Keywords. Mixed boundary conditions, asymptotics of eigenvalues, Aharonov–Bohm eigenval- ues.

Eigenvalue variation under moving mixed Dirichlet–Neumann boundary conditions and applications

L. Abatangelo ^∗ , V. Felli ^† , C. L´ ena ^‡ Revised version, March 27, 2019

Abstract

We deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet–Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion’s leading term. This allows inferring some remarkable consequences for Aharonov–Bohm eigenvalues when the singular part of the operator has two coalescing poles.

Keywords. Mixed boundary conditions, asymptotics of eigenvalues, Aharonov–Bohm eigenval- ues.

MSC classification. Primary: 35P20; Secondary: 35P15, 35J25.

1 Introduction and main results

The present paper deals with elliptic operators with varying mixed Dirichlet–Neumann boundary conditions and their spectral stability under varying of the Dirichlet and Neumann boundary regions. More precisely, we study the behaviour of eigenvalues under a homogeneous Neumann condition on a portion of the boundary concentrating at a point and a homogeneous Dirichlet boundary condition on the complement.

Let Ω be a bounded open set in R ² + := {(x ₁ , x ₂ ) ∈ R ² : x ₂ > 0} having the following properties:

Ω is Lipschitz, (1)

there exists ε 0 > 0 such that Γ ε

₀

:= [−ε 0 , ε 0 ] × {0} ⊂ ∂Ω. (2) We consider the eigenvalue problem for the Dirichlet Laplacian on the domain Ω

( −∆ u = λ u, in Ω,

u = 0, on ∂Ω. (3)

We denote by (λ j ) j≥1 the eigenvalues of Problem (3), arranged in non-decreasing order and counted with multiplicities.

For each ε ∈ (0, ε 0 ], we also consider the following eigenvalue problem with mixed boundary conditions:



 

 

−∆ u = λ u, in Ω, u = 0, on ∂Ω \ Γ ε ,

∂u

∂ν = 0, on Γ ε ,

(4)

∗

Dipartimento di Matematica e Applicazioni, Universit` a di Milano–Bicocca, Via Cozzi 55, 20125 Milano, Italy, laura.abatangelo@unimib.it

†

Dipartimento di Scienza dei Materiali, Universit` a di Milano–Bicocca, Via Cozzi 55, 20125 Milano, Italy, veronica.felli@unimib.it

‡

Matematiska institutionen, Stockholms universitet, 106 91 Stockholm, Sweden, corentin@math.su.se

Neumann

Dirichlet Dirichlet

−ε Γ ε ε

Ω

Figure 1: The mixed boundary condition problem in the domain Ω.

with Γ ε := [−ε, ε] × {0}, see Figure 1. We denote by (λ j (ε)) j≥1 the eigenvalues of Problem (4), arranged in non-decreasing order and counted with multiplicities.

A rigorous weak formulation of the eigenvalue problems described above can be given as follows.

For ε ∈ (0, ε 0 ], we define

Q ε = u ∈ H ¹ (Ω) : χ ∂Ω\Γ

_ε

γ 0 (u) = 0 in L ² (∂Ω) ,

where γ 0 is the trace operator from H ¹ (Ω) to L ² (∂Ω), which is a continuous linear mapping (see for instance [21, Definition 13.2]) and χ _∂Ω\Γ

_ε

is the indicator function of ∂Ω \ Γ _ε in ∂Ω. Furthermore, we define the quadratic form q on H ¹ (Ω) by

q(u) :=

Z

Ω

|∇u| ² dx. (5)

Let us denote by q 0 the restriction of q to H ₀ ¹ (Ω) and by q ε the restriction of q to Q ε . The sequences (λ j ) j≥1 and (λ j (ε)) j≥1 for ε ∈ (0, ε 0 ] can then be defined by the min-max principle:

λ j := min

E⊂H

₀¹

(Ω) subspace

dim (E)=j

max

u∈E

q(u)

kuk ² (6)

and

λ j (ε) := min

E⊂Q

ε

subspace

dim (E)=j

max u∈E

q(u)

kuk ² , (7)

where

kuk ² = Z

Ω

u ² (x) dx.

Since H ¹ (Ω) is compactly embedded in L ² (Ω) (see e.g. [21, Lemma 18.4]), the eigenvalues of q ₀ , defined by Equation (6), and those of q _ε , defined by Equation (7), are of finite multiplicity, and form sequences tending to +∞.

Remark 1.1. Let us fix ε 1 and ε 2 in (0, ε 0 ] such that ε 1 > ε 2 . Since H ₀ ¹ (Ω) ⊂ Q ε

₂

⊂ Q ε

₁

, the definitions given by Formulas (6) and (7) immediately imply that λ j (ε 1 ) ≤ λ j (ε 2 ) ≤ λ j for each integer j ≥ 1. The function (0, ε 0 ] 3 ε 7→ λ j (ε) is therefore non-increasing and bounded by λ j for each integer j ≥ 1.

Remark 1.2. For the sake of simplicity, in the present paper we assume that the domain Ω satisfies

assumption (2), i.e. that ∂Ω is straight in a neighborhood of 0. We observe that, since we are in

dimension 2, this assumption is not restrictive. Indeed, starting from a general sufficiently regular

domain Ω, a conformal transformation leads us to consider a new domain satisfying (2), see e.g.

[10]. The counterpart is the appearance of a conformal weight in the new eigenvalue problem;

however, if Ω is sufficiently regular, the weighted problem presents no additional difficulties.

The purpose of the present paper is to study the eigenvalue function ε 7→ λ j (ε) as ε → 0 ⁺ . The continuity of this map as well as some asymptotic expansions were obtained in [12] (see also Appendix C of the present paper for an alternative proof of some results of [12]). Here we mean to provide some explicit characterization of the leading terms in the expansion given in [12] and of the limit profiles arising from blowing-up of eigenfunctions.

Spectral stability and asymptotic expansion of the eigenvalue variation in a somehow comple- mentary setting were obtained in [3]; indeed, if we consider the eigenvalue problem under homo- geneous Dirichlet boundary conditions on a vanishing portion of a straight part of the boundary, Neumann conditions on the complement in the straight part and Dirichlet conditions elsewhere, by a reflection the problem becomes equivalent to the one studied in [3], i.e. a Dirichlet eigenvalue problem in a domain with a small segment removed.

Related spectral stability results were discussed in [8, Section 4] for the first eigenvalue under mixed Dirichlet-Neumann boundary conditions on a smooth bounded domain Ω ⊂ R ^N (N ≥ 3), both for vanishing Dirichlet boundary portion and for vanishing Neumann boundary portion.

We also mention that some regularity results for solutions to second-order elliptic problems with mixed Dirichlet–Neumann type boundary conditions were obtained in [13, 20], see also the references therein, whereas asymptotic expansions at Dirichlet-Neumann boundary junctions were derived in [10].

Let us assume that

λ N (i.e. the N -th eigenvalue of q 0 ) is simple. (8) Let u N be a normalized eigenfunction associated to λ N , i.e. u N satisfies



 

 

−∆u N = λ N u N , in Ω,

u N = 0, on ∂Ω,

R

Ω u ² _N (x) dx = 1.

(9)

It is known (see [12]) that, under assumption (8), the rate of the convergence λ ^ε _N → λ _N is strongly related to the order of vanishing of the Dirichlet eigenfunction u _N at 0. Moreover u _N has an integer order of vanishing k ≥ 1 at 0 ∈ ∂Ω and there exists β ∈ R \ {0} such that

r ^−k u N (r cos t, r sin t) → β sin(kt) as r → 0 in C ^1,τ ([0, π]) (10) for any τ ∈ (0, 1), see e.g. [9, Theorem 1.1]. Actually, one can see that β is directly linked to the norm of the k-th differential of u N at 0. More precisely, if we consider

kd ^j u(x)k ² :=

2

X

i

1

,...,i

j

=1

   

∂ ^j u

∂x _i

₁

. . . ∂x _i

_j

(x)    

2

,

then

β ² = kd ^k u N (0)k ² (k!) ² 2 ^k−1 .

This follows by differentiating the harmonic homogeneous functions βr ^k sin(kt) and βr ^k cos(kt) with respect to x ₁ and x ₂ and considering d ^k u _N (0).

Our main results provide sharp asymptotic estimates with explicit coefficients for the eigenvalue

variation λ _N − λ _N (ε) as ε → 0 ⁺ under assumption (8) (Theorem 1.3), as well as an explicit

representation in elliptic coordinates of the limit blow-up profile for the corresponding eigenfunction

u ^ε _N (Theorem 1.4).

Theorem 1.3. Let Ω be a bounded open set in R ² satisfying (1) and (2). Let N ≥ 1 be such that the N -th eigenvalue λ _N of q ₀ on Ω is simple with associated eigenfunctions having in 0 a zero of order k with k as in (10). For ε ∈ (0, ε ₀ ), let λ _N (ε) be the N -th eigenvalue of q _ε on Ω. Then

lim

ε→0

⁺

λ N − λ N (ε)

ε ^2k = β ² kπ 2 ^2k−1

 k − 1

 k−1 2



 2

with β 6= 0 being as in (9)–(10).

Theorem 1.4. Let Ω be a bounded open set in R ² satisfying (1) and (2). Let N ≥ 1 be such that the N -th eigenvalue λ N of q 0 on Ω is simple with associated eigenfunctions having in 0 a zero of order k with k as in (10). For ε ∈ [0, ε 0 ), let λ N (ε) be the N -th eigenvalue of q ε on Ω and u ^ε _N be an associated eigenfunction satisfying R

Ω |u ^ε _N | ² dx = 1 and R

Ω u ^ε _N u N dx ≥ 0. Then ε ^−k u ^ε _N (εx) → β(ψ _k + W _k ◦ F ⁻¹ ) as ε → 0 ⁺

in H _loc ¹ (R ² + ), a.e. and in C _loc ² (R ² + \ {(1, 0), (−1, 0)}), where β is as in (9)–(10),

ψ _k (r cos t, r sin t) = r ^k sin(kt), for t ∈ [0, π] and r > 0, (11) F (ξ, η) = (cosh(ξ) cos(η), sinh(ξ) sin(η)), for ξ ≥ 0, η ∈ [0, 2π), (12) and

W _k (ξ, η) = 1 2 ^k−1

b

^k−1₂

c X

j=0

k j



exp(−(k − 2j)ξ) sin((k − 2j)η). (13)

Actually, the fact that lim _ε→0

+

λ

_N

−λ

N

(ε)

ε

^2k

is finite and different from zero and the convergence of ε ^−k u ^ε _N (εx) to some nontrivial profile was proved in the paper [12] with a quite implicit description of the limits (see also Appendix C for an alternative proof). The original contribution of the present paper relies in the explicit characterization of the leading term of the expansion provided by [12] and in its applications to Aharonov–Bohm operators, see Section 2. The key tool allowing us to write explicitly the coefficients of the expansion consists in the use of elliptic coordinates, which turn out to be more suitable to our problem than radial ones, see Section 3.

2 Applications to Aharonov–Bohm operators

The present work is in part motivated by the study of Aharonov–Bohm eigenvalues. In this section we describe some applications of Theorem 1.3 to the problem of spectral stability of Aharonov–

Bohm operators with two moving poles, referring to Section 4 for the proofs.

Let us first review some definitions and known results. For any point a = (a 1 , a 2 ) ∈ R ² , we define the Aharonov–Bohm potential of circulation 1/2 by

A a (x) = 1 2

 −(x 2 − a 2 )

(x ₁ − a ₁ ) ² + (x ₂ − a ₂ ) ² , x 1 − a 1

(x ₁ − a ₁ ) ² + (x ₂ − a ₂ ) ²

 .

Let us consider an open and bounded open set b Ω with Lipschitz boundary, such that 0 ∈ b Ω. For better readability, we denote by H the complex Hilbert space of complex-valued functions L ² (b Ω, C), equipped with the scalar product defined, for all u, v ∈ H, by

hu, vi :=

Z

Ω b

uv dx.

We define, for a ∈ b Ω,

Q ^AB _a :=



u ∈ H ₀ ¹ Ω, C b
; |u|

|x − a| ∈ L ² Ω b



, (14)

the quadratic form q _a ^AB on Q ^AB _a by q _a ^AB (u) :=

Z

Ω b

|(i∇ + A a )u| ² dx,

and the sequence of eigenvalues λ ^AB _j (a)

j≥1 by the min-max principle λ ^AB _j (a) := min

E⊂Q

^AB_a

subspace

dim (E)=j

max

u∈E

q ^AB _a (u) kuk ² .

It follows from the definition in Equation (14) that Q ^AB _a is compactly embedded in H. The above eigenvalues are therefore of finite multiplicity and λ ^AB _j (a) → +∞ as j → +∞.

Remark 2.1. Let us note that, as shown in [16, Lemma 2.1], Q ^AB _a is the completion of the set of smooth functions supported in b Ω \ {a} for the norm k · k _a defined by

kuk ² _a = kuk ² + q _a ^AB (u).

Let us point out that functions in Q ^AB _a satisfy a Dirichlet boundary condition, which is not the case in [16]. However, this difference is unimportant for the compact embedding.

Remark 2.2. We could also consider the Friedrichs extension of the differential operator (i∇ + A a ) ^∗ (i∇ + A a )u = −∆u + 2iA a · ∇u + |A a | ² u

acting on functions u ∈ C _c ^∞ (b Ω \ {a}, C). As shown for instance in [15, Section I] or [7, Section 2]), this defines a positive and self-adjoint operator with compact resolvent, whose eigenvalues, counted with multiplicities, are λ ^AB _j (a)

j≥1 . It is called the Aharonov-Bohm operator of pole a and circulation 1/2.

In recent years, several authors have studied the dependence of eigenvalues on the position of the pole. It has been established in [7, Theorem 1.1], that the functions a 7→ λ ^AB _j (a) are continuous in Ω. In [1, 2], the two first authors obtained the precise rate of convergence λ ^AB _j (a) → λ ^AB _j (0) as a converges to the interior point 0 for simple eigenvalues. In order to state the most complete result, given in [2, Theorem 1.2], we consider an L ² -normalized eigenfunction u ⁰ _N of q ₀ ^AB associated with the eigenvalue λ ^AB _N (0). We additionally assume that λ ^AB _N (0) is simple. From [11, Section 7]

it follows that there exists an odd positive integer k and a non-zero complex number β 0 such that, up to a rotation of the coordinate axes,

r ⁻

^k2

u ⁰ _N (r cos t, r sin t) → β 0 e ⁱ

^2t

sin  k 2 t



in C ^1,τ ([0, 2π], C)

as r → 0 ⁺ , for all τ ∈ (0, 1). The integer k has a simple geometric interpretation: it is the number of nodal lines of the function u ⁰ _N which meet at 0. We say that u ⁰ _N has a zero of order k/2 in 0. Our coordinate axes are chosen in such a way that one of these nodal lines is tangent to the positive x ₁ semi-axis.

Theorem 2.3. Let us define a ε := ε(cos(α), sin(α)), with ε > 0. We have, as ε → 0 ⁺ ,

λ ^AB _N (a _ε ) = λ ^AB _N (0) − kπβ ₀ ² 2 ^2k−1

 k − 1

 _k−1

2



 ²

cos(kα)ε ^k + o ε ^k
.

Remark 2.4. The expansion in [1, 2] involves a constant depending on k, defined as the minimal

energy in a Dirichlet-type problem. We compute this constant in Appendix A in order to obtain

the more explicit result in Theorem 2.3.

Let us now consider, for any ε > 0, an Aharonov-Bohm potential with two poles (ε, 0) and (−ε, 0), of fluxes respectively 1/2 and −1/2:

A _ε := A _(ε,0) − A (−ε,0) . As in the case of one pole, we define the vector space Q ^AB _ε by

Q ^AB _ε :=



u ∈ H ₀ ¹ Ω, C b
; |u|

|x ± ε e| ∈ L ² Ω b



, (15)

where e = (1, 0), the quadratic form q ^AB _ε on Q ^AB _ε by q ^AB _ε (u) :=

Z

Ω b

|(i∇ + A ε )u| ² dx, (16)

and the sequence of eigenvalue λ ^AB _j (ε)

j≥1 by the min-max principle λ ^AB _j (ε) := min

E⊂Q

^AB_ε

subspace

dim (E)=j

max

u∈E

q ^AB _ε (u)

kuk ² . (17)

It follows from [15, Corollary 3.5] that, for any j ≥ 1, λ ^AB _j (ε) converges to the j-th eigenvalue of the Laplacian in b Ω as ε → 0 ⁺ . In [4, 3] the authors obtained in some cases a sharp rate of convergence. In order to state the result, let us introduce some notation. We denote by q b ₀ the quadratic form on H ₀ ¹ (b Ω) defined by Equation (5), replacing Ω with b Ω, and we denote by b λ j

j≥1

the sequence of eigenvalues defined by Equation (6), replacing Ω with b Ω and q with q b ₀ . We fix an integer N ≥ 1 and assume that b λ _N is a simple eigenvalue. We denote by u b _N an associated eigenfunction, normalized in L ² Ω
. b

Theorem 2.5. [4, Theorem 1.2] If u b N (0) 6= 0, we have, as ε → 0, λ ^AB _N (ε) = b λ N + 2π

| log(ε)| b u ² _N (0) + o

 1

| log(ε)|

 .

In the case b u _N (0) = 0, it is well known that there exist k ∈ N \ {0}, b β ∈ R \ {0} and α ∈ [0, π) such that

r ^−k u b _N (r cos t, r sin t) → b β sin (α − kt) in C ^1,τ ([0, 2π], C)

as r → 0 ⁺ for all τ ∈ (0, 1). In particular, there is a nodal line whose tangent makes the angle α/k with the positive x ₁ semi-axis. As above we can characterize b β as | b β| ² = ^kd _(k!)

^k

^{u b}

₂^N

₂ ^(0)k

_k−1²

.

Let us assume that

Ω is symmetric with respect to the x b 1 -axis.

Since b λ N is simple, b u N is either even or odd in the variable x 2 and α is either π/2 or 0 accordingly.

Theorem 2.6. [3, Theorem 1.16] If u b _N is even in x ₂ , which corresponds to α = π/2, we have, as ε → 0 ⁺ ,

λ ^AB _N (ε) = b λ N + kπ b β ² 4 ^k−1

 k − 1

 _k−1

2



 ²

ε ^2k + o ε ^2k
.

Remark 2.7. The statements in [3] contain a constant C _k which we put in a simpler form in Appendix A, in order to obtain Theorem 2.6.

As a corollary of Theorem 1.3, we prove in Section 4 the following result, which complements the previous theorem.

Theorem 2.8. If u b _N is odd in x ₂ , which corresponds to α = 0, we have, as ε → 0 ⁺ , λ ^AB _N (ε) = b λ N − kπ b β ²

4 ^k−1

 k − 1

 k−1 2



 2

ε ^2k + o ε ^2k
.

Remark 2.9. As discussed in Section 4, the assumption that b λ N is simple can be slightly relaxed,

admitting, in some cases, also double eigenvalues.

3 Sharp asymptotics for the eigenvalue variation

3.1 Related results from the literature

As already mentioned in the introduction, some asymptotic expansions for the eigenvalue variation λ N − λ N (ε) were derived in [12]. Let us first recall the results from [12] which are the starting of our analysis.

Let s := {(x ₁ , x ₂ ) ∈ R ² : x ₂ = 0 and x ₁ ≥ 1 or x 1 ≤ −1}. We denote as Q the completion of C _c ^∞ (R ² + \ s) under the norm ( R

R

²₊

|∇u| ² dx) ^1/2 . From the Hardy type inequality proved in [14] and a change of gauge, it follows that functions in Q satisfy the Hardy type inequalities

1 4

Z

R

²+

|ϕ(x)| ²

|x − e| ² dx ≤ Z

R

²+

|∇ϕ(x)| ² dx, for all ϕ ∈ Q, (18)

and

1 4

Z

R

²+

|ϕ(x)| ²

|x + e| ² dx ≤ Z

R

²+

|∇ϕ(x)| ² dx, for all ϕ ∈ Q, (19) where e = (1, 0). Inequalities (18) and (19) allow characterizing Q as the following concrete functional space:

Q = n

u ∈ L ¹ _loc (R ² + ) : ∇u ∈ L ² (R ² + ), _|x±e| ^u ∈ L ² (R ² + ), and u = 0 on s o .

We refer to the paper [12], where the following theorem can be found as a particular case of more general results.

Theorem 3.1 ([12]). Let Ω be a bounded open set in R ² satisfying (1) and (2). Let N ≥ 1 be such that the N -th eigenvalue λ _N of q ₀ on Ω is simple with associated eigenfunction u _N having in 0 a zero of order k with k as in (10). For ε ∈ (0, ε ₀ ), let λ _N (ε) be the N -th eigenvalue of q _ε on Ω and u ^ε _N be its associated eigenfunction, normalized to satisfy R

Ω |u ^ε _N | ² dx = 1 and R

Ω u ^ε _N u _N dx ≥ 0.

Then, as ε → 0 ⁺ ,

λ N − λ N (ε)

ε ^2k → −β ² Z 1

−1

∂w k

∂x ₂ w k dx 1 , (20)

ε ^−k u ^ε _N (εx) → β(ψ k + w k ) in H _loc ¹ (R ² + ), (21) with β 6= 0 being as in (9)–(10), ψ k being defined in (11), and w k being the unique Q-weak solution to the problem



 

 

−∆w _k = 0, in R ² + , w k = 0, on s,

∂w

_k

∂ν = − ^∂ψ _∂ν

^k

, on Γ ₁ .

(22)

Convergence (20) can be obtained combining [12, Equation (4.6)] for simple eigenvalues, [12, Equation (3.4)] together with [12, Lemma 3.3]. As well, (21) is given by [12, Equation (2.3)], which is a consequence of [12, Theorem 5.2], [12, Equation (4.10)], [12, Lemma 3.3]. For the sake of clarity and completeness, we present an alternative proof in Appendix C, which relies on energy estimates obtained by an Almgren type monotonicity argument and blow-up analysis.

We remark that in [12] the author describes the limit profile w k solving (22) with polar coordi- nates. On the contrary, our contribution relies essentially on the use of elliptic coordinates in place of polar ones. This allows us to compute explicitly the right hand side of (20), thus obtaining the following result.

Proposition 3.2. For any positive integer k, Z 1

−1

∂w k

∂x 2

w _k dx ₁ = − kπ 2 ^2k−1

 k − 1

 _k−1

2



 ²

.

The proof of Proposition 3.2 relies in an explicit construction of the limit profile w _k , using a parametrization of the upper half-plane R ² + by elliptic coordinates, a finite trigonometric expansion, and the simplification of a sum involving binomial coefficients.

3.2 Computation of the limit profile w k

Let us first compute w _k . By uniqueness, any function in the functional space Q that satisfies all the conditions of Problem (22) is equal to w _k . In order to find such a function, we use the elliptic coordinates (ξ, η) defined by

( x ₁ = cosh(ξ) cos(η),

x 2 = sinh(ξ) sin(η). (23)

More precisely, we consider the function F : (ξ, η) 7→ (x ₁ , x ₁ ) defined by the equations (23). It is a C ^∞ diffeomorphism from D := (0, +∞) × (0, π) to R ² + . We note that F is actually a conformal mapping. Indeed, if we define the complex variables z := x 1 + ix 2 and ζ := ξ + iη, we have z = cosh(ζ), which proves the claim since cosh is an entire function. Let us denote by h(ξ, η) the scale factor associated with F , expressed in elliptic coordinates. We have

h(ξ, η) = 

cosh ⁰ (ζ) 

 = |sinh(ζ)| = |sinh(ξ) cos(η) + i cosh(ξ) sin(η)| = q

cosh ² (ξ) − cos ² (η).

For any function u ∈ Q, let us define U := u ◦ F . From the fact the F is conformal, it follows that

|∇U | is in L ² (D) with

Z

D

|∇U | ² dξdη = Z

R

²+

|∇u| ² dx.

We also have

∂u

∂ν (x) = − 1 h(0, η)

∂U

∂ξ (0, η) (24)

for any x ∈ Γ 1 , where η ∈ (0, π) satisfies x = F (0, η) = (cos(η), 0). Furthermore, U is harmonic in D if, and only if, u is harmonic in R ² + .

We now give an explicit formula for w _k ◦ F .

Proposition 3.3. For any positive integer k, w _k ◦ F = W _k , where W _k is defined in (13).

Proof. Let us begin by computing the function Ψ k := ψ k ◦ F . We have ψ k (x) = Im z ^k
, so that Ψ _k (ξ, η) = Im (cosh(ζ)) ^k
, where the complex variables z and ζ are defined as above. Using the binomial theorem, we find

Ψ k (ξ, η) = Im



 1 2 ^k

k

X

j=0

 k j



e ^(k−2j)ζ



 = 1 2 ^k

k

X

j=0

 k j



e ^(k−2j)ξ sin ((k − 2j)η) .

This can be written

Ψ k (ξ, η) = 1 2 ^k−1

b

^k−12

c X

j=0

 k j



sinh ((k − 2j)ξ) sin ((k − 2j)η)

by grouping terms of the sum in pairs, starting from opposite extremities. In particular, for all η ∈ (0, π),

∂Ψ k

∂ξ (0, η) = 1 2 ^k−1

b

^k−1₂

c X

j=0

(k − 2j)

 k j



sin ((k − 2j)η) .

We now define

V (ξ, η) = 1 2 ^k−1

b

^k−1₂

c X

j=0

 k j



e ^{−(k−2j)ξ} sin ((k − 2j)η) .

The function |∇V | is in L ² (D) and, for all η ∈ (0, π),

∂V

∂ξ (0, η) = − 1 2 ^k−1

b

^k−1₂

c X

j=0

(k − 2j)

 k j



sin ((k − 2j)η) .

Additionally, V vanishes on half-lines defined by η = 0 and η = π, which are the lower and upper boundary of D, respectively, and are mapped to R × {0} \ Γ ¹ by F . It can be checked directly that V ◦ F ⁻¹ ∈ Q. Finally, V is harmonic in D, since it is a linear combination of functions of the type (ξ, η) 7→ e ^±nξ e ^±inη , which are harmonic. We conclude that V ◦ F ⁻¹ is a solution of Problem (22), and therefore V = w _k ◦ F by uniqueness.

Proof of Theorem 1.4. Theorem 1.4 follows combining Theorem 3.1 and Proposition 3.3.

Corollary 3.4. For any positive integer k ≥ 1, Z 1

−1

∂w k

∂x ₂ w _k dx ₂ = − π 2 ^2k−1

b

^k−1₂

c X

j=0

(k − 2j)

 k j

 ²

. (25)

Proof. Using (13), a direct computation gives

∇W k (ξ, η) = 1 2 ^k−1

b

^k−1₂

c X

j=0

(k − 2j)

 k j



e ^{−(k−2j)ξ} (− sin ((k − 2j)η) , cos ((k − 2j)η) .

Recalling (24), we perform a standard change of variables in the left-hand side of (25) to elliptic coordinates and this yields the thesis.

3.3 Simplification of the sum

We now prove the following result.

Lemma 3.5. For every integer k ≥ 1, b

^k−12

c

X

j=0

(k − 2j)

 k j

 ²

= k

 k − 1

 k−1 2



 ² .

Proof. We will use repeatedly the two following properties of binomial coefficients. First, the Vandermonde identity: for any non-negative integers m, n and r,

r

X

j=0

 m j

  n r − j



=

 m + n r



; (26)

and second, the elementary identity

 n r



= n r

 n − 1 r − 1



(27) with n and r positive integers.

Let us now fix an integer k ≥ 1. To simplify the notation, we write s :=  k − 1

2



and S :=

s

X

j=0

(k − 2j)

 k j

 ² .

Next, we remark that

S = S 0 − 2 k S 1 − 2

k S 2 ,

with

S 0 :=

s

X

j=0

k

 k j

 2

, S 1 :=

s

X

j=0

j(k − j)

 k j

 2

, S 2 :=

s

X

j=0

j ²

 k j

 2

.

Let us compute the previous sums when k = 2p + 1, with p a non-negative integer. We first have S ₀

k = 1 2

k

X

j=0

 k j

 2

= 1 2

 2k k

 ,

where the last equality is a special case of identity (26). We then find

S ₁ = 1 2

k

X

j=0

j

 k j

 (k − j)

 k

k − j



= k ² 2

k−1

X

j=1

 k − 1 j − 1

  k − 1 k − j − 1



= k ² 2

k−2

X

`=0

 k − 1

`

  k − 1 k − 2 − `



= k ² 2

 2k − 2 k − 2



by applying Identity (27) followed by (26). Finally, Identity (27) implies

S 2 = k ²

p

X

j=1

 k − 1 j − 1

 ²

= k ²

p−1

X

`=0

 k − 1

`

 ²

= k ² 2

k−1

X

`=0

 k − 1

`

 2

−

 k − 1 p

 2 !

= k ² 2

 2k − 2 k − 1



− k ² 2

 k − 1 p

 2

.

We obtain

S = k 2

 2k k



− k

 2k − 2 k − 2



− k

 2k − 2 k − 1

 + k

 k − 1 p

 ²

= k 2

 2k k



− k

 2k − 1 k − 1

 + k

 k − 1 p

 ²

= k

 k − 1 p

 ²

where the second equality follows from Pascal’s identity and the third from Identity (27).

Let us now treat the case k = 2p, with p a positive integer. In a similar way as before, we find

S 0 = k 2





k

X

j=0

 k j

 2

−

 k p

 2 

 = k 2

 2k k



− k 2

 k p

 2

,

S 1 = 1 2





k

X

j=0

j

 k j

 (k − j)

 k

k − j



− p ²

 k p

 2 

 = k ² 2

 2k − 2 k − 2



− k ² 8

 k p

 2

and

S 2 = k ²

p−2

X

j=0

 k − 1 j

 ²

= k ² 2





k−1

X

j=0

 k − 1 j

 ²

− 2

 k − 1 p − 1

 ²



 = k ² 2

 2k − 2 k − 1



−k ²

 k − 1 p − 1

 ² .

We finally obtain, after simplifications,

S = k

 k − 1 p − 1

 2

.

This completes the proof of Lemma 3.5.

−ε ε

Figure 2: The domain considered for Aharonov–Bohm eigenvalues with collapsing symmetric poles.

3.4 Conclusions

By the results from the preceding subsections, we can now prove Proposition 3.2 and Theorem 1.3.

Proof of Proposition 3.2. It follows from Corollary 3.4 and Lemma 3.5.

Combining the above results, we can now prove our main theorem.

Proof of Theorem 1.3. Theorem 1.3 follows from the combination of Theorem 3.1 and Proposition 3.2.

4 Asymptotic estimates for Aharonov–Bohm eigenvalues

4.1 Symmetry for the Aharonov–Bohm operator

As in Section 2, we assume b Ω ⊂ R ² to be a bounded open set with a Lipschitz boundary, such that 0 ∈ b Ω. We additionally assume that b Ω is symmetric with respect to the x 1 -axis and that Ω := b Ω ∩ R ² + also has a Lipschitz boundary.

According to [18, Theorem VIII.15], there exists a unique Friedrichs extension H ε of the quadratic form q ^AB _ε , that is to say a self-adjoint operator whose domain D(H ε ) is contained in Q ^AB _ε and which satisfies

hH _ε u, vi = q _ε ^AB (u, v) = Z

Ω b

(i∇ + A _ε )u · (i∇ + A _ε )v dx for all u, v ∈ D(H _ε ),

where we are denoting by q _ε ^AB both the quadratic form defined in (16) and the associated bilinear form (see Figure 2). We recall in this section the results proved in [3] concerning the properties of H ε , in particular the effect of the symmetry of the domain on its spectrum. Since most of the proofs in the present section reduce to a series of standard verifications, we generally only give an indication of them. We use gauge functions Φ ε , for ε ∈ (0, ε 0 ], whose existence is guaranteed by the following result. In the sequel the denote as σ the reflection through the x 1 -axis, i.e.

σ(x 1 , x 2 ) = (x 1 , −x 2 ).

Lemma 4.1. For each ε > 0, there exists a function Φ ε in C ^∞ R ² \ Γ ε
satisfying

(i) Φ ε ◦ σ = Φ ε in R ² \ Γ ε ;

(ii) |Φ _ε | = 1 in R ² \ Γ ε ;

(iii) (i∇ + A ε ) Φ ε = 0 in R ² \ Γ ε ;

(iv) Φ ε = 1 on (R × {0}) \ Γ ^ε and lim _δ→0

+

Φ ε (t, ±δ) = ±i for every t ∈ (−ε, ε).

We define the anti-unitary operators K ε and Σ ^c by K ε u := Φ ² _ε u and Σ ^c u := u ◦ σ. The subspace D(H ε ) ⊂ H is preserved by K ε and Σ ^c . The operators K ε , Σ ^c and H ε mutually commute. In particular, we can define the following subsets

H K,ε := {u ∈ H : K ε u = u};

D(H _K,ε ) := {u ∈ D(H _ε ) : K _ε u = u}.

The scalar product h· , ·i gives H _K,ε the structure of a real Hilbert space. As suggested by the notation, we define H _K,ε as the restriction of H _ε to D(H _K,ε ). It is a positive self-adjoint operator on H _K,ε of domain D(H _K,ε ), with compact resolvent. It has the same eigenvalues as H _ε , with the same multiplicities. The fact that K and Σ ^c commute ensures that H K,ε and D(H K,ε ) are Σ ^c -invariant. We can therefore define

H _K,ε ^s := {u ∈ H K,ε : Σ ^c u = u};

D(H _K,ε ^s ) := {u ∈ D(H _K,ε ) : Σ ^c u = u};

H _K,ε ^a := {u ∈ H K,ε : Σ ^c u = −u};

D(H _K,ε ^a ) := {u ∈ D(H K,ε ) : Σ ^c u = −u}.

We have the following orthogonal decomposition of H K,ε into spaces of symmetric and antisym- metric functions:

H K,ε = H _K,ε ^s ⊕ H _K,ε ^a . (28)

We also define H _K,ε ^s and H _K,ε ^a as the restrictions of H _K,ε to D(H _K,ε ^s ) and D(H _K,ε ^a ) respectively.

The operator H _K,ε ^s is positive and self-adjoint on H ^s _K,ε of domain D(H _K,ε ^s ) , with compact resolvent.

Similar conclusions hold for H ^a _K,ε . Decomposition (28) implies the following result.

Lemma 4.2. The spectrum of H K,ε is the union of the spectra of H _K,ε ^s and H _K,ε ^a , counted with multiplicities.

Remark 4.3. Let us note that we can give an alternative description of the spectra of H _K,ε ^s and H _K,ε ^a . One can check that they are the spectra of the quadratic form q ^AB _ε restricted to Q ^AB _ε ∩ H ^s _K,ε and Q ^AB _ε ∩ H ^a _K,ε respectively. These spectra can therefore be obtained by the min-max principle.

4.2 Isospectrality

In this subsection, we establish an isospectrality result between Aharonov-Bohm eigenvalue prob- lems with symmetry and Laplacian eigenvalue problems with mixed boundary conditions, in the spirit of [6].

To this aim, we define an additional family of eigenvalue problems, similar to Problems (3) and (4). With the notation ∂Ω ₊ := ∂Ω ∩ R ² + and ∂Ω ₀ := ∂Ω ∩ (R × {0}), we consider the eigenvalue problem



 

 

−∆ u = λ u, in Ω, u = 0, on ∂Ω ₊ ,

∂u

∂ν = 0, on ∂Ω 0 .

(29)

We denote by (µ j ) j≥1 the eigenvalues of Problem (29). We also consider, for each ε ∈ (0, ε 0 ],



 

 

−∆ u = λ u, in Ω,

u = 0, on ∂Ω + ∪ Γ ε ,

∂u

∂ν = 0, on ∂Ω 0 \ Γ ε ,

(30)

and denote by (µ _j (ε)) _j≥1 the corresponding eigenvalues. In order to give a rigorous definitions, we use a weak formulation. We define

R 0 = u ∈ H ¹ (Ω) ; χ ∂Ω

₊

γ 0 u = 0 in L ² (∂Ω) , and, for ε ∈ (0, ε 0 ],

R ε = u ∈ H ¹ (Ω) ; χ ∂Ω

₊

∪Γ

ε

γ 0 u = 0 in L ² (∂Ω) .

We denote by r 0 and r ε the restriction of the quadratic form q, defined in Equation (5), to R 0

and R _ε respectively. We then define (µ _j ) _j≥1 and (µ _j (ε)) _j≥1 as, respectively, the eigenvalues of the quadratic forms r ₀ and r _ε ; they are obtained by the min-max principle.

Remark 4.4. We can give another interpretation of the eigenvalues (µ j ) j≥1 and (λ j ) j≥1 . Using the unitary operator Σ : u 7→ u ◦ σ, we obtain a orthogonal decomposition of L ² Ω
into symmetric b and antisymmetric functions:

L ² Ω
= ker (I − Σ) ⊕ ker (I + Σ) . b (31) This decomposition is preserved by the action of the Dirichlet Laplacian − b ∆, and we can therefore define −∆ ^s (resp. −∆ ^a ) as the restriction of − b ∆ to symmetric (resp. antisymmetric) functions in the domain of − b ∆. One can then check that (µ j ) j≥1 is the spectrum of −∆ ^s and (λ j ) j≥1 is the spectrum of −∆ ^a .

It remains to connect the eigenvalues of Problems (30) and (4) to the eigenvalues of H ε . To this end, we define the following linear operator, which performs a gauge transformation:

U ε : H → L ² (Ω, C)

u 7→ √

2 Φ ε u _|Ω .

We recall that L ² (Ω) denotes the real Hilbert space of real-valued L ² functions in Ω. We have the following result.

Lemma 4.5. The operator U ε satisfies the following properties:

(i) U _ε (H _K,ε ) ⊂ L ² (Ω) and U _ε Q ^AB _ε
⊂ H ¹ (Ω, C);

(ii) U _ε induces a real-unitary bijective map from Q ^AB _ε ∩ H _K,ε ^s to R _ε such that q ^AB _ε (u) = q (U _ε u) for all u ∈ Q ^AB _ε ∩ H ^s _K,ε ;

(iii) U ε induces a real-unitary bijective map from Q ^AB _ε ∩ H ^a _K,ε to Q ε such that q ^AB _ε (u) = q (U ε u) for all u ∈ Q ^AB _ε ∩ H ^a _K,ε .

Proof. If u ∈ H _K,ε , then u = Φ ² _ε u, so that Φ _ε u = Φ _ε u, that is to say Φ _ε u is real-valued. This proves the first half of (i). For the second half, let us assume that u ∈ Q ^AB _ε . Using the definition of Q ^AB _ε , given in Equation (15), and Property (iii) of Lemma 4.1, we find the following identity, in the sense of distributions in Ω:

∇ Φ ε u
= Φ ε ∇u + ∇ Φ ε
u = Φ ε (∇ − iA ε ) u in Ω.

This proves that Φ ε u _|Ω ∈ H ¹ (Ω, C) and that Z

Ω

|(∇ − iA ε ) u| ² dx = Z

Ω

 ∇ Φ ε u
 

2 dx.

Let us now additionally assume that u ∈ Q ^AB _ε ∩ H ^s _K,ε . Since Σ ^c u = u, Property (i) of Lemma 4.1 implies that (Φ ε u) ◦ σ = Φ ε u. Therefore,

Z

Ω b

|u| ² dx = 2 Z

Ω

 Φ ε u  

2 dx = Z

Ω

|U ε u| ² dx.

Furthermore, Property (iv) of Lemma 4.1 and the equation Σ ^c u = u imply that u vanishes on Γ _ε , hence U _ε u ∈ R _ε . This implies that Φ _ε u ∈ H ¹ (b Ω) and

Z

Ω b

|(∇ − iA ε ) u| ² dx = Z

b Ω

 ∇ Φ ε u
 

2 dx = 2 Z

Ω

 ∇ Φ ε u
 

2 dx = Z

Ω

|∇ (U ε u)| ² dx.

We conclude that the mapping U _ε : Q ^AB _ε ∩ H ^s _K,ε → R _ε is well-defined, real-unitary, and that q _ε ^AB (u) = q (U ε u). To show that the mapping is bijective, we consider the operator V ε defined in the following way: given v ∈ L ² (Ω), we denote by e v its extension by symmetry to b Ω and we set

V _ε v := 1

√ 2 Φ _ε e v.

It can be checked, in a way similar to what has been done for U ε , that V ε induces the inverse of U ε , from R ε to Q ^AB _ε ∩ H _K,ε ^s . This proves (ii). The proof of (iii) is similar, the difference being that we must check that Φ ε u vanishes on (R × {0}) \ Γ ^ε when u ∈ Q ^AB _ε ∩ H ^a _K,ε .

Corollary 4.6. The spectra of H _K,ε ^s and H _K,ε ^a are (µ j (ε)) j≥1 and (λ j (ε)) j≥1 respectively.

4.3 Eigenvalues variations

Let us first state some auxiliary results, which we prove in Appendix B.

Proposition 4.7. For all N ∈ N ^∗ , µ N (ε) → µ N as ε → 0.

Proposition 4.8. Let µ N be a simple eigenvalue of −∆ ^s (see Remark 4.4) and u N be an associated eigenfunction, normalized in L ² Ω
. If u b N (0) 6= 0, then

µ N (ε) = µ N + 2π

| log(ε)| u ² _N (0) + o

 1

| log(ε)|



as ε → 0.

If

r ^−k u N (r cos t, r sin t) → b β cos (kt) in C ^1,τ ([0, π], R) as r → 0 ⁺ for all τ ∈ (0, 1), with k ∈ N ^∗ and b β ∈ R \ {0}, then

µ N (ε) = µ N + kπ b β ² 4 ^k−1

 k − 1

 k−1 2



 ²

ε ^2k + o ε ^2k

as ε → 0.

We now prove Theorem 2.8. Since u b N is odd in x 2 , b λ N belongs to the spectrum of −∆ ^a . Since b λ N is simple, it does not belong to the spectrum of −∆ ^s , according to the orthogonal decomposition (31). It follows from Remark 4.4 that there exists K ∈ N ^∗ such that b λ N = λ K and that λ K is a simple eigenvalue of q 0 in Ω. By continuity, λ K (ε) → λ K as ε → 0 ⁺ .

From Corollary 4.6, Proposition 4.7 and the fact that b λ _N is simple, it follows that there exists ε ₁ > 0 such that λ ^AB _N (ε) = λ _K (ε) for every ε ∈ (0, ε ₁ ). The conclusion of Theorem 2.8 follows from Theorem 1.3, using the fact that λ _K is simple. Let us note that the eigenfunction u b _N in Theorem 2.8 is normalized in L ² Ω
, while the eigenfunction u b N in Theorem 1.3 is normalized in L ² Ω
.

We therefore have to apply Theorem 1.3 with β = √

2 b β to obtain the correct result.

We can use the results of the preceding sections to study some multiple eigenvalues. Let b λ N

be an eigenvalue of −∆ on b Ω, possibly multiple. We define N 0 := min n

M ∈ N ^∗ ; b λ M = b λ N

o

and N 1 := max n

M ∈ N ^∗ ; b λ M = b λ N

o .

According to Remark 4.4, there exists K ∈ N ^∗ such that b λ N = λ K or there exists L ∈ N ^∗ such

that b λ N = µ L .

Proposition 4.9. Let us assume that b λ _N = λ _K with K ∈ N ^∗ and that λ _K is a simple eigenvalue of q ₀ . Let us denote by u _K an associated normalized eigenfunction for q ₀ , and let us assume that

r ^−k u _K (r cos t, r sin t) → β sin (kt) in C ^1,τ ([0, π], R) as r → 0 ⁺ for all τ ∈ (0, 1), with k ∈ N ^∗ and β ∈ R \ {0}. Then

λ ^AB _N

₀

(ε) = b λ N − kπβ ² 2 ^2k−1

 k − 1

 k−1 2



 ²

ε ^2k + o ε ^2k

as ε → 0.

Proof. Let us set m := N 1 − N 0 + 1, the multiplicity of b λ N . If m = 1, the conclusion follows from Theorem 2.8. We therefore assume m ≥ 2 in the rest of the proof. Remark 4.4 and the fact that λ K is simple imply that there exists L ∈ N ^∗ such that µ L = µ L+1 = · · · = µ L+m−2 = b λ N . From Proposition 4.7, we deduce that there exists ε 1 > 0 such that, for every ε ∈ (0, ε 1 ),

λ ^AB _N

₀

(ε); λ ^AB _N

₀

₊₁ (ε), . . . , λ ^AB _N

₁

(ε) = {λ K (ε), µ L (ε), . . . , µ L+m−2 (ε)} .

The function ε 7→ λ K (ε) is non-increasing, and the function ε 7→ µ j (ε) is non-decreasing for every j ∈ {L, . . . , L + m − 2}, therefore µ _j (ε) ≥ µ _j = b λ _N = λ _K ≥ λ _K (ε). In particular λ ^AB _N

0

(ε) = λ _K (ε) for every ε ∈ (0, ε ₁ ). The conclusion follows from Theorem 1.3.

Proposition 4.10. Let us assume that b λ _N = µ _L with L ∈ N ^∗ and that µ _L is a simple eigenvalue of

−∆ ^s . Let us denote by u _L an associated eigenfunction for −∆ ^s , normalized in L ² (b Ω). If u _L (0) 6= 0, then

λ ^AB _N

1

(ε) = b λ _N + 2π

| log(ε)| u ² _L (0) + o

 1

| log(ε)|



as ε → 0.

If

r ^−k u L (r cos t, r sin t) → b β cos (kt) in C ^1,τ ([0, π], R) as r → 0 ⁺ for all τ ∈ (0, 1), with k ∈ N ^∗ and b β ∈ R \ {0}, then

λ ^AB _N

₁

(ε) = b λ N + kπ b β ² 4 ^k−1

 k − 1

 k−1 2



 ²

ε ^2k + o ε ^2k

as ε → 0.

Proof. In a similar way as in the proof of Proposition 4.9, we show that there exists ε 1 > 0 such that, for every ε ∈ (0, ε 1 ), λ ^AB _N

₁

(ε) = µ L (ε). The conclusion then follows from Proposition 4.8.

4.4 Example: the square

As an application of the preceding results, let us study the first four eigenvalues of the Dirichlet Laplacian for the square

Ω := b 

− π 2 , π

2

 ²

. (32)

The open set b Ω is symmetric with respect to the x 1 -axis. We define Ω := b Ω ∩ R ² + . We denote by (b λ j ) j≥1 the eigenvalues of the Dirichlet Laplacian on the square b Ω and, for ε ∈ (0, π/2), we consider the Aharonov-Bohm eigenvalues λ ^AB _j (ε)

j≥1 defined in Section 2.

It is well known that the eigenvalues of the Dirichlet Laplacian on b Ω are b λ m,n := m ² + n ² ,

with m and n positive integers, and that an associated orthonormal family of eigenfunctions is given by

u m,n (x 1 , x 2 ) = 2

π f m (x 1 )f n (x 2 ), where

f _k (x) =

( sin(kx), if k is even,

cos(kx), if k is odd.

Proposition 4.11. Let us assume that b λ _N is simple. Then b λ _N = b λ _m,m = 2m ² for some positive integer m, and b λ N cannot be written in any other way as a sum of squares of positive integers.

Then we have, as ε → 0 ⁺ ,

λ ^AB _N (ε) = b λ N + 8

π| log(ε)| + o

 1

| log(ε)|



if m is odd and

λ ^AB _N (ε) = b λ N − m ⁴

2π ε ⁴ + o ε ⁴
if m is even.

Proof. In the case where m is odd, an associated eigenfunction, normalized in L ² (b Ω), is u m,m (x 1 , x 2 ) = 2

π cos(mx 1 ) cos(mx 2 ).

The first asymptotic expansion then follows from Theorem 2.5.

In the case where m is even, an associated eigenfunction, normalized in L ² (b Ω), is u _m,m (x ₁ , x ₂ ) = 2

π sin(mx ₁ ) sin(mx ₂ ).

Then b λ N = λ K , where λ K is a simple eigenvalue of q 0 . Furthermore, r ⁻² u _m,m (r cos t, r sin t) → m ²

π sin (2t) in C ^1,τ ([0, π], R)

as r → 0 ⁺ for all τ ∈ (0, 1). An application of Proposition 4.9, taking care of normalizing in L ² (Ω), gives the second asymptotic expansion.

Proposition 4.12. Assume that b λ N = b λ m,n = m ² + n ² with m even and n odd, and that b λ N has no other representation as a sum of two squares of positive integers, up to the exchange of m and n. Then b λ N has multiplicity two; up to replacing N with N − 1, we can assume that b λ N = b λ N +1 . Then, as ε → 0 ⁺ ,

λ ^AB _N (ε) = b λ _N − 4m ²

π ε ² + o ε ²
; λ ^AB _{N +1} (ε) = b λ N + 4m ²

π ε ² + o ε ²
. Proof. The associated eigenfunctions

u m,n (x 1 , x 2 ) = 2

π sin(mx 1 ) cos(nx 2 ) and

u n,m (x 1 , x 2 ) = 2

π cos(nx 1 ) sin(mx 2 )

are normalized in L ² (b Ω) and respectively symmetric and antisymmetric in the variable x ₂ . It follows that b λ N = µ L = λ K , where µ L is a simple eigenvalue of r 0 and λ K a simple eigenvalue of q 0 . Furthermore,

r ⁻¹ u _m,n (r cos t, r sin t) → 2m

π cos (t) in C ^1,τ ([0, π], R) and

r ⁻¹ u _n,m (r cos t, r sin t) → 2m

π sin (t) in C ^1,τ ([0, π], R)

as r → 0 ⁺ for all τ ∈ (0, 1). The asymptotic expansions then follow from Propositions 4.10 and

4.9

Remark 4.13. We note that if b λ _N is even, in any representation b λ _N = m ² +n ² , m and n have the same parity. Therefore, if n 6= m, b λ _N cannot be a simple eigenvalue either of r ₀ or of q ₀ . On the other hand, if b λ _N is odd, in any representation b λ _N = m ² + n ² , m and n have the opposite parity.

Therefore, as soon as b λ _N can be written in at least two different ways as the sum of two squares, b λ N cannot be a simple eigenvalue either of r 0 or of q 0 . The cases described in Propositions 4.11 and 4.12 are thus the only ones in which we can apply the results of Section 4.3 for the square.

The first four eigenvalues of the Dirichlet Laplacian on the square b Ω satisfy the assumptions of either Proposition 4.11 or Proposition 4.12, so we can apply the previous results to derive the following asymptotic expansions of the Aharonov-Bohm eigenvalues λ ^AB _j (ε) for j = 1, 2, 3, 4.

Corollary 4.14. Let λ ^AB _j (ε) be the Aharonov-Bohm eigenvalues defined in (16)–(17) with b Ω being the square defined in (32). Then we have, as ε → 0 ⁺ ,

λ ^AB ₁ (ε) = 2 + 8 π

1

| log(ε)| + o

 1

| log(ε)|



; λ ^AB ₂ (ε) = 5 − 16

π ε ² + o ε ²
; λ ^AB ₃ (ε) = 5 + 16

π ε ² + o ε ²
; λ ^AB ₄ (ε) = 8 − 8

π ε ⁴ + o ε ⁴
.

4.5 Example: the disk

Let (r, t) ∈ [0, 1] × [0, 2π) be the polar coordinates of the disk. It is well known that the eigenvalues of the Dirichlet Laplacian on the disk are given by the sequences

{j _0,k ² } k≥1 ∪ {j _n,k ² } n,k≥1 ,

where j n,k denotes the k-th zero of the Bessel function J n for n ≥ 0, k ≥ 1. We recall that j n,k = j n

⁰

,k

⁰

if, and only if, n = n ⁰ and k = k ⁰ (see [22, Section 15.28]). The first set is therefore made of simple eigenvalues; their eigenfunctions are given by the Bessel functions

u _0,k (r cos t, r sin t) :=

q 1 π

1

|J

₀⁰

(j

_0,k

)| J ₀ (j _0,k r) for k ≥ 1. (33) The second set is made of double eigenvalues whose eigenfunctions are spanned by

u ^s _n,k (r cos t, r sin t) :=

q 2 π

1

|J

_n⁰

(j

_n,k

)| J n (j n,k r) cos nt, (34) u ^a _n,k (r cos t, r sin t) :=

q 2 π

1

|J

_n⁰

(j

n,k

)| J _n (j _n,k r) sin nt, (35) for n, k ≥ 1. We stress that these eigenfunctions have L ² -norm equal to 1 on the disk. It is convenient to recall (see [22, Chapter III]) that for any n ∈ N ∪ {0}

J n (z) =

+∞

X

k=0

(−1) ^k ( ¹ ₂ z) ^n+2k

k! Γ(n + k + 1) . (36)

We denote by b λ j

j≥1 the eigenvalues of the Dirichlet Laplacian on the disk D 1 = {(x 1 , x 2 ) ∈ R ² : x ² ₁ + x ² ₂ < 1}

and, for ε ∈ (0, 1/2), we consider the Aharonov-Bohm eigenvalues λ ^AB _j (ε)

j≥1 defined in Section 2.

Proposition 4.15. If b λ _N is simple, there exists an integer k ≥ 1 such that b λ _N = j _0,k ² . Then

λ ^AB _N (ε) = j _0,k ² + 2

|J ₀ ⁰ (j 0,k )| ² 1

| log(ε)| + o

 1

| log(ε)|



(37)

as ε → 0 ⁺ . If b λ _N is double, there exist integers n ≥ 1 and k ≥ 1 such that b λ _N = j _n,k ² . Up to replacing N by N − 1, we can assume that b λ N = b λ N +1 . Then, as ε → 0 ⁺ ,

λ ^AB _N (ε) = j _n,k ² − 2nj _n,k ²ⁿ (n!) ² 4 ²ⁿ⁻¹ |J _n ⁰ (j n,k )| ²

 n − 1

 _n−1

2



 2

ε ²ⁿ + o ε ²ⁿ
, (38)

λ ^AB _{N +1} (ε) = j _n,k ² + 2nj _n,k ²ⁿ (n!) ² 4 ²ⁿ⁻¹ |J _n ⁰ (j _n,k )| ²

 n − 1

 n−1 2



 ²

ε ²ⁿ + o ε ²ⁿ
. (39)

Proof. We first consider the case where the eigenvalue b λ N = j _0,k ² is simple; then an associated eigenfunction, normalized in the disk, is u 0,k defined by Equation (33). It follows from Equation (36) that

u _0,k (0) = r 1

π 1

|J ₀ ⁰ (j 0,k )| > 0.

Theorem 2.5 gives us the asymptotic expansion (37).

We then consider the case where b λ N is double, with b λ N = b λ N +1 = j _n,k ² , n, k ≥ 1. We note that j _n,k ² is a simple eigenvalue of q 0 , and that the restriction of √

2u ^a _n,k to the upper half-disk is an associated normalized eigenfunction. It follows from Equation (36) that

r ⁻ⁿ u ^a _n,k (r cos t, r sin t) → r 2

π 1

|J _n ⁰ (j n,k )|

1 Γ(n + 1)

 j _n,k 2

 n

sin nt in C ^1,τ ([0, π], R)

as r → 0 ⁺ . The asymptotic expansion (38) then follows from Proposition 4.9. In a similar way, j _n,k ² is a simple eigenvalue of −∆ ^s , and u ^s _n,k is an associated normalized eigenfunction. It follows from Equation (36) that

r ⁻ⁿ u ^s _n,k (r cos t, r sin t) → r 2

π 1

|J _n ⁰ (j n,k )|

1 Γ(n + 1)

 j _n,k 2

 ⁿ

cos nt in C ^1,τ ([0, π], R)

as r → 0 ⁺ . The asymptotic expansion (39) then follows from the second case of Proposition 4.10.

Additionally, there exist relations between the zeros of Bessel functions (to this aim we refer to [22, Chapter XV.22]): in particular, the positive zeros of the Bessel function J n are interlaced with those of the Bessel function J n+1 and by Porter’s Theorem there is an odd number of zeros of J n+2 between two consecutive zeros of J n . Then, we have,

0 < j 0,1 < j 1,1 < j 2,1 < j 0,2 < j 1,2 < . . .

and hence, since j 3,1 > j 2,1 , the first three zeros of Bessel functions are, in order, 0 < j _0,1 < j _1,1 < j _2,1 .

Combining this information with Proposition 4.15, we find for example the following asymptotic

expansions for the first few Aharonov-Bohm eigenvalues λ ^AB _j (ε) on the disk D ₁ as ε → 0 ⁺ :

λ ^AB ₁ (ε) = j _0,1 ² + 2

|J ₀ ⁰ (j 0,1 )| ² 1

| log(ε)| + o

 1

| log(ε)|

 ,

λ ^AB ₂ (ε) = j _1,1 ² − 1 2

j _1,1 ²

|J ₁ ⁰ (j 1,1 )| ² ε ² + o(ε ² ), λ ^AB ₃ (ε) = j _1,1 ² + 1

2 j _1,1 ²

|J ₁ ⁰ (j _1,1 )| ² ε ² + o(ε ² ), λ ^AB ₄ (ε) = j _2,1 ² − 1

64 j _2,1 ⁴

|J ₂ ⁰ (j _2,1 )| ² ε ⁴ + o(ε ⁴ ), λ ^AB ₅ (ε) = j _2,1 ² + 1

64 j _2,1 ⁴

|J ₂ ⁰ (j 2,1 )| ² ε ⁴ + o(ε ⁴ ).

A Computation of the constants

A.1 The Neumann-Dirichlet case

In the present section, we use the above results to compute the quantities appearing in [1, Section 4]. In order to avoid a conflict of notation with the present paper, for any odd positive integer k, we denote here by ψ ⁰ _k , m ⁰ _k and w ⁰ _k what is denoted in [1] by ψ k , m k and w k respectively.

As in [1], we use the notation

s 0 := {(x ⁰ ₁ , 0) ; x ⁰ ₁ ≥ 0} ; and

s := {(x ⁰ ₁ , 0) ; x ⁰ ₁ ≥ 1} . We now define the mapping G : R ² + → R ² \ s 0 by

G(x) := (x ² ₁ − x ² ₂ , 2x ₁ x ₂ ).

The mapping is conformal; indeed, if for x ∈ R ² + we write z := x ₁ + ix ₂ and z ⁰ := x ⁰ ₁ + ix ⁰ ₂ , with x ⁰ = (x ⁰ ₁ , x ⁰ ₂ ) := G(x), we have z ⁰ = z ² . The scale factor associated with G is h(x) = 2|z| = 2|x|.

Let u ⁰ be a function in H ¹ R ² \ s ₀
and u := u ⁰ ◦ G. Since G is conformal, |∇u| is in L ² R ² +
,

with Z

R

²+

|∇u| ² dx = Z

R

²

\s

0

|∇u ⁰ | ² dx ⁰ .

Furthermore, for any x ⁰ in the segment (0, 1) × {0}, which we write as x ⁰ = (x ⁰ ₁ , 0), we have

∂u ⁰

∂ν +

(x ⁰ ) = − 1 2px ⁰ ₁

∂u

∂x 2

 p x ⁰ ₁ , 0 

and ∂u ⁰

∂ν −

(x ⁰ ) = − 1 2px ⁰ ₁

∂u

∂x 2

 − p x ⁰ ₁ , 0 

,

where _∂ν ^∂u

⁰

+

(x ⁰ ) and _∂ν ^∂u

⁰

−

(x ⁰ ) denote the normal derivative at x ⁰ respectively from above and from below. We also note that u is harmonic in R ² + if, and only if, u ⁰ is harmonic in R ² \ s 0 .

Let us now denote by w e _k ⁰ the extension by reflexion to R ² \ s 0 of w _k ⁰ , originally defined on R ² + . We recall that w _k ⁰ is the unique finite energy solution to the problem



 

 

−∆w _k ⁰ = 0, in R ² + , w _k ⁰ = 0, on s,

∂w

⁰_k

∂ν = − ^∂ψ

0 k

∂ν , on ∂R ² + \ s,

where ψ ⁰ _k (r cos t, r sin t) = r ^k/2 sin ^k ₂ t
.