easy since we are in dimension one. For s∈ B_r(π−η), we check that

V(s) =κ²s²+O(s³) and Φ_r(s) =κs²

2 +O(s³).

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

Z s 0

∂_σ√ V −κ 2√

V dσ

! , whereK0 is a normalization constant determined by

1 = Z

Br(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 ofL_h in Section4.

2.3 Agmon estimates and WKB approximation

Let us recall the following lemma (see [8] 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 Q the multiplication by a smooth function.

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

Proposition 2.3 Let Φ be a Lipschitzian function such that

V(s)− |Φ⁰(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 all h∈(0,1),

V(s)− |Φ⁰(s)|² >M h, ∀s∈ B_r(π−η)∩{B_r(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(L_h,r),

chke^Φ/huk_L2(Br(π−η))6ke^Φ/h(L_h,r−z)uk_L2(Br(π−η))+Chkuk_L2(Br(π−η)∩Br(Rh^1/2)), (2.10) and

hDs

e^Φ/hu

2

L²(B_r(π−η))6 C

hke^Φ/h(L_h,r−z)uk²_L2(Br(π−η))+Chkuk²_L2(Br(π−η)∩Br(Rh^1/2)). (2.11) Proof: We apply Lemma 2.2withP =hDs,Q= e^Φ/h and u∈Dom(L_h,r)to get

Re Z

Br(π−η)

hD_su hD_s

e^2Φ/hu ds

!

= Z

Br(π−η)

|hD_s(e^Φ/hu)|²ds− Z

Br(π−η)

|Φ⁰(s)|²e^2Φ/h|u|²ds.

Integrating by parts, adding the electric potentialV, and recalling thatL_h,r =h²D_s²+V, we find

Z

Br(π−η)

|hD_s(e^Φ/hu)|²ds+ Z

Br(π−η)

(V(s)− |Φ⁰(s)|²)e^2Φ/h|u|²ds

= Re Z

B_r(π−η)

L_h,rue^2Φ/huds

!

6ke^Φ/hL_h,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/hL_h,rukke^Φ/huk.

Thanks to (2.9),Φ/his uniformly bounded with respect to honB_r(Rh^1/2) and we deduce khD_s(e^Φ/hu)k²+M hke^Φ/huk² 6ke^Φ/hL_h,rukke^Φ/huk+C_Rhkuk²_L2(Br(π−η)∩Br(Rh^1/2)). For|z|6C0h, we get

khD_s(e^Φ/hu)k²+ (M−C₀)hke^Φ/huk²

6ke^Φ/h(L_h,r−z)ukke^Φ/huk+CRhkuk²_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) Proposition2.3 applies in the following cases:

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

1−εΦ_r with R >0 and M =c₀εR²,

(b) for N ∈ N^∗ and h ∈ (0,1), the precised weight Φe_r,N,h = Φr −N hln max ^Φ_h^r, N , with R=

qN

c0 andM =Ninf_{Br(π−η)Φ}^V

r,

(c) for ε∈(0,1),N ∈N^∗ and h∈(0,1), the intermediate weight

Φb_r,N,h(s) = min (

Φe_r,N,h(s),√

1−ε inf

t∈suppχ⁰_r Φ_r(t) + Z

[s,t]

pV(σ) dσ

!)

, (2.14)

withR= qN

c0 andM =Nmin

ε,inf_{Br(π−η)Φ}^V

r

, where we recall thatχ⁰_r is supported in B_r(π−η)\ B_r(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) ofΦ_r, 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 − |Φ⁰_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|eΦ⁰_r,N,h|² =|Φ⁰_r|² =V. On{Φ_r >N h}, we get

Φe⁰_r,N,h = Φ⁰_r

1−N h Φr

, so that

V − |Φe⁰_r,N,h|² =VN h Φr

2−N h Φr

>N hV

Φr >cN h>0, (2.15) since the function V /Φr is continuous and bounded from below by some c > 0 on B_r(π−η). This proves (2.7). According to (2.13), for allR >0andh∈(0,1), we have Φ_r>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)⊂ {Φ_r>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 |bΦ⁰_r,N,h|=√

1−ε√

V, or |bΦ⁰_r,N,h| =|eΦ⁰_r,N,h|.

Then we apply Proposition 2.4(a) and (b).

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

Lemma 2.5 Let K be a compact with K⊂ B_r(π−2η). We consider the weight defined in Proposition2.4(c). For all N ∈N^∗, there existsε₀ such that for all0< ε < ε₀, there exist h₀ >0 and R >0 such that, for all h∈(0, h₀), we have

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

1−εΦ_r on suppχ⁰_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)6Φ_r(s)6√

1−ε inf

t∈suppχ⁰_r Φ_r(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∈suppχ⁰_r. There exists h₀ >0 (depending on ε) such that for allh∈(0, h₀), we have







inft∈suppχ⁰_r

Φ_r(t) +R

[s,t]

pV(σ) dσ

= Φ_r(s),

Φe_r,N,h(s) = Φ_r(s) +O(hlnh) >√

1−εΦ_r(s).

ThusΦb_r,N,h =√

1−εΦ_r onsuppχ⁰_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

Π_rψ=hψ, φ_h,riφ_h,r.

Proposition 2.6 Let K be a compact set with K ⊂ B_r(π −2η). We have both in the L^∞(K) and in the L²(K) sense

e^Φr/h(ψh,r−Πrψh,r) =O(h^∞), (2.16) e^Φr/hD_s(ψ_h,r−Π_rψ_h,r) =O(h^∞). (2.17) Proof: Let us apply Proposition2.3withu=ψ_h,r−Π_rψ_h,r andz=λ(h)and the weight Φ =Φbr,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/hu

2

L²(Br(π−η))

6Ch⁻¹ke^Φbr,N,h/h(L_h,r−λ(h))ψ_h,rk²_L2(Br(π−η))+Chkuk²_L2(B

r(π−η)∩Br(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(L_h,r−λ(h))ψh,r = e^Φbr,N,h/h(L_h,r−λ(h))χrΨh,r

= e^Φbr,N,h/hχ_r(L_h,r−λ(h))Ψ_h,r+ e^Φbr,N,h/h[L_h,r, χ_r]Ψ_h,r

= e^{(bΦr,N,h−Φr)/h}O_L^∞_{(Br(π−η))}(h^∞) + e^{(bΦr,N,h−Φr)/h}O_L^∞_(suppχ⁰

r)(1). (2.19) Using Lemma 2.5, there exists c₁ >0such that

e^{(bΦr,N,h−Φr)/h}O_L^∞_{(Br(π−η))}(h^∞) =O_L^∞_{(Br(π−η))}(h^∞),

e^{(bΦr,N,h−Φr)/h}= e⁻⁽¹⁻

√1−ε)Φ_r/h

6e^−c1/h=O(h^∞) onsuppχ⁰_r. Putting these estimates in (2.19), we deduce that

Ch⁻¹ke^Φbr,N,h/h(L_h,r−λ(h))ψ_h,rk²_L2(B_r(π−η)) =O(h^∞). (2.20)

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

chkuk_L2(B_r(π−η))=chkψ_h,r−Πrψh,rk_L2(B_r(π−η))

6k(L_h,r−λ(h))uk_L2(B

r(π−η))=k(L_h,r−λ(h))ψ_h,rk_L2(B

r(π−η))=O(h^∞), (2.21) where we have used (2.6) for the last estimate.

Consequently (2.18) becomes chke^Φbr,N,h/huk²_L2(Br(π−η))+

hD_s

e^Φbr,N,h/hu

2

L²(Br(π−η))=O(h^∞). (2.22) By Sobolev embedding, we deduce that, as well as in L^∞(B_r(π−η))as inL²(B_r(π−η)),

he^Φbr,N,h/hu=O(h^∞).

To deduce (2.16), we first recall Lemma 2.5 (2), so that Φb_r,N,h = Φe_r,N,h on K. Then we have, inL^∞(K) and inL²(K),

he^Φer,N,h/hu=O(h^∞). (2.23)

Now the definition ofΦe_r,N,h (given in Proposition2.4 (b)) implies that inL^∞(K) we have

e^{(Φr−eΦr,N,h)/h}=O(h^−N). (2.24)

By using (2.23), we get, in L^∞(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 2.5(2) gives

Φbr,N,h=Φer,N,h onK. (2.26)

We first write that

e^Φbr,N,h/hhD_su

2 L²(K)6

hD_s

e^Φbr,N,h/hu

L²(K)+

Φe⁰_r,N,h

e^Φbr,N,h/hu

L²(K). (2.27) Using that|eΦ⁰_r,N,h|² 6V which is bounded and (2.22), we deduce

e^Φer,N,h/hhDsu

L²(K)=

e^Φbr,N,h/hhDsu

2

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 hDs. We have

khD_s(e^Φr/hu)k_L2(K)6ke^Φr/hhDsuk_L2(K)+kΦ⁰_re^Φr/huk_L2(K). Using (2.16) in L²(K), the fact that|Φ⁰_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(L_h,r−λ(h))ψh,rk²_L2(K)=O(h^∞). (2.31) SinceΠrψh,r is an eigenfunction, we get

e^Φr/h(L_h,r−λ(h))u

2 L²(K)=

e^Φr/h(L_h,r−λ(h)) (ψ_h,r−Π_rψ_h,r)

2

L²(K)=O(h^∞).

By definition ofL_h,r, this provides

e^Φr/h h²D²_s+V(s)−λ(h) u

2

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 2ihΦ⁰_rD_su+|Φ⁰_r|²u+hΦ⁰⁰_ru

. (2.34)

SinceΦ⁰_r andΦ⁰⁰_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)k_L2(K) =O(h^∞). (2.36)

From Sobolev embedding, we deduce from (2.36) and (2.30) that

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

ke^Φr/hhDsuk_L^∞_(K)6khD_s(e^Φr/hu)k_L^∞_(K)+kΦ⁰_re^Φr/huk_L^∞_(K). (2.38) Using then (2.37) for the term with the derivative, the fact thatΦ⁰_r is bounded and (2.16) in theL^∞(K) sense, we get

ke^Φr/hhD_suk_L∞(K)=O(h^∞). (2.39) The proof of the L^∞(K) estimate in (2.17) is complete, and so is the proof of Proposi- tion2.6.

Remark 2.7 The estimate given by Proposition2.6is 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 `the quantities, operators, quasimodes, etc. related to the left-hand side well whose coordinate is s=π.

Let B_{`</sub>(ρ) := B(π, ρ) = (π−ρ, π+ρ), for any ρ ∈ (0, π). The Dirichlet realization of (hD<sub>s</sub>+ξ<sub>0</sub>)<sup>2</sup>+V(s) onL<sup>2</sup>(B<sub>`}(π−η),ds)is denoted L_h,`.

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(B_{`</sub>(ρ),ds). According to Assumption1.1about the symmetry ofV, the two operators L<sub>h,r</sub> and L<sub>h,`} are unitary equivalent:

L_h,`=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,`</sub> are obviously deduced from those ofL<sub>h,r</sub> thanks to the unitary transform U. We let φh,` =U φh,r. Then the function φh,` is a positive L<sup>2</sup>-normalized eigenfunction ofL<sub>h,`} (the Dirichlet realization of h²D²_s+V on L²(B_`(π−η),ds)) associated withλ(h).

Thus we have

L_{h,`</sub>φ<sub>h,`}= (h²D_s²+V)φ_{h,`</sub>=λ(h)φ<sub>h,`} onB_`(π−η).

The functionϕ_{h,`</sub> defined on B<sub>`}(π−η) by

ϕh,`=U ϕh,r, (2.42)

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

ϕh,`(s) = e<sup>iξ0hπ</sup>e<sup>−iξ0hs</sup>φh,`(s), ∀s∈ B_`(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_`(s)), with the Agmon distances defined as in (2.4) by

Φr(s) = Z

[0,s]

pV(σ) dσ, ∀s∈ B_r(π) and Φ_`(s) = Z

[π,s]

pV(σ) dσ, ∀s∈ B_`(π). (3.1) The functionΦis Lipschitzian and satisfies the eikonal equation |Φ⁰|² =V.

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

B(ρ) =b B_r(ρ)∪ B_`(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/huk_L2(S¹)6ke

√1−ε)Φ/h(L_h−z)uk_L2(S¹)+Chkuk_L2(

B(Ahb ^1/2)), (3.2) and

(hDs+ξ0)

e

√1−εΦ/hu

2

L²(S¹)6 C hke

√1−εΦ/h(L_h−z)uk²_L2(S¹)+Chkuk²

L²(B(Ahb ^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 thatΦ is Lipschitzian. After an integration by parts, we obtain

Re Z

S¹

(hDs+ξ0)²ue^2Φε/huds= Z

S¹

(hDs+ξ0)

e^Φε/hu

2

ds− Z

S¹

|Φ⁰_ε|²e^2Φε/h|u|²ds.

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

S¹

(hD_s+ξ₀)

e^Φε/hu

2

ds+ Z

S¹

V − |Φ⁰_ε|²

e^2Φε/h|u|²ds= Re Z

S¹

L_hue^2Φε/huds 6

e^Φε/hL_hu e^Φε/hu

, so that

Z

S¹

(hD_s+ξ₀)

e^Φε/hu

2

ds+ Z

S¹

εVe^2Φε/h|u|²ds6

e^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 of V 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/hϕk_L2(S¹)6Ckϕk_L2(S¹) and khD_s(e

√1−εΦ/hϕ)k_L2(S¹)6Ckϕk_L2(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 operatorsL_h,r and L_h,`

defined respectively onB_r(π−η)andB_`(π−η). Let us consider the quadratic approximation ofL_h,r defined on Rby

h²D_s²+1

2V⁰⁰(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 eigenvalueλ(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 onS¹ and2π-periodic functions of the variables∈R. We also extend by0 onS¹\ B_r(π−η) the functions χ_r andϕ_h,r and by 0on S¹\ B<sub>`</sub>(π−η)the functionsχ` and ϕh,`.

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

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

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

f_h,r=χ_rϕ_h,r and f_{h,`</sub>=χ<sub>`}ϕ_h,`, (3.6) with

χ_`=U χ_r. (3.7)

We have in particular f_h,`=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∈ {`,r},

rh,α = (Lh−λ(h))fh,α = (Lh,α−λ(h))χaϕh,α= [Lh,α, χα]ϕh,α. (3.8) For η sufficiently small, we have

(i) rh,α(s) =O(ee ^−S/h),

(ii) hr_h,α, f_h,αi=O(ee ^−2S/h) and hr_h,α, f_h,βi=O(ee ^−S/h) for α6=β,

(iii) hf_h,α, f_h,αi= 1 +O(ee ^−2S/h) andhf_h,α, f_h,βi=O(ee ^−S/h) for α6=β,

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

Proof:

(i) Thanks to Corollary3.2, we get inL^∞(S¹) and L²(S¹) sense that, for allε >0, e

√1−εΦα(s)/hr_h,α(s) =O(1).

Since the support of [Lh,α, χα]is included in B_α¯(2η), we get:

r_h,α(s) =O(ee ^−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 =O(ee ^−Φα/h), (3.10)

in L²(B_α(π−η)) and H¹(B_α(π−η)). According to Agmon estimates, this gives in particular

hf_h,α, f_h,αi= 1 +O(ee ^−2S/h). (3.11) Forα 6=β, using (3.10), the supports ofχα and χβ and sinceΦα+ Φβ =S, we get

hf_h,α, f_h,βi=O(ee ^−S/h).

(iv) The previous estimates imply that dimF = 2 for hsmall 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 in I_h.

Lemma 3.6 Let us define G = range (1I_h(L_h)). Then dist(sp(L_h), λ(h)) =O(ee ^−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=O(ee ^−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)−O(ee ^−S/h), λ(h) +O(ee ^−S/h)].

Proof:

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

χe²_` +χe²_r = 1 on S¹,

whereχe` =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∈{`,r}

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

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

h(L_h−λ(h))u, ui=X

α∈{`,r}

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

> X

α∈{`,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 suppχeα ⊂ {χbα ≡1} and suppχbα ⊂ {χ_α ≡1}. Thanks to the condition on the support, we have

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

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

6k(χeα−χbα)f_h,αk kuk=O(e^−c/h)kuk, (3.14) thanks to Corollary 3.2. This gives (3.13). From (3.12) and (3.14), we infer

h(L_h−λ(h))u, ui> X

α∈{`,r}

2κhkχeαuk²+O(h^3/2)kuk²

>κhkuk², for h small enough. This gives (i).

(ii) Now using again the first inequality in the preceding computation also gives hL_hu, ui> X

α∈{`,r}

2κhkχeαuk²+λ(h)kuk²+O(h^3/2)kuk²

>2κhkuk²,

from (3.4) and forh small enough. From the min-max principle and since{f_h,`, f_h,r} is a free family, we getdimG 62 and we deduce (ii).

(iii) Eventually using Lemma 3.5(i), we get (iii) and the proof is complete.

3.3 Precised estimates about quasimodes and eigenfunctions

In this section we give precise estimates of the quasimodes fh,α and their projections on the spectral subspaces g_h,α = Πf_h,α where Π denotes the projection on G. Let us first estimate the difference between f_h,α andg_h,α.

Lemma 3.8 We have f_h,α−g_h,α =O(ee ^−S/h) inL²(S¹) and H¹(S¹).

Proof: We write

(L_h−λ(h))(f_h,α−g_h,α) = (L_h−λ(h))f_h,α−(L_h−λ(h))g_h,α.

The first term isO(ee ^−S/h)from Lemma 3.5(i). The second isO(ee ^−S/h)from the exponen- tial localization in Lemma3.7(iii). We therefore get in L²(S¹)

(Lh−λ(h))(fh,α−gh,α) =O(ee ^−S/h).

Since fh,α−gh,α ∈ G^⊥, we can use Lemma 3.7 (i) and the spectral theorem to conclude that

f_h,α−g_h,α=O(ee ^−S/h) inL²(S¹).

By using the two preceding estimates, we get the result inH¹(S¹).

The following obvious lemma will be convenient in the following.

Lemma 3.9 Let (H,h·,·i) be a Hilbert space and Π ∈ L(H) be an orthogonal projection.

Then, for allu, v∈H, we have

hu, vi=hΠu,Πvi+h(Id−Π)u,(Id−Π)vi.

Lemma 3.10 Let us define the matrixT= (T_α,β)_{α,β∈{`,r}} withT_α,β=hf_h,α, f_h,βiifα6=β and0 otherwise. Then T=O(ee ^−S/h) and we have

(i) (hf_h,α, fh,βi)_{α,β∈{`,r}</sub>=Id+T+O(ee <sup>−2S/h</sup>), (ii) hg<sub>h,α</sub>, g<sub>h,β</sub>i=hf<sub>h,α</sub>, f<sub>h,β</sub>i+O(ee <sup>−2S/h</sup>), (iii) (hg<sub>h,α</sub>, g<sub>h,β</sub>i)<sub>α,β∈{`,r}} =Id+T+O(ee ^−2S/h).

Proof: The fact that T = O(ee ^−S/h) and (i) follow from Lemma 3.5 (iii). (ii) is a consequence of Lemma3.9and Lemma 3.8. (iii) is then obvious.

3.4 Interaction matrix

From Lemma3.10(iii), the basis(g_{h,`</sub>, g<sub>h,r</sub>)is quasi orthonormal but not exactly orthonor- mal. Therefore we introduce the new basis g = gG<sup>−1/2</sup>, where G is the Gram-Schmidt matrix (hg<sub>h,α</sub>, gh,βi)<sub>α,β∈{`,r}} and g the row vector (gh,`, gh,r). The basis g is orthonormal since

(hg_h,α,g_h,βi)_{α,β∈{`,r}</sub>=<sup>t</sup>G<sup>−1/2</sup>(hg<sub>h,α</sub>, g<sub>h,β</sub>i)<sub>α,β∈{`,r}}G^−1/2 =G^−1/2GG^−1/2 =Id.

Proposition 3.11 The matrix M of the restriction to L_h in the basis g is given by M:= (hL_hg_α,g_βi)_{α,β∈{`,r}} =D+W+O(ee ^−2S/h),

where

(a) D=λ(h)Id,

(b) the “interaction matrix” W= (w_α,β(h))_{α,β∈{`,r}} is defined, recalling (3.8), by w_α,β(h) =hr_h,α, f_h,βi if α6=β, and 0 otherwise.

In particular, the gap between the two first eigenvalues, denoted by λ₁(h) andλ₂(h), of L_h (or ofM) satisfies

λ2(h)−λ1(h) = 2|w_`,r(h)|+O(ee ^−2S/h). (3.15) For the proof of Proposition 3.11 we begin by two lemmas. First, we notice that W is indeed an Hermitian matrix by using the symmetries of our constructions.

Lemma 3.12 The matrix W is Hermitian.

Proof: By definition, we have w_α,α(h) = 0 forα ∈ {r, `} and w<sub>`,r</sub>(h) =h[L_{h,`</sub>, χ<sub>`}]ϕ_h,`, χrϕ_h,ri. By using (2.41), (2.42) and (3.7), we deduce that

w`,r(h) =

[UL_h,rU⁻¹, U χr]U ϕh,r, U⁻¹(χ`ϕh,`)

=

UL_h,rU⁻¹(U χ_rU ϕ_h,r)−U χ_rUL_h,rU⁻¹(U ϕ_h,r), U⁻¹(χ_{`</sub>ϕ<sub>h,`})

=

UL_h,r(χrϕh,r)−U χrUL_h,r(ϕh,r), U⁻¹(χ`ϕh,`)

=

U(L_h,r(χ_rϕ_h,r)−χ_rL_h,r(ϕ_h,r)), U⁻¹(χ_{`</sub>ϕ<sub>h,`})

=

U([L_h,r, χr]ϕ_h,r), U⁻¹(χ_{`</sub>ϕ<sub>h,`})

=h[L_h,r, χ_r]ϕ_h,r, χ_{`</sub>ϕ<sub>h,`}i=w_r,`(h), since U is anti-hermitian.

Then, we write the matrix ofL_h in the quasi orthonormal basisg.

Lemma 3.13 We have

(i) hL_hg_h,α, g_h,βi=hL_hf_h,α, f_h,βi+O(ee ^−2S/h),

(ii) (hL_hf_h,α, f_h,βi)_{α,β∈{`,r}</sub>=D+DT+W+O(ee <sup>−2S/h</sup>), (iii) (hL<sub>h</sub>g<sub>h,α</sub>, g<sub>h,β</sub>i)<sub>α,β∈{`,r}} =D+DT+W+O(ee ^−2S/h).

Proof:

(i) With Lemma3.9, we get

hL_hfh,α, fh,βi − hL_hgh,α, gh,βi=hL_h(fh,α−gh,α), fh,β−gh,βi.

From Lemma 3.8applied inH¹, we get directly that

hL_hgh,α, gh,βi − hL_hfh,α, fh,βi=O(ee ^−2S/h).

(ii) We can write

hL_hf_h,α, f_h,βi=λ(h)hf_h,α, f_h,βi+hr_h,α, f_h,βi.

The result follows from the definition of D,W, Lemma3.5(ii) and Lemma3.10 (i).