On the semiclassical Laplacian with magnetic field having self-intersecting zero set

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On the semiclassical Laplacian with magnetic field having self-intersecting zero set

Monique Dauge, Jean-Philippe Miqueu, Nicolas Raymond

To cite this version:

Monique Dauge, Jean-Philippe Miqueu, Nicolas Raymond. On the semiclassical Laplacian with mag- netic field having self-intersecting zero set. Journal of Spectral Theory, European Mathematical Society, 2020, 10 (4), pp.1211-1252. �10.4171/JST/325�. �hal-01846603�

ON THE SEMICLASSICAL LAPLACIAN WITH MAGNETIC FIELD HAVING SELF-INTERSECTING ZERO SET

MONIQUE DAUGE, JEAN-PHILIPPE MIQUEU, AND NICOLAS RAYMOND

ABSTRACT. This paper is devoted to the spectral analysis of the Neumann realization of the 2D magnetic Laplacian with semiclassical parameterh >0in the case when the magnetic field vanishes along a smooth curve which crosses itself inside a bounded domain. We investigate the behavior of its eigenpairs in the limit h → 0. We show that each crossing point acts as a potential well, generating a new decay scale of h^3/2 for the lowest eigenvalues, as well as exponential concentration for eigenvectors around the set of crossing points. These properties are consequences of the nature of associated model problems inR²for which the zero set of the magnetic field is the union of two straight lines. In this paper we also analyze the spectrum of model problems when the angle between the two straight lines tends to0.

1. INTRODUCTION

1.1. The magnetic Laplacian. LetΩbe a bounded, smooth and simply connected open set of R², andA ∈ C^∞(Ω,R²)be a regular potential vector. Forh > 0, we consider the self-adjoint operator

Ph,Ω^A = (−ih∇+A)², with domain

Dom(P_h,Ω^A ) = {u∈H²(Ω),(−ih∇+A)u·ν = 0 on ∂Ω}, whereνis the outward pointing normal at the boundary ofΩ.

The operator Ph,Ω^A has compact resolvent and is associated with the quadratic form Q^Ah,Ω

defined on the form domainDom(Qh,Ω^A ) =H¹(Ω)by

(1.1) Qh,Ω^A (u) =

Z

Ω

|(−ih∇+A)u(x)|²dx.

Herex= (x₁, x₂)denotes Cartesian coordinates inR². We have the gauge invariance (1.2) e^−iφ/h(−ih∇+A)²e^iφ/h = (−ih∇+A+∇φ)² ,

for anyφ∈H²(Ω). Therefore, the spectrum ofPh,Ω^A only depends on the magnetic field

(1.3) B=∇ ×A=∂₂A₁−∂₁A₂.

Notation 1.1. We denote byλn(h)then-th eigenvalue (with multiplicity) ofPh,Ω^A . In all of the paper,S(P)denotes the spectrum of any operatorP.

We are interested in the behavior of the eigenvaluesλ_n(h)and their associated eigenfunctions in the semiclassical limith→0for special configurations of the magnetic fieldB.

2010Mathematics Subject Classification. 35P20, 35Q40, 65M60.

Key words and phrases. Magnetic Laplacian, Semiclassical asymptotics, Exponential concentration of eigenvectors.

1

1.2. Motivations and context. The spectral analysis of the magnetic Laplacian comes from the theory of superconductivity in which the magnetic Laplacian appears in the study of the third critical field of the Ginzburg-Landau functional (see for instance [34] and also the books [15] and [32], and the references therein). The regime whenhgoes to0(called the semiclassical limit) is equivalent to the strong magnetic field limit which is often involved in applications. In this paper, we restrict to dimension two.

1.2.1. Overview of the literature. In the past two decades, most of the contributions dealt with non-vanishing magnetic fields. We can refer for instance to the works by Bolley & Helffer [5], Bauman, Phillips & Tang [3], del Pino, Felmer & Sternberg [13], Helffer & Morame [21], Bonnaillie-No¨el [6], Lu & Pan [25], Raymond [29], Bonnaillie-No¨el & Dauge [7], Bonnaillie- No¨el & Fournais [8], Raymond & Vu-Ngoc [33].

The present paper is devoted to the case of vanishing magnetic fields. Such an investigation was initially motivated by a paper of Montgomery [27], followed by the contributions of Helf- fer & Morame [20], Helffer & Kordyukov [18, 16], and Dombrowski & Raymond [14]. The aforementioned papers do not investigate the case when the zero set of the magnetic field B intersects the boundary. This was the purpose of the work by Pan & Kwek [28] and [26]. We can find in [28] a one term asymptotics of the first eigenvalue λ₁(h). The paper [26] estab- lishes a sharper result by giving an explicit control of the remainder as well as expansions of all the eigenvalues and eigenfunctions, as the semiclassical parameterh goes to0, under suitable assumptions when the zero set ofBdoes not self-intersect.

1.2.2. When the zero set of B self-intersects. In the present paper, we want to include non- degenerate quadratic cancellations inside the domain, which is a new configuration in the inves- tigations about vanishing magnetic fields.

Assumption 1.2. Let

Γ ={x∈Ω :B(x) = 0}, and assume thatΓ6=∅. We work under the following assumptions

i) The set

Σ = {x∈Γ,∇B(x) = 0}

is non-empty, finite and such that∂Ω∩Σ =∅.

ii) For anyx∈Σ, the Hessian matrixHessB(x)of the magnetic field at the pointxhas two non-zero eigenvalues with opposite signs.

iii) The setΓ∩∂Ωis finite, and in each of these intersection points,Γis non tangent to∂Ω.

Note that assumptionsi)-ii)imply that the setΓis a simple curve in a neighborhood of each of its intersection points with ∂Ω. Moreover, the set Σ is made of isolated points which are locally the intersection point of two smooth curves.

Notation 1.3. Choosex₀ ∈Σ.

i) B^x0 denotes the second order Taylor expansion of the magnetic field atx₀. So B^x0(x) = ¹₂(x−x₀)HessB(x₀)(x−x₀)^>.

ii) A^x0 denotes the Taylor expansion of the magnetic potential Ato the third order at the pointx₀. ThusB^x0 =∇ ×A^x0.

iii) Letα(x₀), β(x₀) be the eigenvalues of ¹₂HessB(x₀), agreeing that |α(x₀)| ≤ |β(x₀)|.

Set

ε(x₀) = p

|α(x₀)|/|β(x₀)| and Ξ(x₀) =|β(x₀)|,

so that in a suitable local system of orthogonal coordinatesy= (s, t)centered atx₀ B^x0(x) =α(x0)s²+β(x0)t² =−β(x0) ε²(x0)s² −t²

.

Hence the zero set ofB^x0 has the equationε²(x₀)s²−t² = 0: It is the union of the two lines {t =±ε(x₀)s}. These lines are the two tangents to the setΓat the pointx₀.

The main novelty in this paper is related to the presence ofΣand to the role of the following family of model operators indexed byεand acting onL²(R²), defined as

(1.4) Xε =

D_σ− τ³

3 +ε²σ²τ2

+D²_τ, (σ, τ) =: Y∈R²,

with D_σ = −i∂_σ andD_τ = −i∂_τ. Note that the magnetic field associated with the operator Xε isB(σ, τ) =−τ²+ε²σ². That is why the operatorXε forε =ε(x₀)will serve as a model magnetic operator at pointx₀.

As a consequence of [19,22], the operatorXεhas a compact resolvent inR². Then it follows that the eigenfunctions ofXε have an exponential decay (see [1], [2], [15, Theorem B.5.1] and the example in [30, p. 100]). To sum up:

Proposition 1.4. Letε > 0. The spectrum of the operatorXε is formed by a non-decreasing unbounded sequence of positive eigenvalues denoted by (κn(ε))n≥1. Moreover, for any eigen- functionΨ_εofXε, there existsc >0such thate^c|Y|Ψ_ε∈L²(R²).

1.3. Semiclassical expansions of the magnetic eigenvalues. The model operatorsXε have the following homogeneity property, due to their magnetic potential of degree3. By rescaling, we find immediately:

Lemma 1.5. Set A_ε := (−^τ₃³ +ε²σ²τ,0)the magnetic potential ofXε. Let(κ,Ψ)be a nor- malized eigenpair ofXε. Setting forh >0andΞ>0

ψ_h(y) = Ξ^1/4h^−1/4Ψ(Ξ^1/4h^−1/4Y)

we obtain that(h^3/2Ξ^1/2κ, ψh)is a normalized eigenpair for the semiclassical magnetic oper- ator(−ih∇+ ΞAε)²onR².

After the scalehfor non-vanishing magnetic fields, the scaleh^4/3for magnetic field vanishing at order 1, we note the apparition of the new scaleh^3/2.

Since for each crossing pointx₀ ∈Σthe operator(−ih∇+ Ξ(x₀)A_ε(x0))²is unitarily equiv- alent to the “tangent” magnetic operator(−ih∇+A^x0)², we can guess that the behavior of the low lying spectrum of the operatorPh,Ω^A corresponds to the low lying spectrum of the operator

(1.5) P_h^A,Σ :=M

x∈Σ

−ih∇+ Ξ(x)A_ε(x)2

,

onL²(R²)^]Σ, where]Σis the cardinal of the finite setΣ. Lemma1.5then gives that S(P_h^A,Σ) =a

x∈Σ

h^3/2Ξ(x)^1/2S(Xε(x)).

This leads to introduce, for allx∈Σand alln∈N^∗, the enumeration of the eigenvalues

(1.6a) Λ^x_n= Ξ(x)^1/2κn(ε(x)),

and the ordered set

(1.6b)

Λ^B_n ,n ∈N^∗ =a

x∈Σ

{Λ^x_n, n∈N^∗},

for which the same value can possibly appear several times (for instance, if we haveΛ^x_n = Λ^x_n⁰0

for(n,x)6= (n⁰,x⁰)). Note in particular that the smallest element of this set is given by

(1.7) Λ^B₁ = min

x∈Σ Ξ(x)^1/2κ1(ε(x)),

where we recall thatκ1(ε)is the first eigenvalue ofXεandε(x)is defined in Notation1.3.

We are ready to state the main two results relating to the low lying spectrum of the operator Ph,Ω^A . The first result provides a localized estimate from below of the energy functional Q^Ah,Ω

(defined in (1.1)) and exponential decay estimates for the eigenvectors ofPh,Ω^A . The second result states an asymptotic expansion for the eigenvalues ofPh,Ω^A .

1.3.1. Estimate from below of the energy and Agmon estimates. Let the functionx 7→ d(x,Σ) be the Euclidean distance between the pointx∈Ωand the setΣ.

Theorem 1.6. Under Assumption 1.2, for any exponent d ∈ (₁₆³,¹₄), there exist cd, Cd > 0, h_d >0, such that for allu∈Dom(Q_h,Ω^A )and allh∈(0, h_d),

(1.8a) Qh,Ω^A (u)≥

Z

Ω

Ih,d^Σ (x)|u(x)|²dx, with

(1.8b) Ih,d^Σ (x) =

Λ^B₁h^3/2−C_dh^{min{2−2d,3/4+4d}} if d(x,Σ)≤h^d

c_dh^4/3+2d/3 if d(x,Σ)> h^d,

whereΛ^B₁ is defined in(1.7). Note that the exponents present in the definition ofI_h,d^Σ satisfy (1.9) min{2−2d, ³₄ + 4d}> ³₂ and ⁴₃ + ^2d₃ < ³₂.

Agmon estimates are an a priori result of exponential decay of the eigenfunctions. The following theorem states that the eigenfunctions associated with eigenvalues of orderh^4/3+2d/3 are localized nearΣ.

Theorem 1.7. LetL > 0andd ∈(₁₆³ ,¹₄). There existC,h_d >0such that for allh ∈(0, h_d), and all eigenpair(λ(h), ψh)ofPh,Ω^A withλ(h)≤Lh^3/2, we have

(1.10)

Z

Ω

e^2d(x,Σ)/hd|ψ_h(x)|²dx+h^−3/2Qh,Ω^A e^d(·,Σ)/hdψ_h

≤Ckψ_hk².

In fact estimate (1.10) would also hold for the relaxed conditionλ(h)≤Lh^4/3+2d/3 ifL <cd. 1.3.2. Expansion of lowest eigenvalues. Let us recall that

λ₁(h)≤λ₂(h)≤. . .≤λ_n(h). . .

denote the increasing sequence of the eigenvalues of the operatorPh,Ω^A while Λ^B₁ ≤Λ^B₂ ≤. . .≤Λ^B_n . . .

denote the elements defined in (1.6a)-(1.6b) from the eigenvalues of the model operatorsXε. Theorem 1.8. Under Assumption 1.2, for allN ∈ N^∗, there existC_N > 0 andh₀ > 0such that, for allh∈(0, h₀)and alln= 1, . . . , N

|λ_n(h)−h^3/2Λ^B_n| ≤C_Nh^7/4.

In Section3, an extended version of this theorem is proved, providing full expansions of all lowest eigenvaluesλ_n(h), see Theorem3.6.

1.4. Low lying eigenvalues of the magnetic cross in the small angle limit. Whenεtends to 0, the angle between the two lines {τ = ±εσ} tends to 0. It is interesting to understand the behavior of κn(ε)in such a limit. One could naively expect that, whenεgoes to0,infS(Xε) goes toinfS(M^[2])where

M^[2] =D²_τ +

D_σ− τ³ 3

2

acting onL²(R²).

The operator M^[2] is sometimes called Montgomery operator of order two. By Fourier trans- form, we have

infS(M^[2]) = inf

ξ∈R

infS(Mξ^[2])

, whereMξ^[2] =D²_τ+

ξ− τ³ 3

2

acting onL²(R).

The quantityinfS(M^[2])has been numerically estimated, see [23, Table 1].

Actually, the limitε →0is singular: The operatorXεis partially semiclassical. Indeed, after the scalingσ =ε⁻¹s, τ =t, the operatorXεbecomes

(1.11) D_t²+

εD_s− t³

3 +s²t2

with D_t =−i∂_t, D_s =−i∂_s. The operator (1.11) is the Weyl quantizationOp^w_ε(X_α,ξ)of the symbolX_α,ξ where

(1.12) X_α,ξ =D²_t +

ξ− t³

3 +α²t2

is a self-adjoint operator acting onL²(R)depending on the two real parametersαandξ. In the small angle regime, the spectral analysis of the operatorXεis related to the spectral analysis of the family(X_α,ξ)_(α,ξ)∈

R². The asymptotic expansions of the first eigenvalues ofXε is related to the“band function”%₁defined by the ground state energy ofX_α,ξ

%₁(α, ξ) = minS(X_α,ξ), (α, ξ)∈R²,

see for instance [31] and [9], where such operators and reductions are considered. One of the requirements to apply the theory developed in [9] (and that relates %₁ to the eigenvalue asymptotic expansions whenε→0for the operatorXε) is that%₁ has a minimum.

Theorem 1.9. The function%₁ reaches its infimumS₀inR², so S₀ = min

(α,ξ)∈R²

%₁(α, ξ), and this minimum is reached on the set

n

(α, ξ)∈R², |ξ| ≤ 2 3|α|³o

.

By construction, S₀ ≤ infS(M^[2]). The numerical simulations that we have performed provide the upper bound

(1.13) S₀ ≤%₁(α₀,0)'0.4941 for α₀ = 0.786.

We note that the value given in [23, Table 1] for infS(M^[2])is ' 0.66, corresponding to the valueξ = 0of the Fourier parameter. Hence we have obtained the strict inequality

S₀ <infS(M^[2]).

Our numerical results for the band function%₁, see Figures1and2, suggest that the minimum S₀ is attained at the sole two values±(α₀,0)of(α, ξ).

The following result gives the convergence of the eigenvaluesκn(ε)of the model operator Xεasε→0.

Theorem 1.10. For alln ≥1, there existC > 0andε₀ >0such that for allε∈(0, ε₀)

|κn(ε)−S₀| ≤Cε.

1.5. Organization of the paper. In Section2, we prove the preliminary Theorems1.6and1.7.

In Section3, we establish the full asymptotic expansions of the eigenvalues and eigenfunctions.

Section4is devoted to the proofs of Theorems1.9and1.10and to the presentation of numerical simulations concerning the band function%₁and the eigenpairs ofXεasε→0.

2. BOUNDS FROM BELOW AND EXPONENTIAL DECAY

In this section we prove the preliminary bounds from below (1.8a)-(1.8b) and the Agmon estimates (1.10). We will use several times a perturbation formula for the magnetic Laplacian which we first state.

2.1. Perturbation of the magnetic potential. Let A? be another smooth magnetic potential defined onΩ. In practiceA?will be the Taylor expansion ofAat some pointx₀and to various orders (2, 3 or 4). By expanding the square, we get

(2.1) Qh,Ω^A (u) =Qh,Ω^A?(u) + 2Re

(−ih∇+A?)u,(A−A?)u

+k(A−A?)uk². This yields Q_h,Ω^A (u) ≥ Q_h,Ω^A?(u)−2Q_h,Ω^A?(u)^1/2k(A−A?)uk by Cauchy-Schwarz inequality, leading to the parametric estimate (based on the inequality2ab≤ηa²+η⁻¹b², for allη >0) (2.2) ∀η >0, Q^Ah,Ω(u)≥(1−η)Q^Ah,Ω^?(u)−η⁻¹k(A−A?)uk².

Such a lower bound is used, for instance, in the seminal paper [20, p. 51], and in the book [15, Chap. 8].

2.2. Lower bound. The proof of (1.8a) is based on a quadratic partition of unity. Let us recall that such a partition is given for each relevant h > 0 by a finite collection of smooth cutoff functions(χ^h_j)satisfying onΩ

(2.3) X

j

|χ^h_j|² = 1.

Then we have the following well-known localization formula (see [11])

(2.4) Q_h,Ω^A (u) =X

j

Q_h,Ω^A (χ^h_ju)−h²X

j

ku|∇(χ^h_j)|k²_L2(Ω). We introduce three setsΣ^[1],Σ^[2](h)andΣ^[3](h)coveringΩ.

Notation 2.1. LetB(x₀, r)denote the open ball of centerx₀and radiusrand B(Σ, r) = [

x∈Σ

B(x, r).

LetR₁ >0such thatB(Σ,3R₁)⊂Ω. Choosed∈(₁₆³,¹₄)and introduce (i) Σ^[1] = Ω\B(Σ, R₁).

(ii) Σ^[2](h) = B(Σ,2R₁)\B(Σ,¹₂h^d).

(iii) Σ^[3](h) = B(Σ, h^d).

Then we consider a partition of unity composed of three cutoff functions(X^[1],X^h,[2],X^h,[3]) associated with this covering ofΩin the sense that

|X^[1]|²+|X^h,[2]|²+|X^h,[3]|² = 1, and

suppX^[1] ⊂Σ^[1], suppX^h,[2] ⊂Σ^[2](h), suppX^h,[3] ⊂Σ^[3](h).

The distance between∂Σ^[2](h)∩Σ^[1](h)and∂Σ^[1](h)∩Σ^[2](h) isR₁. The distance between

∂Σ^[3](h)and∂Σ^[2](h)∩Σ^[3](h)is ¹₂h^d. Hence we can choose the cutoff functions so that (2.5) |∇X^[1]|² ≤K_loc, |∇X^h,[2]|² ≤K_loch^−2d, |∇X^h,[3]|² ≤K_loch^−2d.

with a constant K_loc independent of h and d. Combined with the localization formula (2.4) associated with the partition(X^[1],X^h,[2],X^h,[3]), the estimates (2.5) yield for allu∈H¹(Ω), (2.6) Q^Ah,Ω(u)≥Q^Ah,Ω(X^[1]u) +Qh,Ω^A (X^h,[2]u) +Qh,Ω^A (X^h,[3]u)−K_loch^2−2dkuk²_L2(Ω), We setu₁ =X^[1]u,u₂ =X^h,[2]uandu₃ =X^h,[3]u, and are going to bound eachQ^A_h,Ω(u_k)from below (fork= 1,2,3).

Lemma 2.2(Lower bound onΣ^[1]). There existc₁ andh₀ >0such that for all0< h < h₀ Q_h,Ω^A (u₁)≥c₁h^4/3ku₁k².

Proof. This result is a consequence of [26, Theorem 1.9].

Lemma 2.3(Lower bound onΣ^[2](h)). There existc₂andh₀ >0such that for all0< h < h₀ Qh,Ω^A (u2)≥c2h^4/3+2d/3ku2k².

Proof. Setρ= ¹₄. We introduce a second partition of unity(χ^h_j)j∈JonΣ^[2](h)associated with a family of ballsB(xj, djh^ρ)for some constantsdj. We first cover the zero setΓ∩Σ^[2](h)and choose the centersx_j ∈Γwithd_j = 1so that the distances between consecutive points alongΓ is' ¹₂h^ρ. Hence, setting

gΣ^[2](h) = Σ^[2](h)\ [

xj∈Γ

B(x_j, h^ρ)

we obtain that the distance between Γ and gΣ^[2](h)is larger than ¹₂h^ρ. We then cover gΣ^[2](h) with balls B(x_j, d_jh^ρ)choosingd_j = ¹₄ and the mutual distances between the centers bounded from below by ¹₈h^ρ. Finally we can choose the functionsχ^h_j so that|∇χ^h_j|² ≤C2,loch^2ρand the localization formula (2.4) yields

(2.7) Q^A_h,Ω(u₂)≥X

j

Q_h,Ω^A (χ^h_ju₂)−C_2,loch^2−2ρku₂k². We have, by [26, Lemma 2.3] (for the case(`) = (3)of Table 2.1), (2.8) Qh,Ω^A (χ^h_ju₂)≥1

2M₀|∇B(x_j)|^2/3h^4/3−2Ch^6ρ

kχ^h_ju₂k² (x_j ∈Γ).

As a consequence of the non degeneracy of∇BonΓoutsideΣ(namely∇B| 6= 0onΓ\Σ) and the non degeneracy ofHessBonΣ(meaning that the eigenvalues of the Hessian matrix are not equal to0, according to Assumption1.2) there exists a positive constantC(B)such that

|∇B(x)| ≥C(B) d(x,Σ), ∀x∈Ω. Sinced(x_j,Σ)is larger than ¹₂h^dby construction, from (2.8) we get

Q^Ah,Ω(χ^h_ju₂)≥1

4M₀C(B)^2/3h^4/3+2d/3−2Ch^6ρ

kχ^h_ju₂k².

We note that, with ρ = ¹₄ and d < ¹₄, the exponent 6ρ = ³₂ is (strictly) larger than ⁴₃ + ²₃d.

Therefore, forhsmall enough (2.9) Q^Ah,Ω(χ^h_ju₂)≥ 1

8M₀C(B)^2/3h^4/3+2d/3kχ^h_ju₂k² (x_j ∈Γ).

Whenx_j does not belong toΓ, in the ballB(x_j, d_jh^ρ)we use the lower bound (see [15, Lemma 1.4.1])

(2.10) Qh,Ω^A (χ^h_ju₂)≥ inf

x∈B(xj,djh^ρ)|B(x)|hkχ^h_ju₂k² (x_j 6∈Γ).

For anyx∈ B(x_j, d_jh^ρ), its distance toΓis larger than ¹₂h^ρby construction. Letg∈Γbe such thatd(x,Γ) = d(x,g). Thend(g,Σ)is larger than ¹₂h^d. As a consequence of the Morse lemma

|B(x)| ≥C⁰(B) d(x,g) d(g,Σ)≥Ch^ρ+d. Hence, with (2.10) we obtain

(2.11) Qh,Ω^A (χ^h_ju2)≥Ch^1+ρ+dkχ^h_ju2k² (xj 6∈Γ).

Withρ= ¹₄ andd < ¹₄, the exponent1 +ρ+dis< ⁴₃ +²₃d. Therefore, forhsmall enough, as a result of (2.9) and (2.11) we have, for a positive constantcindependent ofh

Q^Ah,Ω(χ^h_ju₂)≥ch^4/3+2d/3kχ^h_ju₂k² (∀j ∈J).

With (2.7) this yields

Qh,Ω^A (u2)≥ch^4/3+2d/3X

j

kχ^h_ju2k²−C2,loch^2−2ρku2k²

≥c2h^4/3+2d/3ku2k²,

since2−2ρ= ³₂ >4/3 + 2d/3for anyd < ¹₄. The lemma is proved.

Lemma 2.4(Lower bound onΣ^[3]). There existC₃ andh₀ >0such that for all0< h < h₀ Q^Ah,Ω(u₃)≥ Λ^B₁h^3/2−C₃h^3/4+4d

ku₃k².

Proof. Forhsmall enough the setΣ^[3]is the union of the ballsB(x₀, h^d), withx₀spanningΣ. It suffices to consider each pointx₀separately. We denote byu^x₃⁰ the restriction ofu₃toB(x₀, h^d) and use the perturbative lower bound (2.2) withA? =A^x0 the third order Taylor expansion of Aatx0:

∀η >0, Q_h,Ω^A (u^x₃⁰)≥(1−η)Q_h,Ω^Ax0(u^x₃⁰)−η⁻¹k(A−A^x0)u^x₃⁰k².

Recall from the introduction that the magnetic field associated with A^x0 is B^x0, and that the magnetic operator(−ih∇+A^x0)² is unitarily equivalent to(−ih∇+ Ξ(x₀)A_ε(x0))² whose first eigenvalue ish^3/2Λ^x₁⁰. Besides,

|A(x)−A^x0(x)| ≤Ch^4d, ∀x∈B(x₀, h^d).

So we have

∀η >0, Q^Ah,Ω(u^x₃⁰)≥ (1−η)h^3/2Λ^x₁⁰ −η⁻¹Ch^8d

ku^x₃⁰k².

Choosing η = h^4d−3/4 to equalize the remainders, we deduce the inequality Qh,Ω^A (u^x₃⁰) ≥ h^3/2Λ^x₁⁰−Ch^3/4+4d

ku^x₃⁰k². Taking the infimum over all the points of the finite setΣgives the

result.

We can now conclude with the proof of Theorem1.6.

Proof of Theorem1.6. We gather the estimates provided by Lemmas2.2–2.4and combine them with the localization estimate (2.6) and obtain thatQh,Ω^A (u)is bounded from below by

c₁h^4/3ku₁k²+c₂h^4/3+2d/3ku₂k²+ Λ^B₁ h^3/2−C₃h^3/4+4d

ku₃k²−K_loch^2−2dkuk². Sinced < ¹₄, thenh^2−2d h^4/3+2d/3and the lower bound above can be replaced by

ch^4/3+2d/3(ku₁k²+ku₂k²) + Λ^B₁h^3/2−C₃h^3/4+4d−K_loch^2−2d ku₃k²

≥ch^4/3+2d/3k(1−χ)uk²+ Λ^B₁ h^3/2−Chmin{3/4+4d,2−2d}

kχuk² withχthe characteristic function of the set{x∈Ω,d(x,Σ)< h^d}. The theorem is proved.

2.3. Agmon estimates. Letd ∈ (₁₆³,¹₄)and L > 0. We consider an eigenpair(λ(h), ψ_h) of Ph,Ω^A such thatλ(h)≤Lh^3/2. For all functionΦ∈W^1,∞(Ω), we havee^Φψ_h ∈Dom(Q^Ah,Ω)and (2.12) Qh,Ω^A (e^Φψ_h) = λ(h)ke^Φψ_hk²+h²

|∇Φ|e^Φψ_h

2 . DefiningΦ(x) = d(x,Σ)h^−d, we have

|∇Φ(x)|² =h^−2d. Hence

(2.13) h²k|∇Φ|e^Φψ_hk² ≤h^2−2dke^Φψ_hk². We denote

Z_h^near ={x∈Ω, d(x,Σ)≤h^d} and Z_h^far ={x ∈Ω, d(x,Σ)> h^d}.

We introduceδ such thatδ = min{3/4 + 4d,2−2d} − ³₂. Sinced ∈ (₁₆³,¹₄), the numberδ is positive. Using (1.8a)-(1.8b) we obtain

Qh,Ω^A (e^Φψ_h)≥c_dh^4/3+2d/3 e^Φψ_h

2

L²(Z_h^far)+ Λ^B₁h^3/2−C_dh^3/2+δ) e^Φψ_h

2 L²(Z_h^near)

≥c_dh^4/3+2d/3 e^Φψ_h

2

L²(Z_h^far)−C_dh^3/2+δ e^Φψ_h

2

L²(Z_h^near) .

Combining the latter inequality with (2.12) and (2.13), we obtain (using the fact that λ(h) ≤ Lh^3/2)

c_dh^4/3+2d/3 −Lh^3/2−h^2−2d

ke^Φψ_hk²_L2(Z_h^far) ≤(Lh^3/2+h^2−2d+ C_dh^3/2+δ)ke^Φψ_hk²_L2(Z_h^near)

from which we immediately deduce that forhsmall enough e^Φψ_h

2

L²(Z_h^far) ≤ e^Φψ_h

2

L²(Z_h^near) . Since by construction|Φ|is bounded by1onZ_h^near, there holds

e^Φψ_h

2

L²(Z_h^near) ≤ kψ_hk²_L2(Z_h^near) ,

we finally getke^Φψ_hk²_L2(Ω) ≤ kψ_hk²_L2(Ω), which implies the Agmon estimates (1.10).

3. ASYMPTOTIC EXPANSIONS OF EIGENVALUES

In this section, we prove Theorem1.8in two steps. First, the localization around each cross- ing pointx₀ ∈Σ, and, second, a perturbation argument for the magnetic potential around each crossing point. We conclude the section by stating a full asymptotic expansion for eigenvalues and a sketch of the proof.

3.1. Preliminaries. We will use several times an argument based on reciprocal quasimodes between two operators. We state this in a general form.

Lemma 3.1. LetPandP?two positive operators with discrete spectra associated with sesqui- linear forms a and a?, respectively. Let (µ_n) and (µ?_n) be the increasing sequences of their eigenvalues (counted with multiplicity). Let(ϕ_n)be an associated orthonormal basis of eigen- vectors forP. LetN be a positive integer and assume that for eachn= 1, . . . , N, there exists ϕ?_n∈Dom(a?)such that

(3.1a) hϕ?_n, ϕ?_mi=hϕ_n, ϕ_mi+ν_n,m and a?(ϕ?_n, ϕ?_m) =a(ϕ_n, ϕ_m) +µ_n,m and set

(3.1b) ν = max

n,m |ν_n,m| and µ= max

n,m |µ_n,m| Assume thatν < _N¹. Then

(3.2) µ?_n≤ µ_n+nµ

1−nν , n = 1, . . . , N.

Proof. LetM ≤N. By the min-max formula,

µ?_M ≤ max

ϕ∈span{ϕ?_n, n=1,...,M}

a?(ϕ, ϕ) hϕ, ϕi We writeϕas a sumP

nγnϕ?_n. Then a?(ϕ, ϕ) = X

n,m

γ_nγ¯_ma?(ϕ?_n, ϕ?_m)

=X

n

|γ_n|²µ_n+X

n,m

γ_nγ¯_mµ_n,m ≤(µ_M +M µ)X

n

|γ_n|². Likewise

hϕ, ϕi=X

n,m

γ_n¯γ_mhϕ?_n, ϕ?_mi

=X

n

|γ_n|²+X

n,m

γ_n¯γ_mν_n,m ≥(1−M ν)X

n

|γ_n|².

Whence formula (3.2)

We will also need a simple but useful consequence of Agmon estimates.

Lemma 3.2. Assume that the family of function (ψ_h)_h>0 satisfies (for someγ and δ > 0) the estimate

Z

R²

e^γ|y|/hδ|ψ_h(y)|²dy ≤Ckψ_hk²,

forhsmall enough andC independent ofh. Then for allm >0, there existsC_msuch that for h >0small enough

Z

R²

|y|^m|ψ_h(y)|²dy≤C_mh^mδkψ_hk². Proof. It suffices to write

Z

R²

|y|^m|ψ_h(y)|²dy≤max

ρ>0 ρ^me^−γρ/hδ Z

R²

e^γ|y|/hδ|ψ_h(y)|²dy

and notice thatmax_ρ>0ρ^me^−γρ/hδ = max_ρ>0(h^δρ)^me^−γρ=h^mδmax_ρ>0ρ^me^−γρ.

3.2. Localization. Introduce a positive radius r₀ such that the collection of balls B(x₀, r₀), x₀ ∈ Σ, are pairwise disjoint. Then consider the collection of operatorsP_h,^A_B(x0,r0)forx₀ ∈Σ.

The main result of this section is:

Lemma 3.3. Recall thatλ_n(h)is the increasing sequence of eigenvalues ofPh,Ω^A (counted with multiplicity). Denote byλ^loc_n (h)the increasing sequence of eigenvalues of

M

x0∈Σ

Ph,^AB(x0,r0).

Letd∈(₁₆³ ,¹₄)and letN be a positive integer. Ifλ^loc_N (h)≤Lh^3/2 forh >0small enough, then there existCN andhN >0such that for allh∈(0, hN)

|λ_n(h)−λ^loc_n (h)| ≤C_Ne^−r0/hd, ∀n= 1, . . . , N.

Proof. Choose for each x₀ ∈ Σa smooth cut-off function χ_x0 with support in B(x₀, r₀) and equal to1onB(x₀,¹₂r₀).

i)Use Lemma3.1withP?=Ph,Ω^A andP=⊕x0∈ΣP_h,^A_B(x0,r0)(acting on⊕x0∈ΣL²(B(x0, r0))):

For each eigenvectorϕ_nofP, one crossing pointx₀ ∈Σis selected and we set ϕ?_n =χx0ϕn defined on Ω.

Relying on the assumption that λ^loc_n (h) ≤ Lh^3/2 for n = 1, . . . , N, we may apply the Agmon estimates (1.10) to the operator P_h,^A_B(x0,r0). This yields conditions (3.1a)-(3.1b) considering ν =µ=Ce^−r0/hd. Henceλ_n(h)≤λ^loc_n (h) +C_Ne^−r0/hd.

ii)Conversely, we swap the roles of the two operators:P=Ph,Ω^A andP? =⊕_x0∈ΣP_h,^A_B(x0,r0). For each eigenvectorϕ_nofP, we consider

ϕ?_n := χ_x0ϕ_n

x0∈Σ defined on a

x0∈Σ

B(x₀, r₀).

Note that, as a consequence of the previous step of the proof, we have thatλ_n(h)≤ 2Lh^3/2 for all n = 1, . . . , N. As above, we conclude with the help of Agmon estimates for the operator Ph,Ω^A thatλ^loc_n (h)≤λ_n(h) +C_Ne^−r0/hd. The lemma is proved.

3.3. Taylor approximation of a localized operator. With Lemma3.3at hand, we can assume that Ω = B(x0, r0). ThusΣ is reduced to one element, x0. Recall thatA^x0 denotes the third order Taylor expansion of Aat the point x₀. We are going to consider the operatorPh^Ax0 :=

(−ih∇+A^x0)² posed onR². We have seen in the introduction that the eigenvalues ofPh^Ax0

are theh^3/2Λ^x_n⁰, see (1.6a), and that its eigenvectorsψ_h,n^x0 are scaled from the eigenvectors ofXε

withε =ε(x₀)andΞ = Ξ(x₀). As a consequence of the exponential decay of the eigenvectors ofXε(Proposition1.4) and the scaling provided by Lemma1.5, we find that, for some positive constantγ

(3.3)

Z

R²

e^2γ|y|/h1/4|ψ_h,n^x0 (y)|²dy+h^−3/2Q^Ah^x0 e^γ|y|/h1/4ψ_h,n^x0

≤Ckψ^x_h,n⁰ k².

Lemma 3.4. WithΩ = B(x₀, r₀), we denote byλ^x_n⁰(h)the eigenvalues ofPh,Ω^A . The eigenvalues ofPh^Ax0 are given byh^3/2Λ^x_n⁰. For anyd₀ < ¹₄ and any positive integerN, there existC_N and h_N >0such that for allh∈(0, h_N)

(3.4) h^3/2Λ^x_n⁰ −C_Nh^3/2+d0 ≤ λ^x_n⁰(h) ≤ h^3/2Λ^x_n⁰ +C_N h^7/4, n = 1, . . . , N.

Proof. The proof combines Lemma 3.1 with the perturbation identity (2.1). We still use the cut-off functionχ_x0 as in the proof of Lemma3.3.

i)Use Lemma3.1withP=Ph^Ax0 andP?=Ph,Ω^A . For each eigenvectorϕ_nofP, we consider the quasimodeϕ?_n = χ_x0ϕ_n forP_h^Ax0. The localization error is exponentially decaying thanks to (3.3) but the principal part of the discrepancy for the quasimodes arises from the difference betweenAandA^x0. Thus the boundνin (3.1b) satisfies (for somer0 >0)

ν ≤Ce^−r0/h1/4.

For estimate the diagonal termsµ_n,nin (3.1a), we use the identity (2.1) forA?=A^x0. Then Q^Ah,Ω(χ_x0ψ_h,n^x0 ) =Qh^Ax0(χ_x0ψ_h,n^x0 ) + 2Re

(−ih∇+A^x0)χ_x0ψ_h,n^x0 ,(A−A^x0)χ_x0ψ^x_h,n⁰ +k(A−A^x0)χ_x0ψ_h,n^x0 k².

Hence the differenceµ_n,n :=Q_h,Ω^A (χ_x0ψ_h,n^x0 )−Q_h^Ax0(χ_x0ψ_h,n^x0 )is estimated by

|µ_n,n| ≤2 Q^Ah^x0(χ_x0ψ^x_h,n⁰ )1/2

k(A−A^x0)χ_x0ψ^x_h,n⁰ k+k(A−A^x0)χ_x0ψ^x_h,n⁰ k² Using the Agmon estimates (3.3) and Lemma3.2withδ = ¹₄ andm= 8, we find

|µn,n| ≤C(h^3/4h+h²).

The reasonning is similar forµ_n,m,n6=m. Hence the right part of inequalities (3.4).

ii)For the left part of (3.4), we swap the roles ofPh^Ax0 andPh,Ω^A . The sole difference consists in Agmon estimates for the truncated eigenvectors ofPh,Ω^A . Thanks to the right part of (3.4), Agmon estimates (1.10) hold and then Lemma3.2withδ =d∈(₁₆³,¹₄)andm= 8, from which we deduce

|µn,n| ≤C(h^3/4h^4d+h^8d).

Choosingd= ₁₆³ +^d₄⁰, we obtain the left part of (3.4).

3.4. Expansion of eigenvalues. Putting together Lemmas3.3and3.4, we can see that Lemma 3.4 provides the boundλ^loc_n (h) ≤Lh^3/2 which validates the application of Lemma3.3. There- fore, we have now proved:

Lemma 3.5. Under Assumption 1.2, for any d < ¹₄ and any positive integer N, there exist C_N >0andh₀ >0such that, for allh∈(0, h₀)and alln= 1, . . . , N

(3.5) h^3/2Λ^B_n −C_Nh^3/2+d ≤ λ_n(h) ≤ h^3/2Λ^B_n +C_N h^7/4, n = 1, . . . , N.

To go further, we are going to exhibit, for each n, series expansions of eigenpairs. Owing to the exponential localization given by Lemma 3.3, it suffices to restrict the construction to any chosen localized operator P_h,^A_B(x0,r0). In order to alleviate notations, we will remove the mention ofx₀in general, and work in the Cartesian coordinatesyfor which the crossing point is at the origin. Then the domainΩis the ballB(0, r₀)and the magnetic field cancels to the order 2at0. After a possible change of gauge, we can assume that the magnetic potential Acancels to the order3at0. We write its Taylor formal series as

A∼X

j≥0

Aj where Aj is polynomial and homogeneous of degree3 +j.

The first nonzero term isA₀ formerly denoted by A^x0. We retrieve the principal part ofPh,Ω^A

at 0, and its natural expansion in powers of h^1/4 by considering, via the change of variables Y=y/h^1/4:

h^−3/2Ph,Ω^A [y,∇_y] =h^−3/2P_h,Ω/h^{A 1/4}[h^1/4Y, h^−1/4∇_Y].

We expand the right hand side as a formal series of operatorsP

jh^j/4Lj[Y,∇_Y]defined onR². We have

X

j

h^j/4Lj[Y,∇_Y] =h^−3/2

−ih^3/4∇_Y+X

j≥0

h^(3+`)/4A_j(Y)2

. Hence the seriesP

jh^j/4Lj starts withL0 given by

L0 = −i∇Y+A0(Y)2

and the other termsLj forj ≥ 1are partial differential operators of degree 1with polynomial coefficients. The main term L0 is isospectral to (−i∇ + Ξ(x₀)A_ε(x₀))², see (1.5), and its eigenvalues are given by theΛ^x_n⁰ (n∈N^∗), see (1.6a).

Choose a normalized eigenpair ofL0, which we denote by(`0,Ψ0). We look for

`_j ∈R and Ψ_j ∈Dom(L0), j = 1,2, . . . solving P

jh^j/4Lj

P

kh^k/4Ψ_k

= P

jh^j/4`_j P

kh^k/4Ψ_k

in the sense of formal series, i.e., solving the sequence of equations,

m

X

j=0

LjΨm−j =

m

X

j=0

`jΨm−j m= 1,2, . . . . We write the first equation (form= 1) as

(L0−`<sub>0</sub>)Ψ<sub>1</sub> =`₁Ψ₀−L1Ψ₀, and the next ones as

(L0−`<sub>0</sub>)Ψ<sub>m</sub> =`_mΨ₀−LmΨ₀+

m−1

X

j=1

(`_j−Lj)Ψm−j.

If `₀ is a simple eigenvalue of L0, the solution of such a sequence of equations is classical, resulting from the Fredholm alternative for the self-adjoint operatorL0. For instance, we get

(3.6) `1 =hL1Ψ0,Ψ0i

ifm = 1. If`<sub>0</sub>is amultipleeigenvalue ofL0, we cannot choose a priori an associated eigenvec- tor, but have to work in the whole associated eigenspaceE<sub>`0</sub>. Then identity (3.6) is replaced by an eigen-equation for a finite dimensional hermitian matrix acting on the eigenspace E_`0. The process can be pursed as well, see [12] for details on this procedure. The termsΨm belong to the domain ofL0 and have, furthermore, exponential decay. Setting form ≥1

`^[m](h) = h^3/2

m

X

j=0

h^j/4`j and ψ_h^[m](y) =χx0(y)

m

X

j=0

h^j/4Ψj(y/h^1/4) we have constructed a quasimode forP_h,Ω^A to the orderh3/2+(m+1)/4, i.e.

Ph,Ω^A (ψ_h^[m]) =`^[m](h)ψ_h^[m]+ρ^[m+1]_h with kρ^[m+1]_h k ≤C_mh3/2+(m+1)/4kψ_h^[m]k.

Combining this with (3.5), we deduce by the spectral theorem that there holds

Theorem 3.6. Under Assumption1.2, for all integersN ≥1andM ≥0, there exist coefficients Λ^B_n,m ∈R⁺ for 1≤n ≤N, 0≤m≤M, with Λ^B_n,0 = Λ^B_n

and constantsC_N,M >0andh₀ >0such that, for allh∈(0, h₀)and alln= 1, . . . , N

λn(h)−h^3/2

M

X

m=0

h^m/4Λ^B_n,m

≤CNh3/2+(M+1)/4

.