Asymptotics for the low-lying eigenstates of the Schroedinger operator with magnetic field near corners

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Asymptotics for the low-lying eigenstates of the Schroedinger operator with magnetic field near corners

Virginie Bonnaillie-Noël, Monique Dauge

To cite this version:

Virginie Bonnaillie-Noël, Monique Dauge. Asymptotics for the low-lying eigenstates of the

Schroedinger operator with magnetic field near corners. Annales Henri Poincaré, Springer Verlag,

2006, 7, pp.899-931. �hal-00012081�

Asymptotics for the low-lying eigenstates of the Schr¨odinger operator with magnetic field near corners

V. Bonnaillie-No¨el and M. Dauge

Abstract

The Neumann realization for the Schr¨odinger operator with magnetic field is con- sidered in a bounded two-dimensional domain with corners. This operator is associated with a small semi-classical parameter h or, equivalently, with a large magnetic field.

We investigate the behavior of its eigenpairs as h tends to zero, like in a semi-classical limit. We prove, in the situation where the domain is a polygon and the magnetic field is constant, that the lowest eigenvalues are exponentially close to those of model prob- lems associated with the corners. We approximate the corresponding eigenvectors by linear combinations of functions concentrated in corners at the scale √

h. If the do- main has curved sides and the magnetic field is smoothly varying, we exhibit a full asymptotics for eigenpairs in powers of √

h.

Contents

1 Introduction 2

2 Outline 3

3 Model operators in infinite sectors 4

3.1 Spectrum . . . . 4 3.2 Solvability and exponential decay . . . . 5 3.3 Generalization to any affine magnetic potential . . . . 10

4 Quasi-modes for the Schr¨odinger operator with constant magnetic field 11

4.1 Definition of corner quasi-modes . . . . 11

4.2 Properties of quasi-modes . . . . 12

4.3 Partition of unity . . . . 14

5 Spectral asymptotics in a polygon (constant magnetic field) 15

5.1 Approximation of eigenvalues of P

_h

by corner model operators . . . . 15

5.2 Eigenspaces . . . . 17

6 Quasi-modes for the Schr¨odinger operator in a curvilinear polygon 20 6.1 Change of variables . . . . 21

6.2 Gauge transform . . . . 22

6.3 Scaling and formal series expression . . . . 23

6.4 Solutions of the formal series equation (53) . . . . 24

6.5 Sequences of quasi-modes for P

_h

near the corner s . . . . 25

7 Spectral asymptotics in a curvilinear polygon 29 7.1 Eigenvalue asymptotics . . . . 30

7.2 Eigenspaces . . . . 31

8 Conclusion 32

1 Introduction

The topic of our paper takes its origin from the Ginzburg-Landau theory modeling supercon- ducting properties in presence of an external magnetic field [11, 28]: The study of the Hes- sian of the Ginzburg-Landau functional leads to analyze the ground state of the Schr¨odinger operator with magnetic field [12, 18]. A small semi-classical parameter h = (κ B )

⁻¹

ap- pears as the magnetic field B is large or the physical characteristic κ of the superconducting material is large. When compared with most of the literature about Schr¨odinger operators, the unusual feature of the resulting problem is that it is posed on subdomains of R

²

or R

³

, and subject to Neumann or impedance boundary conditions.

Motivated by this, and also by other works about the spectrum of Schr¨odinger operator in the semi-classical limit, see [19, 20] for instance, we deal with the asymptotics for the low-lying eigenstates of the Schr¨odinger operator with magnetic field in a bounded two- dimensional domain, with focus on the influence of convex corners.

Let Ω denote a bounded domain in R

²

and A = ( A

1

, A

2

) a smooth magnetic potential associated with its magnetic field B = curl A . It is assumed that B > 0 on Ω . We investi- gate the behavior of the eigenpairs of the Neumann realization P

_h

on Ω for the Schr¨odinger operator − (h ∇ − i A )

²

as h → 0.

Many papers are devoted to the analysis of the first eigenpair when Ω is a smooth domain.

We can quote works of Bernoff-Sternberg [3], Lu-Pan [22, 23], Helffer-Morame [16, 17]:

It is proved that the fundamental state is localized near points of the boundary where the

curvature is maximal, and a two-term asymptotics of the fundamental state energy of P

_h

is

given. More recently, Fournais-Helffer [14] prove a complete asymptotic expansion for low- lying eigenvalues of P

_h

on domains such that the boundary curvature reaches its maximum in only one point.

Although the interest for non smooth domain is often mentioned in physical literature [8, 13, 26], quite few mathematical papers are devoted to that problem: Let us mention con- tributions of Jadallah [21], Pan [24] which deal with very particular domains like a square or a quarter plane. More recently, [5] gives a systematic analysis for infinite sectors of R

²

, proving an asymptotics of the smallest eigenvalue of − ( ∇ − i A )

²

when the aperture α of the sector tends to 0 , and exponential decay estimates for the corresponding eigenfunctions.

The limit as h → 0 of the first eigenvalue of P

_h

for domains with corners is deduced.

In this paper, we prove sharper results, exhibiting the complete asymptotic expansion of low-lying eigenstates for curvilinear polygonal domains. We also prove refined results in the case when the domain has straight sides and the magnetic field is constant: The convergence of the eigenpairs to their limits is then exponential, behaving as exp( − β/ √

h) for a positive β depending on their rank.

2 Outline

Let us sketch our results. The behavior of the first eigenstates of P

_h

depends on the spectrum of model problems associated with each point of the boundary, in particular, those associated with the corners s of Ω. Section 3 is devoted to spectral and solvability properties of such model operator Q

^α

:= − ( ∇ − i A

0

)

²

on an infinite sector of opening α and vertex at the origin. Here A

0

is the canonical magnetic potential

¹₂

( − X

₂

, X

₁

) corresponding to the magnetic field B = 1 . For any opening α , the essential spectrum of the operator Q

^α

is equal to [Θ

₀

, + ∞ ) , with the universal constant Θ

₀

≃ 0.590125. Depending on the value of α , the discrete spectrum of Q

^α

is empty or consists of K

α

eigenvalues. The corresponding eigenvectors are exponentially decreasing and, moreover, solutions Ψ of Q

^α

Ψ = L with Neumann conditions and exponentially decreasing right hand side L , are exponentially decreasing, too.

Sections 4-5 are devoted to the Schr¨odinger operator P

_h

when the domain Ω is a poly- gon, i.e. its sides are segments on lines, and the magnetic field B is equal to 1. To fix ideas, the magnetic potential is taken as

¹₂

( − x

₂

, x

₁

) . The eigenvectors of the model operators Q

^αs

corresponding to the aperture α

s

at each corner s of Ω allow the construction of quasi- modes in Section 4. These quasi-modes generate a space of dimension K

_Ω

:= P

s

K

_αs

, the

sum of the contributions of each corner. In Section 5, we prove that the first K

_Ω

eigenvalues

of P

_h

, when divided by h , converge exponentially fast towards the eigenvalues of the model

operators ⊕

^s

Q

^αs

. We also prove the localization of their eigenfunctions in corresponding

corners. Let us emphasize that, when several corners have the same aperture, clustering of

eigenvalues appear, and that each of the corresponding eigenvectors may concentrate in the

vicinity of several corners.

In Sections 6-7, we analyze more general domains (curvilinear polygons) with smoothly varying magnetic fields B . Again, we use the model operators Q

^αs

to construct quasi-modes for P

_h

, but now in combination with a formal series calculus. We obtain asymptotics series in powers of √

h for a finite number of low-lying eigenstates of P

_h

. In Section 8, we conclude our paper by commenting on numerical approximation issues: The eigenmodes have a two-scale structure, in the form of the product of a corner layer at scale √

h with an oscillatory term at scale h . The latter makes the numerical approximation delicate, see [1, 2, 7]. A finite element method using high degree polynomials is being investigated by the authors, together with the tunneling effect in presence of symmetries.

3 Model operators in infinite sectors

The model problem associated with a corner of opening α in the domain Ω is a Schr¨odinger operator Q

^α

in an infinite sector G

^α

of same opening, with a model magnetic potential A

0

corresponding to a constant field equal to 1 . After recalling results from [5] on the spectrum of this operator, we study its solvability in spaces of exponentially decreasing functions. We end this section by stating the relation between this model problem and a more general Schr¨odinger operator Q

^α,A

associated with any affine magnetic potential A .

3.1 Spectrum

We denote by X = ( X

₁

, X

₂

) the Cartesian coordinates in R

²

, and by R = | X | and θ the polar coordinates. Let G

^α

be the sector in R

²

with opening α :

G

^α

= { X ∈ R

²

, θ ∈ (0, α) } . We consider the model magnetic potential A

0

defined on R

²

by

A

0

( X ) = 1

2 ( − X

₂

, X

₁

). (1)

Then the magnetic field B given by curl A

0

is equal to 1 . Let Q

^α

be the Neumann real- ization of the Schr¨odinger operator − ( ∇ − i A

0

)

²

on the sector G

^α

. The operator Q

^α

is associated with the sesquilinear form q

^α

defined on the variational space V (q

^α

) as follows:

V (q

^α

) = n

Ψ ∈ L

²

(G

^α

), ( ∇ − i A

0

)Ψ ∈ L

²

(G

^α

) o

, (2)

q

^α

(Ψ, Φ) = Z

G^α

( ∇ − i A

0

)Ψ( X ) · ( ∇ − i A

0

)Φ( X ) d X , Ψ, Φ ∈ V (q

^α

). (3) The norm attached with the space V (q

^α

) is

|| Ψ ||

²_V(q^α₎

= || Ψ ||

²_L²_(G^α₎

+ || ( ∇ − i A

0

)Ψ ||

²_L²_(G^α₎

.

Note that if Ψ ∈ V (q

^α

) , then for any ball B , Ψ ∈ H

¹

(G

^α

∩ B ). Conversely, any Ψ in L

²

(G

^α

) such that ∇ Ψ and | X | Ψ are in L

²

(G

^α

) , belongs to V (q

^α

) .

Then the operator Q

^α

associated with the form q

^α

writes

Q

^α

= − ( ∇ − i A

0

)

²

= − ∆ + i( X

₁

∂

X₂

− X

₂

∂

X₁

) + 1

4 | X |

²

. (4) It is defined on its domain D (Q

^α

) :

D (Q

^α

) = n

Ψ ∈ V (q

^α

), ( ∇ − i A

0

)

²

Ψ ∈ L

²

(G

^α

), ν · ( ∇ − i A

0

)Ψ

∂G^α

= 0 o . Here ν is the outward unit normal on the boundary of G

^α

.

The operator Q

^α

is hermitian and positive. The lowest part of its spectrum can be defined by Rayleigh quotients.

Definition 3.1. Let µ

_k

(α) be the k − th smallest element of the spectrum of Q

^α

, given by the max-min principle:

µ

_k

(α) = max

Ψ1,...,Ψ_k−1

min

q

^α

(Ψ, Ψ)

h Ψ, Ψ i , Ψ ∈ V (q

^α

), Ψ ∈ [Ψ

₁

, . . . , Ψ

_k−1

]

^⊥

. (5) Here h· , ·i denotes the hermitian scalar product of L

²

(G

^α

).

Let us quote some results of [5] about the spectrum of Q

^α

. Theorem 3.2.

(i) The infimum of the essential spectrum of Q

^α

is equal to Θ

0

:= µ

1

(π).

(ii) For all α ∈ (0,

^π₂

], µ

1

(α) < Θ

0

and, therefore, µ

1

(α) is an eigenvalue.

(iii) Let k be a positive integer and α > 0 such that µ

_k

(α) < Θ

₀

. We denote by Ψ

^α_k

a normalized eigenfunction associated with µ

_k

(α) . Then Ψ

^α_k

satisfies the following exponential decay estimate:

∀ ε > 0, ∃ C

_ε,α

, e √

Θ0−µ_k(α)−ε

|X|

Ψ

^α_k

V(q^α)

≤ C

_ε,α

. (6) Remark 3.3. Based on the asymptotics of µ

₁

(α) as α → 0, see [5], and numerical com- putations, see [4, 1], we conjecture that µ

₁

is increasing from (0, π] onto (0, Θ

₀

] and equal to Θ

0

on [π, 2π) .

3.2 Solvability and exponential decay

We firstly prove the Fredholm alternative for the operator Q

^α

− µId, if µ is an eigenvalue.

Then, we prove the exponential decay of solutions if the right hand side is itself exponen-

tially decaying. We recall notation partially introduced in Theorem 3.2.

Notation 3.4.

Let α ∈ (0, 2π) and K

_α

be the largest integer such that µ

_Kα

(α) < Θ

₀

.

• We denote by (Ψ

^α_j

)

1≤j≤Kα

an orthonormalized system of eigenfunctions respectively associated with µ

_j

(α) for the operator Q

^α

.

• Let k ∈ N with k = 1, or 2 ≤ k ≤ K

_α

and such that µ

_k−1

(α) < µ

_k

(α) . Let l be the multiplicity of µ

_k

(α). Thus, we have

µ

_k−1

(α) < µ

_k

(α) = . . . = µ

_k+l−1

(α) < µ

_k+l

(α) ≤ Θ

₀

.

Lemma 3.5. With Notation 3.4, let L be a linear form defined and continuous on V (q

^α

) , and such that

L (Ψ

^α_j

) = 0, ∀ j = k, . . . , k + l − 1. (7) Then, there exists a unique Ψ ∈ V (q

^α

) such that

( h Ψ, Ψ

^α_j

i = 0, ∀ j = k, . . . , k + l − 1,

q

^α

(Ψ, Φ) − µ

_k

(α) h Ψ, Φ i = L (Φ), ∀ Φ ∈ V (q

^α

), (8) with q

^α

defined by (3) and h· , · , i the L

²

-scalar product on G

^α

. If we assume that, more- over, L (Ψ

^α_j

) = 0 for all j = 1, . . . , k − 1 , the solution of (8) is orthogonal to Ψ

^α₁

, . . . , Ψ

^α_k−1

. Proof. Let N ≥ k+ l − 1 such that µ

_N

(α) < Θ

₀

. With Notation 3.4, it is enough to choose k + l − 1 ≤ N ≤ K

_α

. We split the linear form L as

L = L

0

+

k−1

X

j=1

c

_j

Ψ

^α_j

+ X

N

j=k+l

c

_j

Ψ

^α_j

with L

0

(Ψ

^α_j

) = 0, ∀ j = 1, . . . , N.

We define the space

V

^N

= { Ψ ∈ V (q

^α

), h Ψ, Ψ

^α_j

i = 0, ∀ j = 1, . . . , N } .

Let us prove that the sesquilinear form q

^α

− µ

_k

(α) h· , ·i is coercive on V

^N

: Let κ ∈ (0, 1) and Ψ ∈ V

^N

, then

q

^α

(Ψ, Ψ) − µ

_k

(α) h Ψ, Ψ i ≥ (1 − κ)q

^α

(Ψ, Ψ) + (κµ

_N+1

(α) − µ

_k

(α)) h Ψ, Ψ i

≥ min(1 − κ, κµ

_N+1

(α) − µ

_k

(α)) || Ψ ||

²_V(q^α₎

.

It suffices to choose κ ∈ (0, 1) such that κµ

_N+1

(α) − µ

_k

(α) > 0 to deduce the coercivity.

Therefore, by the Lax-Milgram theorem, there exists a unique Ψ

₀

∈ V

^N

such that

q

^α

(Ψ

₀

, Φ) − µ

_k

(α) h Ψ

₀

, Φ i = L

0

(Φ), ∀ Φ ∈ V

^N

.

By orthogonality, Ψ

₀

is the unique solution in V (q

^α

) for the problem ( h Ψ

₀

, Ψ

^α_j

i = 0, ∀ j = 1, . . . , N,

q

^α

(Ψ

₀

, Φ) − µ

_k

(α) h Ψ

₀

, Φ i = L

0

(Φ), ∀ Φ ∈ V (q

^α

). (9) Furthermore, for j ∈ { 1, . . . , k − 1, k + l, . . . , N } , the unique function u

_j

∈ V (q

^α

) or- thogonal to Ψ

^α_k

, . . . , Ψ

^α_k+l−1

such that (Q

^α

− µ

_k

(α))u

j

= Ψ

^α_j

is given by

u

j

= 1

µ

_j

(α) − µ

_k

(α) Ψ

^α_j

. Consequently,

Ψ := Ψ

₀

+

k−1

X

j=1

c

j

µ

_j

(α) − µ

_k

(α) Ψ

^α_j

+ X

N

j=k+l

c

j

µ

_j

(α) − µ

_k

(α) Ψ

^α_j

(10) is the unique solution of (8).

Let us now analyze the decay of the solution Ψ as split in (10). Theorem 3.2 gives the decay of Ψ

^α_j

. Therefore, it is enough to study the decay of Ψ

₀

.

Lemma 3.6. With Notation 3.4, let N be an integer, k + l − 1 ≤ N ≤ K

_α

. Let L

0

be a linear form continuous on V (q

^α

) . We assume

L

0

(Ψ

^α_j

) = 0, ∀ j = 1, . . . , N, (11) and that, moreover, there exists δ

₀

> 0 such that L

0

is defined on { e

^δ0|X|

Ψ, Ψ ∈ V (q

^α

) } with the estimate

∃ C > 0, ∀ Φ ∈ V (q

^α

),

L

0

(e

^δ0|X|

Φ)

≤ C || Φ ||

V(q^α)

. (12) Then, the solution Ψ

₀

∈ V (q

^α

) of (9) satisfies e

^δN|X|

Ψ

₀

∈ V (q

^α

) for some positive number δ

_N

≤ δ

₀

, independent of L

0

.

Proof. Let δ ≤ δ

₀

. We define Ψ

_δ

= e

^δ|X|

Ψ

₀

, and we check that for any Φ ∈ V (q

^α

) , q

^α

(Ψ

₀

, Φ) = q

^α

(e

^−δ|X|

Ψ

_δ

, Φ)

= Z

G^α

( ∇ − i A

0

− δ I )Ψ

_δ

· ( ∇ − i A

0

+ δ I )(e

^−δ|X|

Φ) d X , with I =

1 1

. Let us define the space V

_δ^N

and the form a

_δ

on V

_δ^N

× V

_δ^N

by V

δ^N

= n

Ψ ∈ V (q

^α

), h Ψ, e

^−δ|X|

Ψ

^α_j

i = 0, ∀ j = 1, . . . , N o , a

_δ

(Ψ, Φ) =

Z

G^α

( ∇ − i A

0

− δ I )Ψ · ( ∇ − i A

0

+ δ I )Φ − µ

_k

(α)Ψ Φ

d X .

Then, for any Φ ∈ V (q

^α

) , we have

L

0

(Φ) = q

^α

(Ψ

₀

, Φ) − µ

_k

(α) h Ψ

₀

, Φ i = a

_δ

(Ψ

_δ

, e

^−δ|X|

Φ) = L

δ

(e

^−δ|X|

Φ), (13) where, thanks to (12), the linear form L

δ

can be defined on V (q

^α

) by

L

δ

(Φ) = L

0

(e

^δ|X|

Φ). (14) Due to the compatibility condition (11), we have

L

δ

(e

^−δ|X|

Ψ

^α_j

) = 0, ∀ j = 1, . . . , N.

Solving the problem of finding Φ

_δ

∈ V

_δ^N

such that

a

_δ

(Φ

_δ

, Φ) = L

δ

(Φ), ∀ Φ ∈ V

δ^N

, (15) will provide exponential decay of the solution Ψ

₀

for problem (8).

We verify easily that the form a

_δ

is sesquilinear and continuous on V

δ^N

× V

δ^N

. Let us prove its coercivity. Let Ψ ∈ V

_δ^N

, then

| a

_δ

(Ψ, Ψ) | ≥ Re a

_δ

(Ψ, Ψ) = q

^α

(Ψ, Ψ) − µ

_k

(α) || Ψ ||

²

− δ

²

|| Ψ ||

²

. (16) We decompose Ψ such that

Ψ = X

N

j=1

h Ψ, Ψ

^α_j

i Ψ

^α_j

+ Ψ

^⊥

,

then, the definition of V

_δ^N

and the decay of Ψ

^α_j

give for any j = 1, . . . , N , h Ψ, Ψ

^α_j

i

= h Ψ, (1 − e

^−δ|X|

)Ψ

^α_j

i

≤ || Ψ ||

L²(G^α)

Z

G^α

δ

²

| X |

²

| Ψ

^α_j

( X ) |

²

d X

1/2

≤ C

_j

δ || Ψ ||

L²(G^α)

. (17) Furthermore, due to the decomposition of Ψ and (17), it follows

q

^α

(Ψ, Ψ) = q

^α

(Ψ

^⊥

, Ψ

^⊥

) + X

N

j=1

h Ψ, Ψ

^α_j

i

²

q

^α

(Ψ

^α_j

, Ψ

^α_j

)

≥ µ

_N+1

(α) || Ψ

^⊥

||

²

+ X

N

j=1

µ

_j

(α)

h Ψ, Ψ

^α_j

i

²

≥ µ

N+1

(α) || Ψ ||

²

− X

N

j=1

(µ

N+1

(α) − µ

j

(α))

h Ψ, Ψ

^α_j

i

²

≥

(µ

N+1

(α) − δ

²

X

N

j=1

C

_j²

(µ

N+1

(α) − µ

j

(α))

|| Ψ ||

²

. (18)

Defining M

_N

= P

_N

j=1

C

_j²

(µ

_N+1

(α) − µ

_j

(α)), we deduce from (16)-(18), for κ ∈ (0, 1) ,

| a

_δ

(Ψ, Ψ) | ≥ (1 − κ) || ( ∇ − i A

0

)Ψ ||

²

+

κ µ

N+1

(α) − δ

²

M

N

− µ

_k

(α) − δ

²

|| Ψ ||

²

. We define

δ ¯

_N

= min

δ

₀

, s

µ

_N+1

(α) − µ

_k

(α) 1 + M

N

. For δ ∈ (0, δ ¯

_N

) , we choose κ ∈ (0, 1) such that κ µ

_N+1

(α) − δ

²

M

_N

− µ

_k

(α) − δ

²

> 0 . This choice proves the coercivity of a

_δ

on V

_δ^N

for any δ ∈ [0, δ ¯

_N

).

Applying the Lax-Milgram theorem, we find that there exists a unique Φ

_δ

∈ V

_δ^N

solu- tion of the variational problem (15). Thanks to the orthogonality conditions, Φ

_δ

satisfies, moreover, a

_δ

(Φ

_δ

, Φ) = L

δ

(Φ), ∀ Φ ∈ V (q

^α

). Since δ is positive, Φ

_δ

satisfies, a fortiori,

a

_δ

(Φ

_δ

, e

^−δ|X|

Φ) = L

δ

(e

^−δ|X|

Φ), ∀ Φ ∈ V (q

^α

).

Therefore Φ

_δ

coincides with Ψ

_δ

= e

^δ|X|

Ψ

₀

, compare with (13). We deduce that e

^δ|X|

Ψ

₀

belongs to V (q

^α

) for all δ ∈ [0, δ ¯

_N

) , which ends the proof.

Using Lemmas 3.5 and 3.6, we deduce:

Lemma 3.7. With Notation 3.4, let L be a linear form continuous on V (q

^α

). We assume L (Ψ

^α_j

) = 0, ∀ j = k, . . . , k + l − 1, (19) and that, moreover, there exists δ

₀

> 0 such that L is defined on e

^δ0|X|

V (q

^α

) with the estimate:

∃ C > 0, ∀ Φ ∈ V (q

^α

),

L (e

^δ0|X|

Φ)

≤ C || Φ ||

V(q^α)

. (20) Then, there exists a unique Ψ ∈ V (q

^α

) such that

( h Ψ, Ψ

^α_j

i = 0, ∀ j = k, . . . , k + l − 1,

q

^α

(Ψ, Φ) − µ

_k

(α) h Ψ, Φ i = L (Φ), ∀ Φ ∈ V (q

^α

). (21) Furthermore, e

^δ|X|

Ψ ∈ V (q

^α

) for some δ ≤ δ

₀

independent of L .

Proof. The existence and uniqueness is clear according to Lemma 3.5. Combining Lemma 3.6 and Theorem 3.2 with the decomposition of Ψ into

Ψ

₀

+

k−1

X

j=1

c

_j

µ

_j

(α) − µ

_k

(α) Ψ

^α_j

+ X

N

j=k+l

c

_j

µ

_j

(α) − µ

_k

(α) Ψ

^α_j

, we obtain the decay of Ψ for any δ < min

δ ¯

_N

, p

Θ

₀

− µ

_N

(α)

, with any integer N such

that k + l − 1 ≤ N ≤ K

_α

.

3.3 Generalization to any affine magnetic potential

To conclude this section about model problems, we deal with an arbitrary real-valued affine magnetic potential, thus of the form

A ( X ) =

a

₁₁

X

₁

+ a

₁₂

X

₂

+ a

₁₀

, a

₂₁

X

₁

+ a

₂₂

X

₂

+ a

₂₀

. (22)

The associated magnetic field is

B = curl A = a

₂₁

− a

₁₂

. (23) The quadratic function (the gauge function) defined by

G ( X ) = 1 2

a

11

X

²

1

+ a

22

X

²

2

+ (a

12

+ a

21

) X

₁

X

₂

+ a

10

X

₁

+ a

20

X

₂

, (24) is such that

A = BA

0

+ ∇G , (25)

with the model magnetic potential A

0

defined in (1).

Proposition 3.8. Let α ∈ (0, 2π), and let A be an affine magnetic potential as in (22)-(25).

We assume that the associated magnetic field B is positive. Then the Neumann realization, Q

^α,A

, of the Schr¨odinger operator − ( ∇ − i A )

²

on G

^α

has K

_α

eigenvalues strictly less than B Θ

₀

. For k ≤ K

_α

, the k -th eigenvalue of − ( ∇ − i A )

²

is equal to B µ

_k

(α) and its corresponding normalized eigenvector, Ψ

^α,A_k

, is given on G

^α

by

Ψ

^α,A_k

( X ) = √

B exp (i G ( X )) Ψ

^α_k

( √ B X ).

Proof. We verify easily that Ψ

^α,A_k

is L

²

-normalized. The operator Q

^α,A

is defined on D (Q

^α,A

) with

D (Q

^α,A

) =

Ψ ∈ L

²

(G

^α

), ( ∇ − i A )Ψ ∈ L

²

(G

^α

),

( ∇ − i A )

²

Ψ ∈ L

²

(G

^α

), ν · ( ∇ − i A )Ψ

_|∂Gα

= 0 . (26) Using the transformation

D (Q

^α

) → D (Q

^α,A

)

Ψ 7→ Ψ

^A

with Ψ

^A

( X ) = √

B exp(i G ( X )) Ψ( √ B X ), we see that the change of variables Y = √

B X leads to ( ∇

^X

− i A ( X ))Ψ

^A

( X ) = √

B exp(i G ( X )) √

B ∇

^Y

− i( A ( X ) − ∇G ( X )) Ψ( Y )

= √

B exp(i G ( X )) √

B ∇

^Y

− i B A

0

Y

√ B

Ψ( Y )

= B exp(i G ( X )) ( ∇

^Y

− i A

0

( Y ))Ψ( Y ).

It follows

Q

^α,A

Ψ

^A

( X ) = B √

B exp(i G ( X )) Q

^α

Ψ( Y ).

Remark 3.9. Note that ( ∇ − i A )

²

Ψ = ( ∇ + i A )

²

Ψ for any real-valued smooth magnetic potential A and any Ψ ∈ D (Q

^α,A

). Thus, Proposition 3.8 is still valid when the magnetic field B is negative and Ψ

^α,A_k

is given on G

^α

by

Ψ

^α,A_k

( X ) = p

|B| exp(i G ( X )) Ψ

^α_k

( p

|B| X ).

4 Quasi-modes for the Schr¨odinger operator with constant mag- netic field in a polygonal domain

Before considering in a further step a more general situation (Sections 6-7), we suppose that our domain Ω is a convex bounded polygon with straight edges, and that the magnetic potential is equal to A

0

(x) =

¹₂

( − x

₂

, x

₁

) . We are interested in the behavior of the low- est eigenvalues of the Neumann realization P

_h

on Ω, for the Schr¨odinger operator with magnetic potential A

0

and semi-classical parameter h > 0 .

The associated sesquilinear form p

_h

is defined on H

¹

(Ω) by p

_h

(u, v) =

Z

Ω

(h ∇ − i A

0

)u(x) · (h ∇ − i A

0

)v(x) dx. (27) The operator P

_h

= − (h ∇ − i A

0

)

²

is well defined on its domain D (P

_h

) , with

D (P

_h

) =

u ∈ H

²

(Ω), ν · (h ∇ − i A

0

)u

∂Ω

= 0 . (28) In this section, we introduce suitable corner quasi-modes which will allow to construct limit spectral problems for P

_h

.

4.1 Definition of corner quasi-modes

Let Σ be the set of the vertices s of Ω , and α

s

be the opening of Ω at s ∈ Σ. The spectrum of P

_h

is in close relation with the spectra of the model operators Q

^αs

, as defined in (4), for s describing the set of corners Σ.

As a first step in the explanation of this relation, we introduce, for each vertex s , the infinite plane sector G ˘

^s

which coincides with Ω near the vertex s : For d > 0 small enough, we have

Ω ∩ B( s , d) = ˘ G

s

∩ B( s , d).

There exists a rotation R

^s

such that

{ X = R

^s

(x − s ), x ∈ G ˘

^s

} = G

^αs

. As a consequence of Proposition 3.8, we obtain:

Lemma 4.1. For all integer k, 1 ≤ k ≤ K

_αs

, the function ψ ˘

_h,s,k

defined by ψ ˘

_h,s,k

(x) = 1

√ h exp i

2h x ∧ s

Ψ

^α_k^s

R

^s

(x − s )

√ h

on G ˘

s

, (29) is a normalized eigenvector for the operator − (h ∇ − i A

0

)

²

with Neumann boundary con- ditions on G ˘

s

, associated with the eigenvalue hµ

_k

(α

s

) .

Thus we construct quasi-modes for P

_h

from the eigenpairs (µ

_k

(α

s

), Ψ

^α_k^s

) of Q

^αs

for each corner s of Ω and each k ≤ K

_αs

via translation, rotation and cut-off according to:

Notation 4.2. • Let s ∈ Σ and ρ

s

be the distance to other vertices:

ρ

^s

= dist( s , Σ \ { s } ).

Let ρ

^′

∈ (0, ρ

^s

) and χ

^s

be a radial smooth cut-off function with support in B( s , ρ

^s

) , equal to 1 in B( s , ρ

^′

) and such that 0 ≤ χ

s

≤ 1 .

• Let k ≤ K

_αs

. Applying the cut-off χ

s

to the function ψ ˘

_h,s,k

in (29) we define

ψ

_h,s,k

(x) = χ

s

(x) ˘ ψ

_h,s,k

(x) on Ω. (30) 4.2 Properties of quasi-modes

We gather in the following lemma the main properties of the functions ψ

_h,s,k

. Lemma 4.3. For any ε > 0, there exists C

ε

such that (31)-(33) hold.

(i) The L

²

norm of ψ

_h,s,k

is nearly 1:

1 − C

_ε

exp

− 2

√ h

ρ

^′

p

Θ

₀

− µ

_k

(α

s

) − ε

≤ || ψ

_h,s,k

||

²_L²_(Ω)

≤ 1. (31) (ii) The Rayleigh quotient of ψ

_h,s,k

is nearly hµ

_k

(α

s

):

p

_h

(ψ

_h,s,k

, ψ

_h,s,k

)

|| ψ

_h,s,k

||

²_L²_(Ω)

− hµ

_k

(α

s

)

≤ C

_ε

exp

− 2

√ h ρ

^′

p

Θ

₀

− µ

_k

(α

s

) − ε

. (32) (iii) The pair (hµ

_k

(α

^s

), ψ

_h,^s_,k

) is an approximate eigenpair of P

_h

:

|| P

_h

ψ

_h,s,k

− hµ

_k

(α

s

)ψ

_h,s,k

||

L²(Ω)

|| ψ

_h,s,k

||

L²(Ω)

≤ C

_ε

exp

− 1

√ h

ρ

^′

p

Θ

₀

− µ

_k

(α

^s

) − ε

. (33)

Proof. (i) Since, by construction, || ψ ˘

_h,s,k

||

_L²_{( ˘G}s)

= 1 and thanks to the decay properties (6) of Ψ

^α_k^s

, we have

1 ≥ || χ

^s

ψ ˘

_h,^s_,k

||

²_L²_(Ω)

≥ Z

Ω∩B(s,ρ^′)

| ψ ˘

_h,^s_,k

|

²

dx

= Z

G^αs∩B“ 0,√^ρ′

h

”

| Ψ

^α_k^s

( X ) |

²

d X

= 1 − Z

G^αs\B“ 0,√^ρ′

h

”

| Ψ

^α_k^s

( X ) |

²

d X

≥ 1 − C

_ε

exp

− 2ρ

^′

√ h

p Θ

₀

− µ

_k

(α

^s

) − ε . (ii) Let us prove the estimate about the quadratic form. We have ψ

_h,s,k

∈ H

¹

(Ω) and

p

_h

(ψ

_h,s,k

, ψ

_h,s,k

) = Z

Ω

| (h ∇ − i A

0

)ψ

_h,s,k

|

²

dx

= Z

Ω

| χ

s

(h ∇ − i A

0

) ˘ ψ

_h,s,k

|

²

dx + h

²

Z

Ω

| ψ ˘

_h,s,k

|

²

|∇ χ

s

|

²

dx + 2h Re

Z

Ω

χ

s

(h ∇ − i A

0

) ˘ ψ

_h,s,k

· ∇ χ

s

ψ ˘

_h,s,k

dx. (34) Due to the properties of χ

^s

and to the decay estimate (6) again, we have

Z

Ω

| χ

^s

(h ∇ − i A

0

) ˘ ψ

_h,^s_,k

|

²

dx ≤ Z

Ω

| (h ∇ − i A

0

) ˘ ψ

_h,^s_,k

|

²

dx

≤ hµ

_k

(α

s

) || ψ ˘

_h,s,k

||

²_L2( ˘G^s)

(35) Z and

Ω

| χ

^s

(h ∇ − i A

0

) ˘ ψ

_h,s,k

|

²

dx ≥ Z

Ω∩B(s,ρ^′)

| (h ∇ − i A

0

) ˘ ψ

_h,s,k

|

²

dx

≥ Z

G˘^s

| (h ∇ − i A

0

) ˘ ψ

_h,s,k

|

²

dx − Z

G˘^s\B(s,ρ^′)

| (h ∇ − i A

0

) ˘ ψ

_h,s,k

|

²

dx

≥ hµ

_k

(α

^s

) || ψ ˘

_h,^s_,k

||

²_L2( ˘G^s)

− C

ε

exp

− 2ρ

^′

√ h

p Θ

0

− µ

_k

(α

^s

) − ε . (36) Still using Theorem 3.2, we deduce also the estimate

2h Re

Z

Ω

χ

s

(h ∇ − i A

0

) ˘ ψ

_h,s,k

· ∇ χ

s

ψ ˘

_h,s,k

dx + h

²

Z

Ω

| ψ ˘

_h,s,k

|

²

|∇ χ

s

|

²

dx

≤ Ch Z

Ω\B(s,ρ^′)

| (h ∇ − i A

0

) ˘ ψ

_h,s,k

|

²

+ | ψ ˘

_h,s,k

|

²

dx

≤ C

_ε

exp

− 2ρ

^′

√ h

p Θ

₀

− µ

_k

(α

s

) − ε

. (37)

Putting together relation (34) with estimates (35), (36), (37) and using the estimate (31) for

|| ψ

_h,s,k

||

L²(Ω)

, we deduce (32).

(iii) Let us now prove the last estimate. Since χ

^s

is radial, we can check that ψ

_h,s,k

satisfies the Neumann boundary condition and, thus, belongs to D (P

_h

). We have

P

_h

(ψ

_h,^s_,k

) = χ

^s

P

_h

( ˘ ψ

_h,^s_,k

) − 2h ∇ χ

^s

· (h ∇ − i A

0

) ˘ ψ

_h,^s_,k

− h

²

ψ ˘

_h,^s_,k

∆χ

^s

. On the support of χ

s

, we have, thanks to Lemma 4.1,

P

_h

ψ ˘

_h,s,k

(x) = hµ

_k

(α

s

) ˘ ψ

_h,s,k

(x).

Therefore χ

s

P

_h

( ˘ ψ

_h,s,k

) = hµ

_k

(α

s

)ψ

_h,s,k

. The same arguments as above for the proof of (37) lead to the estimate

|| 2h ∇ χ

^s

· (h ∇ − i A

0

) ˘ ψ

_h,^s_,k

+ h

²

ψ ˘

_h,^s_,k

∆χ

^s

||

²_L²_(Ω)

≤ C

_ε

exp

− 2ρ

^′

√ h

p Θ

₀

− µ

_k

(α

s

) − ε . (38) This ends the proof of Lemma 4.3.

4.3 Partition of unity

We end this section by a useful lemma which will allow to achieve the proof of the spectral asymptotics which can be obtained from the quasi-modes.

Lemma 4.4. For any s ∈ Σ, let χ

^s

be a real-valued cut-off function supported in B( s , ρ

^s

) . We assume moreover that for any s 6 = s

^′

, suppχ

^s

∩ suppχ

^′s

= ∅ . We define χ

₀

on Ω by χ

²₀

= 1 − P

s∈Σ

χ

²s

. By convention χ

s

with s = 0 refers to χ

₀

. Then, for any ψ ∈ H

¹

(Ω), p

_h

(ψ, ψ) = X

s∈Σ∪{0}

p

_h

(χ

s

ψ, χ

s

ψ) − h

²

X

s∈Σ∪{0}

|| ψ ∇ χ

s

||

²_L²_(Ω)

.

Proof. Let s ∈ Σ ∪ { 0 } , then

| (h ∇ − i A

0

)(χ

s

ψ) |

²

= | χ

s

|

²

| (h ∇ − i A

0

)ψ |

²

+ h

²

| ψ |

²

|∇ χ

s

|

²

+ 2h Re χ

^s

ψ (h ∇ − i A

0

)ψ · ∇ χ

^s

= | χ

s

|

²

| (h ∇ − i A

0

)ψ |

²

+ h

²

| ψ |

²

|∇ χ

s

|

²

+ h

²

Re ψ ∇ ψ · ∇| χ

s

|

²

. Let us sum up this relation for s ∈ Σ ∪ { 0 } , it follows

X

s∈Σ∪{0}

| (h ∇ − i A

0

)(χ

s

ψ) |

²

= X

s∈Σ∪{0}

| χ

s

|

²

| (h ∇ − i A

0

)ψ |

²

+ h

²

X

s∈Σ∪{0}

| ψ |

²

|∇ χ

s

|

²

+ h

²

X

s∈Σ∪{0}

Re ψ ∇ ψ · ∇| χ

^s

|

²

.

Since P

s∈Σ∪{0}

| χ

^s

|

²

= 1 on Ω, we notice that on Ω X

s∈Σ∪{0}

| χ

^s

|

²

| (h ∇ − i A

0

)ψ |

²

= | (h ∇ − i A

0

)ψ |

²

, X

s∈Σ∪{0}

Re ψ ∇ ψ · ∇| χ

^s

|

²

= Re ψ ∇ ψ · ∇ X

s∈Σ∪{0}

| χ

^s

|

²

= 0.

Integrating on Ω ends the proof.

5 Spectral asymptotics in a polygon (constant magnetic field)

In this section, we prove that, provided that some of the model operators Q

^αs

have eigenval- ues λ below their essential spectrum, a corresponding number of eigenvalues µ

_h

of P

_h

are exponentially close to hλ as h tends to 0. We also prove the related results for eigenspaces.

5.1 Approximation of eigenvalues of P

h

by corner model operators We first make precise the notations about eigenvalues.

Notation 5.1. • We denote by µ

_h,n

the n − th eigenvalue of P

_h

counted with multiplicity.

• We denote by λ

n

the n − th eigenvalue of ⊕

^s∈Σ

Q

^αs

counted with multiplicity as defined by the min-max principle, and let K

_Ω

be the largest integer such that λ

_KΩ

< Θ

₀

. With Notation 3.4, we have K

_Ω

= P

s∈Σ

K

_αs

. We assume that K

_Ω

≥ 1 . For any n ≤ K

_Ω

, we denote by Σ

n

the subset of vertices:

Σ

n

=

s ∈ Σ, λ

n

is an eigenvalue for Q

^αs

, and by r

_n

the distance

r

_n

= r(λ

_n

) = min

s∈Σn

d( s , Σ \ { s } ).

Theorem 5.2. With Notation 5.1, for any ε > 0, there exists C

ε

such that µ

_h,1

≤ hλ

₁

+ C

_ε

exp

− 2

√ h

r

₁

p

Θ

₀

− λ

₁

− ε ,

| µ

_h,n

− hλ

_n

| ≤ C

_ε

exp

− 1

√ h r

_n

p

Θ

₀

− λ

_n

− ε

, ∀ n ≤ K

_Ω

.

Proof. (i) Estimate (32) of Lemma 4.3 applied with µ

_k

(α

^s

) = λ

₁

and ρ

^′

= r

₁

− ε

^′

and the min-max principle (recalled in Definition 3.1) lead to

µ

_h,1

≤ hλ

₁

+ C

_ε

exp

− 2

√ h (r

₁

− ε

^′

) p

Θ

₀

− λ

₁

− ε

.

(ii) Let n ≤ K

_Ω

and s ∈ Σ

_n

. Let Ψ

^αs

be a normalized eigenvector for Q

^αs

associated with λ

_n

and let ψ ˘

_h,s

be the function deduced from Ψ

^αs

by (29). Then ψ ˘

_h,s

is a normalized eigenfunction of − (h ∇ − i A

0

)

²

on G ˘

s

associated with the eigenvalue hλ

_n

. Let ε > 0. Let χ

^s

∈ C

0^∞

(Ω, [0, 1]) be a smooth cut-off function as in (30), with ρ

^′

< r

n

−

₂^ε

, and define ψ

_h,s

= χ

s

ψ ˘

_h,s

as in (30).

We deduce from estimate (33) that there exists C

_ε

> 0 such that

|| P

_h

(ψ

_h,^s

) − hλ

n

ψ

_h,^s

||

L²(Ω)

|| ψ

_h,^s

||

L²(Ω)

≤ C

ε

exp

− 1

√ h

r

n

p Θ

0

− λ

n

− ε

. (39) Due to the spectral theorem (cf [25, Chap. VII]), it follows

d(σ(P

_h

), hλ

_n

) ≤ C

_ε

exp

− 1

√ h

r

_n

p

Θ

₀

− λ

_n

− ε

. (40)

(iii) Let us prove a lower bound for the eigenvalues of P

_h

using ideas of [9, 27]. Let n ≤ K

_Ω

+ 1 be such that λ

_n−1

6 = λ

_n

. With the cut-off functions χ

s

already introduced for s ∈ Σ, let us define χ

0

on Ω by χ

²₀

= 1 − P

s∈Σ

χ

²s

. Due to Lemma 4.4, we know that for any u ∈ H

¹

(Ω),

p

_h

(u, u) = X

s∈Σ∪{0}

p

_h

(χ

^s

u, χ

^s

u) − h

²

X

s∈Σ∪{0}

|| u ∇ χ

^s

||

²_L²_(Ω)

. (41)

Since suppχ

₀

∩ Σ = ∅ , we can apply the result of [12] for smooth domains: there exists c > 0 so that

p

_h

(χ

₀

u, χ

₀

u) ≥ (hΘ

₀

− ch

²

) || χ

₀

u ||

²_L²_(Ω)

. (42) For any s ∈ Σ, let T

^s

be the restriction of Q

^αs

to the space spanned by the eigenfunctions Ψ

^α_k^s

corresponding to eigenvalues µ

_k

(α

^s

) ≤ λ

n−1

. We denote by R

^∗_h,s

the application

R

^∗_h,^s

D (P

_h

) → D (Q

^αs

)

˘

u

_h,^s

7→ u such that u ˘

_h,^s

(x) = 1

√ h exp i

2h x ∧ s

u

R

^s

(x − s )

√ h

. Then we have

p

_h

(χ

^s

u, χ

^s

u) = h q

^αs

R

^∗h,s

(χ

^s

u), R

^∗h,s

(χ

^s

u)

≥ h λ

_n

||R

^∗h,s

(χ

^s

u) ||

²L²(G^αs)

− h

R

^∗h,s

(χ

^s

u), T

^s

R

^∗h,s

(χ

^s

u) . (43) Let T

_h,s

be the restriction of − (h ∇ − i A

0

)

²

on G ˘

^s

to the space spanned by the eigenfunc- tions ψ ˘

_h,s,k

, see (29), corresponding to eigenvalues µ

_k

(α

s

) ≤ λ

_n−1

. We set

T = X

s∈Σ

χ

^s

T

_h,^s

χ

^s

.

Combining (41)-(43) we obtain, since λ

_n

≤ Θ

₀

,

p

_h

(u, u) ≥ hλ

_n

|| u ||

²_L²_(Ω)

− h u, T u i − ch

²

|| u ||

²_L²_(Ω)

. Since for any s ∈ Σ ,

rank(χ

^s

T

_h,s

χ

^s

) ≤ rank(T

^s

) = card

eigenvalues of Q

^αs

less than λ

_n−1

,

we deduce, thanks to the assumption λ

_n−1

< λ

_n

, that the rank of T is not greater than n − 1 .

To obtain a lower bound for µ

_h,n

, we use the max-min principle. Let u

₁

, . . . , u

_n−1

be- long to the orthogonal space of ker(T) . Then for all ψ in { u

₁

, . . . , u

_n−1

}

^⊥

, ψ belongs to ker(T ) and we have

µ

_h,n

≥ h ψ, P

_h

ψ i

|| ψ ||

²

≥ hλ

_n

− ch

²

. (44) (iv) Now we reach the conclusion: According to (40), we know that there exists µ

_h

∈ σ(P

_h

) such that

| µ

_h

− hλ

n

| ≤ C

ε

exp

− 1

√ h (r

n

p Θ

0

− λ

n

− ε)

.

Using (44), we obtain that µ

_h

belongs to the set { µ

_h,k

, λ

_k

= λ

_n

} , which ends the proof of the theorem.

Remark 5.3. In particular the lower bound (44) is valid for n = K

_Ω

+ 1: In this case, λ

K_Ω+1

= Θ

0

. Thus (44) yields

µ

_KΩ+1,h

≥ hΘ

₀

− ch

²

. 5.2 Eigenspaces

It results from the previous theorem that, according to repetitions of the same values in λ ∈ { λ

₁

, . . . , λ

_KΩ

} , the corresponding eigenvalues µ

_h,n

are gathered into clusters, because they are exponentially close to the same value hλ . We are going to prove that the corresponding eigenvectors are exponentially close to linear combinations of quasi-modes. Let us first introduce the definition of distance between subspaces E and F of a Hilbert space. The (a priori) non-symmetric distance d(E, F ) is defined as

d(E, F ) = || Π

E

− Π

F

Π

E

||

H

,

where Π

_E

and Π

_F

denote the orthogonal projections on E and F respectively. If both E and F are finite dimensional, and if they have the same dimension, then d(E, F ) = d(F, E) .

To prove that eigenvectors of P

_h

are close to linear combinations of quasi-modes, we

use the following refinement of [29] more adapted to clustered eigenvalues:

Theorem 5.4 ([15, Prop. 4.1.1] [19]). Let A be an unbounded self-adjoint operator in a Hilbert space H . Let ψ

₁

, . . . , ψ

_N

be N linearly independent vectors in D (A) and µ

1

, . . . , µ

N

be N real numbers such that

Aψ

_j

= µ

_j

ψ

_j

+ r

_j

with || r

_j

||

H

≤ η. (45) Let I ⊂ R be a compact interval containing µ

₁

, . . . , µ

_N

. We assume that there exists a > 0 such that σ(A) ∩ (I + B(0, 2a) \ I ) = ∅ . Then, if E is the space spanned by ψ

1

, . . . , ψ

N

and if F is the spectral space associated with σ(A) ∩ I , we have d(E, F ) ≤ η √

N a

q κ

^min_S

, (46)

where κ

^min_S

is the smallest eigenvalue of the Gram matrix S =

h ψ

_j

, ψ

_k

i

H

.

Let us now introduce some notation for the cluster of eigenspaces and quasi-modes.

Notation 5.5. • Using Notation 5.1, we denote by { Λ

₁

< · · · < Λ

_M

} the set of distinct values in { λ

₁

, . . . , λ

_KΩ

} . For any m ≤ M , we define the distance

R

_m

= r(Λ

_m

).

• For any n ≤ K

_Ω

, we denote by (µ

_h,n

, u

_h,n

) the n -th eigenpair of P

_h

.

• For any m ≤ M , we define the m -th cluster of eigenspaces of P

_h

by F

_h,m

= span

u

_h,n

for any n such that λ

_n

= Λ

_m

, and the corresponding cluster of quasi-modes, cf. (29)-(30),

E

_h,m

= span

ψ

_h,^s_,k

= χ

^s

ψ ˘

_h,^s_,k

for any s ∈ Σ, k ≥ 1 such that µ

_k

(α

^s

) = Λ

m

. A positive real number δ is attached to these spaces of quasi-modes: δ is such that for all s ∈ Σ, the cut-off function χ

^s

is equal to 1 on B( s , R

_m

− δ) .

Theorem 5.6. With Notation 5.1 and 5.5, for any ε > 0, there exists C

_ε

such that for any m ≤ M ,

d(E

_h,m

, F

_h,m

) ≤ C

_ε

exp

− 1

√ h

(R

_m

− δ) p

Θ

₀

− Λ

_m

− ε . Proof. For any m ≤ M , we define

Σ

^∗_m

=

( s , k) ∈ Σ × { 1, · · · , K

_Ω

} , µ

_k

(α

^s

) = Λ

_m

.

(i) If the set Σ

^∗_m

is reduced to one element, then Theorem 5.6 comes from Lemma 4.3.

(ii) We assume that Σ

^∗_m

is not reduced to one element. We denote by κ

^min_m

the smallest eigenvalue of the matrix

h ψ

_h,s,k

, ψ

_h,s′,k^′

i

((s,k),(s′,k^′))∈Σ^∗_m×Σ^∗_m

.

Let ( s , k) 6 = ( s

^′

, k

^′

) ∈ Σ

^∗_m

. If s = s

^′

, ψ ˘

_h,s,k

and ψ ˘

_h,s,k^′

are orthogonal, and thus Z

Ω

ψ

_h,^s_,k

ψ

_h,s,k^′

dx = Z

Ω

( | χ

^s

|

²

− 1) ˘ ψ

_h,^s_,k

ψ ˘

_h,s,k^′

dx.

Then, by Cauchy-Schwarz inequality and decay of eigenfunctions,

Z

Ω

ψ

_h,s,k

ψ

_h,s,k^′

dx ≤

Z

Ω\B(s,Rm−δ)

ψ ˘

_h,s,k

ψ ˘

_h,s,k^′

dx

≤ C

_ε

exp

− 1

√ h

2(R

_m

− δ) p

Θ

₀

− Λ

_m

− ε . If s 6 = s

^′

, since eigenfunction ψ ˘

_h,s,k

and ψ ˘

_h,s′,k^′

are localized near distinct corner, we have

Z

Ω

ψ

_h,^s_,k

ψ

_h,s′,k^′

dx ≤

Z

Ω

χ

^s

ψ ˘

_h,^s_,k

χ

^s′

ψ ˘

_h,^s′,k^′

dx

≤ Z

Ω\(B(s,Rm−δ)∪B(s′,Rm−δ))

ψ ˘

_h,s,k

ψ ˘

_h,s,k^′

dx

≤ C

_ε

exp

− 1

√ h

2(R

_m

− δ) p

Θ

₀

− Λ

_m

− ε . Using also (31), we deduce that

| κ

^min_m

− 1 | ≤ C

_ε

exp

− 1

√ h

2(R

_m

− δ) p

Θ

₀

− Λ

_m

− ε .

Let us now apply Theorem 5.4 with A = P

_h

. Relation (33) in Lemma 4.3 gives (45) with η = C

_ε

exp( − ((R

_m

− δ) √

Θ

₀

− Λ

_m

− ε)/ √

h) . Let us define I = [hΛ

_m

− η, hΛ

_m

+ η].

According to Theorem 5.2, there exists C > 0 such that for h small enough, σ(P

_h

) ∩ I + B(0, 2Ch) \ I

= ∅ . For example, we choose C = min n

Λm−Λ_m−1

4

,

^Λm+1₄^−Λm

o

with the convention Λ

₀

= 0 . Assumptions of Theorem 5.4 are filled and this ends the proof of Theorem 5.6.

Remark 5.7. 1. Theorem 5.6 shows that any eigenfunction of P

_h

associated with µ

_h,n

is exponentially close to a linear combination of the quasi-modes corresponding to

Λ

_m

= λ

_n

for the model operators. This result is particularly interesting when the

polygon Ω has several angles with the same opening. When the polygon presents

symmetries, these linear combinations are non trivial, as exhibited by numerical ex-

periments on a square, see [6].

2. Using Proposition 3.8, Theorems 5.2 and 5.6 can easily be generalized to the Schr¨odin- ger operator with constant magnetic field equal to B . The eigenvalues are multiplied by B and we use Proposition 3.8 to construct adapted quasi-modes.

3. Theorems 5.2 and 5.6 are still valid for a non convex polygon, even if the domain of the operator P

_h

is not contained in H

²

(Ω) any more (a finite number of singular functions have to be added to H

²

(Ω) if non convex angles are present). Moreover, we do not expect the first eigenfunctions of P

_h

to be localized in a non convex corner, since it is reasonable to conjecture that the bottom of the spectrum of Q

^α

is equal to Θ

₀

for openings α between π and 2π (cf. Remark 3.3).

6 Quasi-modes for the Schr¨odinger operator in a curvilinear polygon

Let Ω be a bounded curvilinear polygon with a piecewise smooth boundary. As previously, we denote by Σ the set of the vertices s of Ω, and by α

s

the opening of Ω at s . For the sake of simplicity (cf. Remark 5.7), we assume that α

s

∈ (0, π) for any s ∈ Σ.

Let B be a smooth positive magnetic field and let A be a potential associated with B , i.e.

B = curl A on Ω. As in Sections 4-5, we are interested in the behavior of the eigenpairs of the Neumann realization P

_h

on Ω, for the Schr¨odinger operator − (h ∇ − i A )

²

as h → 0 .

The associated sesquilinear form p

_h

is defined on H

¹

(Ω) by p

_h

(u, v) =

Z

Ω

(h ∇ − i A )u(x) · (h ∇ − i A )v(x) dx. (47) The operator P

_h

= − (h ∇ − i A )

²

is defined on its domain D (P

_h

) , with

D (P

_h

) =

u ∈ H

²

(Ω), ν · (h ∇ − i A )u

∂Ω

= 0 . (48)

Now, the values of the magnetic field B ( s ) at corners s play a key role in the eigenvalue asymptotics. We introduce:

Notation 6.1. • We denote by µ

_h,n

the n -th eigenvalue of P

_h

counted with multiplicity.

• We define infimum numbers for the magnetic field by b = inf

x∈Ω

B (x) and b

^′

= inf

x∈∂Ω

B (x).

• We denote by λ

_n

the n-th eigenvalue of the model operator M

s∈Σ

B ( s ) Q

^αs

,