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BY-NC-ND 4.0 license Open Access Published by De Gruyter March 18, 2017

On Aharonov–Bohm Operators with Two Colliding Poles

Laura Abatangelo , Veronica Felli EMAIL logo and Corentin Léna

Abstract

We consider Aharonov–Bohm operators with two poles and prove sharp asymptotics for simple eigenvalues as the poles collapse at an interior point out of nodal lines of the limit eigenfunction.

MSC 2010: 35P20; 35P15; 35J10

1 Introduction

The present paper is concerned with asymptotic estimates of theeigenvalue variation for magnetic Schrödinger operators with Aharonov–Bohm potentials.These special potentials generatelocalized magnetic fields, as they are produced byinfinitely long thin solenoids intersecting perpendicularly the planeat fixed points (poles), as the radius of the solenoids goes tozero and the magnetic flux remains constant.

The aim of the present paper is the investigation of eigenvalues of these operators as functions of the poles on the domain.This study was initiated by the set of papers [1, 2, 4, 10, 19],where a single point moving in the domain was considered, providing sharp asymptotics as it goes to an interior point or to a boundary point.On the other hand, to the best of our knowledge, the only paper considering different poles is [18], providing a continuity result for the eigenvalues and an improved regularity for simple eigenvalues as the poles are distinct and far from the boundary.

Additional motivations for the study of eigenvalue functions of these operators appear in the theory of spectral minimal partitions.We refer the interested reader to [7, 9, 14, 20] and references therein.

For a=(a1,a2)2, the Aharonov–Bohm magnetic potential with pole a and circulation 12 is defined as

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 ) , x = ( x 1 , x 2 ) 2 { a } .

In this paper we consider potentials which are the sum of two different Aharonov–Bohm potentials whose singularities are located at two different points in the domain moving towards each other.For a>0 small, let a-=(-a,0) and a+=(a,0) be the poles of the following Aharonov–Bohm potential:

(1.1) 𝐀 a - , a + ( x ) := - 𝐀 a - ( x ) + 𝐀 a + ( x ) = - 1 2 ( - x 2 , x 1 + a ) ( x 1 + a ) 2 + x 2 2 + 1 2 ( - x 2 , x 1 - a ) ( x 1 - a ) 2 + x 2 2 .

Let Ω be an open, bounded and connected set in 2 such that 0Ω.We consider the Schrödinger operator

(1.2) H a - , a + Ω = ( i + 𝐀 a - , a + ) 2

with homogeneous Dirichlet boundary conditions (see Section 3.1 for the notion of magnetic Hamiltonians) and its eigenvalues (λka)k1, counted with multiplicities.We denote by (λk)k1 the eigenvalues of the Dirichlet Laplacian -Δ in Ω.As already mentioned, we know from [18] that for every k1,

(1.3) lim a 0 λ k a = λ k .

The main result of the present paper is a sharp asymptotic for the eigenvalue variation λka-λk as the two poles a- and a+ coalesce towards a point where the limit eigenfunction does not vanish.

A first result in this direction was given in [3], under a symmetry assumption on the domain.

Theorem 1.1

Theorem 1.1 ([3, Theorem 1.13])

Let σ:R2R2, σ(x1,x2)=(x1,-x2).Let Ω be an open, bounded and connected set in R2 satisfying σ(Ω)=Ω and 0Ω.Let λN be a simple eigenvalue of the Dirichlet Laplacian on Ω, and let uN be a L2(Ω)-normalized eigenfunction associated to λN.Let kN{0} be the order of vanishing of uN at 0, and let α[0,π) be such that the minimal slope of nodal lines of uN is equal to αk, so that

u N ( r ( cos t , sin t ) ) r k β sin ( α - k t ) as r 0 + for all t ,

for some βR{0} (see, e.g., [12]).Let us assume that α0 (by symmetry of Ω, this forces α=π2, i.e, the x1-axis is the bisector of two nodal lines of uN).

For a>0 small, let a-=(-a,0), a+=(a,0)Ω, and let λNabe the N-th eigenvalue for (i+Aa-,a+)2.Then

λ N a - λ N = { 2 π | log a | | u N ( 0 ) | 2 ( 1 + o ( 1 ) ) if k = 0 , C k π β 2 a 2 k ( 1 + o ( 1 ) ) if k 1

as a0+, with Ck>0 being a positive constant depending only on k.

In the present paper we are able to remove, in the case k=0(i.e., when the limit eigenfunction uN does not vanish at the collision point), the assumption on the symmetry of the domain, proving the following result.

Theorem 1.2

Theorem 1.2 ([3, Theorem 1.17])

Let Ω be an open, bounded and connected set in R2 such that 0Ω.Let us assume that there exists N1 such that the N-th eigenvalue λN of the Dirichlet Laplacian in Ω is simple.Let uN be a L2(Ω)-normalized eigenfunction associated to λN.If uN(0)0, then

λ N a - λ N = 2 π u N 2 ( 0 ) | log a | ( 1 + o ( 1 ) ) as a 0 + .

It is worthwhile mentioning that in [18] simple magnetic eigenvalues are proved to be analytic functions of the configuration of the poles, provided the limit configuration is made of interior distinct poles.A consequence of our result is that the latter assumption is even necessary, and simple eigenvalues are not analytic in a neighborhood of configurations of poles collapsing outside nodal lines of the limit eigenfunction.

The proof of Theorem 1.2 relies essentially onthe characterization of the magnetic eigenvalue asan eigenvalue of the Dirichlet Laplacian in Ω with a small set removed,in the flavor of [3] (see Section 3.2 below). In [3] only the case of symmetric domains was consideredand the magnetic problem was shown to be spectrallyequivalent to the eigenvalue problem for the Dirichlet Laplacian in the domain obtained by removing the segment joining the poles.In the general non-symmetric case, we can still derive a spectral equivalence with a Dirichlet problem in the domain obtained by removing from Ω the nodal lines of magnetic eigenfunctions close to the collision point.The general shape of this removed set (which is not necessarily a segment as in the symmetric case) creates some further difficulties.In particular, precise information about the diameter of such a set is needed in order to apply the following resultfrom [3].

Theorem 1.3

Theorem 1.3 ([3, Theorem 1.7])

Let ΩR2 be a bounded connected open set containing 0.Let λN be a simple eigenvalue of the Dirichlet Laplacian in Ω, and let uN be a L2(Ω)-normalized eigenfunction associated to λN such that uN(0)0.Let (Kε)ε>0 be a family of compact connected sets contained in Ω such that for every r>0, there exists ε¯ such that KεDr for every ε(0,ε¯) (Dr denoting the disk of radius r centered at 0).Then

λ N ( Ω K ε ) - λ N = u N 2 ( 0 ) 2 π | log ( diam K ε ) | + o ( 1 | log ( diam K ε ) | ) as ε 0 ,

where λN(ΩKε) denotes the N-th eigenvalue of the DirichletLaplacian in ΩKε.

In order to apply Theorem 1.3, a crucial intermediate step in the proof of Theorem 1.2 is the estimate of the diameter of nodal lines of magnetic eigenfunctions near the collision point.More precisely, we prove that when a is sufficiently small, locally near 0 suitable (magnetic-real) eigenfunctions have a nodal set consisting in a single regular curve connecting a- and a+.If da denotes the diameter of such a curve, we obtain that

(1.4) lim a 0 + | log a | | log d a | = 1 ,

see Section 4.

The paper is organized as follows.In Section 2 we obtain some preliminary upper bounds for the eigenvalue variation λNa-λN, testing the Rayleigh quotient for eigenvalues with proper test functions constructed by suitable manipulation of limit eigenfunctions.In Section 3 we prove that, as the two poles of the operator (1.2) move towards each other colliding at 0, λNa is equal to the N-th eigenvalue of the Laplacian in Ω with a small piece of nodal line of the magnetic eigenfunction removed.Combining the upper estimates of Section 2 with Theorem 1.3, in Section 4 we succeed in estimating the diameter of the removed small set as in (1.4); we then conclude the proof of Theorem 1.2 by combining (1.4) and Theorem 1.3.

2 Estimates from Above

We denote by a the closure of Cc(Ω{a+,a-},) with respect to the norm

u a = ( Ω | ( i + 𝐀 a - , a + ) u | 2 d x ) 1 / 2 .

We observe that, by Poincaré and diamagneticinequalities together with the Hardy type inequality proved in [16],aH01(Ω) with continuous inclusion. In orderto estimate from above the eigenvalue λNa, we recall thewell-known Courant–Fisher minimax characterization:

(2.1) λ N a = min { max u F { 0 } Ω | ( i + 𝐀 a - , a + ) u | 2 d x Ω | u | 2 d x : F is a subspace of a , dim F = N } .

Lemma 2.1

Let τ(0,1).For every 0<ε<1, there exists a continuous radial cut-off function ρε,τ:R2R such that ρε,τHloc1(R2), which also has the following properties:

  1. 0 ρ ε , τ ( x ) 1 for all x 2 ,

  2. ρ ε , τ ( x ) = 0 if | x | ε , and ρ ε , τ ( x ) = 1 if | x | ε τ ,

  3. 2 | ρ ε , τ | 2 d x = 2 π ( τ - 1 ) log ε ,

  4. 2 ( 1 - ρ ε , τ 2 ) 𝑑 x = O ( ε 2 τ ) as ε 0 + .

Proof.

We set

ρ ε , τ ( x ) = { 0 if | x | ε , log | x | - log ( ε ) log ( ε τ ) - log ( ε ) if ε < | x | < ε τ , 1 if | x | ε τ .

The function ρε,τ is continuous and locally in H1,with 0ρε,τ1.The function 1-ρε,τ2 is supported in the disk of radius ετ centered at 0.We therefore have

2 ( 1 - ρ ε , τ 2 ( x ) ) 𝑑 x π ε 2 τ ,

which proves (iv).We have ρε,τ(x)=0 if |x|<ε or |x|>ετ, and

ρ ε , τ ( x ) = x ( τ - 1 ) log ( ε ) | x | 2

if ε<|x|<ετ. From this we directly obtain identity (iii).∎

Lemma 2.2

For all a>0, there exists a smooth function ψa:R2saR satisfying

ψ a = 𝐀 a - , a + ,

where sa is the segment in R2 defined by sa:={(t,0):-ata}.Furthermore, for every xR2{(0,0)}, lima0+ψa(x)=0.

Proof.

See [3, Lemma 3.1].∎

The first step in the proof of Theorem 1.2 is the following upper bound for the eigenvalue λNa.

Proposition 2.3

For every τ(0,1),

λ N a λ N + 2 π ( 1 - τ ) | log a | ( u N 2 ( 0 ) + o ( 1 ) ) as a 0 + .

The proof of Proposition 2.3 is based on estimates from above of the Rayleigh quotient for λNa computed at some proper test functions constructed by suitable manipulation of limit eigenfunctions.To this end, let us consider, for each j{1,,N}, a real eigenfunction uj of -Δ with homogeneous Dirichlet boundary conditionsassociated with λj, with ujL2(Ω)=1.Furthermore, we choose these eigenfunctions so that

(2.2) Ω u j u k 𝑑 x = 0 for j k .

For j{1,,N} and a>0 small enough, we set

(2.3) v j , τ a := e i ψ a ρ 2 a , τ u j .

We have that vj,τaa.Lemma 2.1 and the Dominated Convergence Theorem imply that vj,τa tends to uj in L2(Ω) when a0+.This implies, in particular, that the functions vj,τa are linearly independent for a small enough.

Hence, for a>0 small enough, EN,τa=span{v1,τa,,vN,τa} is anN-dimensional subspace of a, so that, in view of (2.1),

(2.4) λ N a max u E N , τ a { 0 } Ω | ( i + 𝐀 a - , a + ) u | 2 d x Ω | u | 2 d x = Ω | ( i + 𝐀 a - , a + ) v τ a | 2 d x Ω | v τ a | 2 d x

with

(2.5) v τ a = j = 1 N α j , τ a v j , τ a for some α 1 , τ a , , α N , τ a such that j = 1 N | α j , τ a | 2 = 1 .

Lemma 2.4

For a>0 small, let vτa be as in (2.4)–(2.5).Then

(2.6) Ω | v τ a | 2 d x = 1 + O ( a 2 τ ) as a 0 + .

Proof.

Taking into account (2.5), (2.3) and (2.2), we can write

Ω | v τ a | 2 d x = j , k = 1 N α j , τ a α k , τ a ¯ Ω ρ 2 a , τ 2 u j u k d x
= 1 + j = 1 N | α j , τ a | 2 Ω ( ρ 2 a , τ 2 - 1 ) u j 2 𝑑 x + j k α j , τ a α k , τ a ¯ Ω ( ρ 2 a , τ 2 - 1 ) u j u k 𝑑 x .

Hence, the conclusion follows from Lemma 2.1 (iv).∎

Lemma 2.5

For a>0 small, let vτa be as in (2.4)–(2.5).Then

(2.7) Ω | ( i + 𝐀 a - , a + ) v τ a | 2 d x = j , k = 1 N α j , τ a α k , τ a ¯ ( λ j + λ k 2 Ω D 2 a ρ 2 a , τ 2 u j u k d x + D ( 2 a ) τ D 2 a u j u k | ρ 2 a , τ | 2 d x ) ,

where, for all r>0, Dr={(x1,x2)R2:x12+x22<r}denotes the disk of center (0,0) and radius r.

Proof.

Let us fix j and k in {1,,N} (possibly equal).In ΩD2a, we have that

( i + 𝐀 a - , a + ) v j , τ a ( i + 𝐀 a - , a + ) v k , τ a ¯ = ( ρ 2 a , τ u j ) ( ρ 2 a , τ u k )
= ρ 2 a , τ 2 u j u k + u j u k | ρ 2 a , τ | 2 + ( u j u k + u k u j ) ρ 2 a , τ ρ 2 a , τ

and, since ρ2a,τρ2a,τ=12(ρ2a,τ2),

Ω ( i + 𝐀 a - , a + ) v j , τ a ( i + 𝐀 a - , a + ) v k , τ a ¯ 𝑑 x
(2.8) = Ω D 2 a ρ 2 a , τ 2 u j u k d x + D ( 2 a ) τ D 2 a u j u k | ρ 2 a , τ | 2 𝑑 x + 1 2 Ω D 2 a ( u j u k + u k u j ) ( ρ 2 a , τ 2 ) d x .

An integration by parts on the last term of (2.8) gives

Ω ( i + 𝐀 a - , a + ) v j , τ a ( i + 𝐀 a - , a + ) v k , τ a ¯ 𝑑 x
= Ω D 2 a ρ 2 a , τ 2 u j u k d x + D ( 2 a ) τ D 2 a u j u k | ρ 2 a , τ | 2 𝑑 x - 1 2 Ω D 2 a ( u j Δ u k + 2 u k u j + Δ u j u k ) ρ 2 a , τ 2 𝑑 x .

After cancellations, we get

(2.9) Ω ( i + 𝐀 a - , a + ) v j , τ a ( i + 𝐀 a - , a + ) v k , τ a ¯ 𝑑 x = λ k + λ j 2 Ω D 2 a ρ 2 a , τ 2 u j u k 𝑑 x + D ( 2 a ) τ D 2 a u j u k | ρ 2 a , τ | 2 𝑑 x .

From bilinearity, (2.5) and (2.9), we obtain (2.7).∎

From (2.4) and (2.7), it follows that

(2.10) λ N a - λ N 1 Ω | v τ a | 2 d x [ 𝒬 a ( α 1 , τ a , α 2 , τ a , , α N , τ a ) + λ N ( 1 - Ω | v τ a | 2 d x ) ] ,

where 𝒬a:N is the quadratic form defined as

(2.11) 𝒬 a ( z 1 , z 2 , , z N ) = j , k = 1 N M j k a z j z k ¯ ,

where

(2.12) M j k a = λ j + λ k 2 Ω D 2 a ρ 2 a , τ 2 u j u k 𝑑 x + D ( 2 a ) τ D 2 a u j u k | ρ 2 a , τ | 2 𝑑 x - λ N δ j k ,

with δjk being the Kronecker delta.

To estimate the largest eigenvalue of the quadratic form 𝒬a, we will use the following technical lemma.

Lemma 2.6

For every ε>0, let us consider the quadratic form

Q ε : N , Q ε ( z 1 , z 2 , , z N ) = j , k = 1 N m j , k ( ε ) z j z k ¯ ,

with mj,k(ε)C such that mj,k(ε)=mk,j(ε)¯.Assume that there exist real numbers C>0 and K1,K2,,KN-1<0 such that

m N , N ( ε ) = C ε ( 1 + o ( 1 ) ) as ε 0 + ,
m j , j ( ε ) = K j + o ( 1 ) as ε 0 + for all j < N ,
m j , k ( ε ) = m k , j ( ε ) ¯ = O ( ε ) as ε 0 + for all j k .

Then

max { Q ε ( z 1 , , z N ) : ( z 1 , , z N ) N , j = 1 N | z j | 2 = 1 } = C ε ( 1 + o ( 1 ) ) as ε 0 + .

Proof.

The result is contained in [1, Lemma 6.1], hence we omit the proof.∎

Lemma 2.7

For a>0 small, let Qa:CNR be the quadratic form defined in (2.11)–(2.12).Then

max { 𝒬 a ( z 1 , , z N ) : ( z 1 , , z N ) N , j = 1 N | z j | 2 = 1 } = 2 π u N 2 ( 0 ) ( 1 - τ ) | log ( a ) | ( 1 + o ( 1 ) ) as a 0 + .

Proof.

Since ΩuN2=1, we can write

M N N a = λ N Ω ( ρ 2 a , τ 2 - 1 ) u N 2 𝑑 x + D ( 2 a ) τ D 2 a u N 2 | ρ 2 a , τ | 2 𝑑 x .

Since uNLloc(Ω), from Lemma 2.1 (iv), it follows that

Ω ( ρ 2 a , τ 2 - 1 ) u N 2 𝑑 x = D ( 2 a ) τ ( ρ 2 a , τ 2 - 1 ) u N 2 𝑑 x = O ( a 2 τ ) as a 0 + .

Since uNCloc(Ω), we have that uN2(x)-uN2(0)=O(|x|) as |x|0+. Then Lemma 2.1 (iii) implies that

D ( 2 a ) τ D 2 a u N 2 | ρ 2 a , τ | 2 d x = u N 2 ( 0 ) D ( 2 a ) τ D 2 a | ρ 2 a , τ | 2 d x + D ( 2 a ) τ D 2 a ( u N 2 ( x ) - u N 2 ( 0 ) ) | ρ 2 a , τ ( x ) | 2 d x
= ( u N 2 ( 0 ) + O ( a τ ) ) D ( 2 a ) τ D 2 a | ρ 2 a , τ | 2 d x
= 2 π ( τ - 1 ) log ( 2 a ) ( u N 2 ( 0 ) + O ( a τ ) )
= 2 π ( τ - 1 ) log ( a ) u N 2 ( 0 ) ( 1 + o ( 1 ) ) as a 0 + .

Then

(2.13) M N N a = 2 π ( τ - 1 ) log ( a ) u N 2 ( 0 ) ( 1 + o ( 1 ) ) as a 0 + .

For all 1j<N, we have that

M j j a = λ j Ω D 2 a ρ 2 a , τ 2 u j 2 𝑑 x + D ( 2 a ) τ D 2 a u j 2 | ρ 2 a , τ | 2 𝑑 x - λ N
= ( λ j - λ N ) + λ j Ω ( ρ 2 a , τ 2 - 1 ) u j 2 d x + Ω u j 2 | ρ 2 a , τ | 2 d x ,

and hence, since ujCloc(Ω) and in view of Lemma 2.1,

(2.14) M j j a = ( λ j - λ N ) + O ( 1 | log a | ) = ( λ j - λ N ) + o ( 1 ) as a 0 + .

Moreover, for all j,k=1,,N with jk, in view of(2.2) and Lemma 2.1, we have that

(2.15) M j k a = λ j + λ k 2 Ω D 2 a ( ρ 2 a , τ 2 - 1 ) u j u k 𝑑 x + D ( 2 a ) τ D 2 a u j u k | ρ 2 a , τ | 2 𝑑 x = O ( 1 | log a | ) as a 0 + .

In view of estimates (2.13), (2.14) and (2.15), we have that 𝒬a satisfies the assumption of Lemma 2.6 (with ε=1|loga|), hence the conclusion follows from Lemma 2.6.∎

Proof of Proposition 2.3.

Combining (2.10), Lemma 2.7 and estimate (2.6), we obtain that

λ N a - λ N 1 1 + O ( a 2 τ ) [ 2 π u N 2 ( 0 ) ( 1 - τ ) | log ( a ) | ( 1 + o ( 1 ) ) + O ( a 2 τ ) ]
= 2 π u N 2 ( 0 ) ( 1 - τ ) | log ( a ) | ( 1 + o ( 1 ) ) as a 0 + ,

thus completing the proof.∎

3 Gauge Invariance, Nodal Sets and Reduction to the Dirichlet–Laplacian

In the following, by a path γ we mean a piecewise C1 map γ:I2, with I=[a,b] being a closed interval.It follows from the definition of 𝐀a-,a+ (see (1.1))that for any closed path γ (i.e., γ(a)=γ(b)),

(3.1) 1 2 π γ 𝐀 a - , a + 𝑑 𝐬 = 1 2 ind γ ( a + ) - 1 2 ind γ ( a - ) ,

where indγ(a+) (resp. indγ(a-)) is the winding number of γ around a+ (resp. a-).

3.1 Gauge Invariance

Let us give some results concerning the gauge invariance of our operators.In view of applying them to several different situations, we give statements valid for a magnetic Hamiltonian in an open and connected domain D, without restricting ourselves to the Aharonov–Bohm case.

In the following, the term vector potential (in an openconnected domain D)stands for a smooth real vector field 𝐀:D2. Inorder to define the quantum mechanical Hamiltonian for a particle inD, under the action of the magnetic field derived from the vectorpotential 𝐀, we first consider the differential operator

P = ( i + 𝐀 ) 2

acting on smooth functions compactly supported in D.Using integration by parts (Green’s formula), one can easily see that P is symmetric and positive.This is formally the desired Hamiltonian, but to obtain a self-adjoint Schrödinger operator, we have to specify the boundary conditions on D, which we choose to be Dirichlet boundary conditions everywhere.More specifically, our Hamiltonian is the Friedrichs extension of the differential operator P.We denote it by H𝐀D, and we call it the magnetic Hamiltonian on D associated with 𝐀.

We observe that the Aharonov–Bohm operator Ha-,a+Ω,introduced in (1.2) with the poles a-=(-a,0) and a+=(a,0) in Ω, can be defined as the magnetic Hamiltonian H𝐀a-,a+Ω˙ on Ω˙, where Ω˙=Ω{a-,a+}, and that the spectrum of Ha-,a+Ω consists of the eigenvalues defined by (2.1).The space a is the form domain of Ha-,a+Ω.

Definition 3.1

We call gauge function a smooth complex valued function ψ:D such that |ψ|1.To any gauge function ψ, we associate a gauge transformation acting as (𝐀,u)(𝐀*,u*), with

𝐀 * = 𝐀 - i ψ ψ , u * = ψ u ,

where ψ=(Reψ)+i(Imψ).We notice that, since |ψ|=1, iψψ is a real vector field.Two magnetic potentials are said to be gauge equivalent if one can be obtained from the other by a gauge transformation (this is an equivalence relation).

The following result is a consequence of [17, Theorem 1.2].

Proposition 3.2

If A and A* are two gauge equivalent vector potentials, then the operators HAD and HA*D are unitarily equivalent.

The equivalence between two vector potentials (which is equivalent to the fact that their difference is gauge-equivalent to 0) can be determined using the following criterion.

Lemma 3.3

Let A be a vector potential in D.This is gauge equivalent to 0 if and only if

(3.2) 1 2 π γ 𝐀 ( s ) 𝑑 𝐬

for every closed path γ contained in D.

Remark 3.4

The reverse implication in Lemma 3.3 is contained in[13, Theorem 1.1] for the Neumann boundary condition.

Proof.

Let us first prove the direct implication. We assume that 𝐀 is gauge equivalent to 0, that is to say that there exists a gauge function ψ such that

𝐀 i ψ ψ .

Fix a closed path γ:I=[a,b]D and consider the mapping z=ψγ from I to 𝕌, where 𝕌={z:|z|=1}.By the lifting property, there exists a piecewise C1 function θ:I such that z(t)=exp(iθ(t)) for all tI.This implies that

ψ ( γ ( t ) ) γ ( t ) = ( ψ γ ) ( t ) = z ( t ) = i θ ( t ) exp ( i θ ( t ) ) ,

and therefore

i ψ ψ ( γ ( t ) ) γ ( t ) = - θ ( t ) .

This implies that

γ 𝐀 ( s ) 𝑑 𝐬 = a b i ψ ψ ( γ ( t ) ) γ ( t ) 𝑑 t = - a b θ ( t ) 𝑑 t = θ ( a ) - θ ( b ) .

Since γ is a closed path, exp(iθ(a))=exp(iθ(b)), and therefore

θ ( a ) - θ ( b ) 2 π .

Let us now consider the reverse implication.We define a gauge function ψ in the following way.We fix an (arbitrary) point X0=(x0,y0)D.Let us show that for X=(x,y)D, the quantity

exp ( - i γ 𝐀 ( s ) 𝑑 𝐬 )

does not depend on the choice of the path γ from X0 toX. Indeed, let γ1 and γ2 be two such paths, and letγ3 be the closed path obtained by going from X0 to Xalong γ1 and then from X to X0 along γ2. On theone hand, we have

γ 3 𝐀 ( s ) 𝑑 𝐬 = γ 1 𝐀 ( s ) 𝑑 𝐬 - γ 2 𝐀 ( s ) 𝑑 𝐬 .

On the other hand, if (3.2) holds, we have

γ 3 𝐀 ( s ) 𝑑 𝐬 2 π .

This implies that

exp ( - i γ 1 𝐀 ( s ) 𝑑 𝐬 ) = exp ( - i γ 2 𝐀 ( s ) 𝑑 𝐬 ) .

By the connectedness of D, there exists a path from X0 to X for any XΩ (we can even choose it piecewise linear).We can therefore define, without ambiguity, a function ψ:Ω by

ψ ( X ) = exp ( - i γ 𝐀 ( s ) 𝑑 𝐬 ) .

It is immediate from the definition that |ψ|1 and that ψ is smooth, with

ψ ( X ) = - i ψ ( X ) 𝐀 ( X ) .

It is therefore a gauge function sending 𝐀 to 0.∎

Lemma 3.3 can be used to define a set of eigenfunctions for Ha-,a+Ω having special nice properties, as was done in [13, Section 3] for the Neumann boundary condition.It is analogous to the set of real eigenfunctions for the usual Dirichlet–Laplacian.To define it, we will construct a conjugation, that is, an antilinear antiunitary operator, which commutes with Ha-,a+Ω.To simplify the notation, we denote 𝐀a-,a+ by 𝐀 and Ha-,a+Ω by H in the rest of this section.

According to (3.1), the vector potential 2𝐀 satisfies condition (3.2) of Lemma 3.3 on Ω˙,and therefore is gauge equivalent to 0.Therefore, there exists a gauge function ψ in Ω˙ such that

2 𝐀 = - i ψ ψ in Ω ˙ .

We now define the antilinear antiunitary operator K by

K u = ψ u ¯ .

For all uC0(Ω˙,),

( i + 𝐀 ) ( ψ u ¯ ) = ψ ( i + i ψ ψ + 𝐀 ) u ¯ = ψ ( i - 𝐀 ) u ¯ = - ψ ( i + 𝐀 ) u ¯ .

The above formula and the fact that K is antilinear and antiunitary,imply that for all u and v in C0(Ω˙,),

K - 1 H K u , v = K v , H K u = Ω ( i + 𝐀 ) ( ψ v ¯ ) ( i + 𝐀 ) ( ψ u ¯ ) ¯ 𝑑 x = Ω ( i + 𝐀 ) v ¯ ( i + 𝐀 ) u 𝑑 x = H u , v ,

where f,g=Ωfg¯𝑑x denotes the standard scalar product on the complex Hilbert space L2(Ω,).By density, we conclude that

K - 1 H K = H .

Definition 3.5

We say that a function uL2(Ω,) is magnetic-realwhen Ku=u.

Let us denote by the set of magnetic-real functions in L2(Ω,). The restriction of the scalar product to gives it the structure of a real Hilbert space.The commutation relation HK=KH implies that is stable under the action of H.We denote by HR the restriction of H to .There exists an orthonormal basis of formed by eigenfunctions of HR.Such a basis can be seen as a basis of magnetic-real eigenfunctions of the operator H in the complex Hilbert space L2(Ω,).

Let us now fix an eigenfunction u of HR (or, equivalently, a magnetic-real eigenfunction of H). We define its nodal set𝒩(u) as the closure in Ω¯ of the zero-set u-1({0}).Let us describe the local structure of 𝒩(u). In the sequel, by a regular curve or regular arc we mean a curve admitting a C1,α parametrization for some α(0,1).

Theorem 3.6

The set N(u) has the following properties:

  1. 𝒩 ( u ) is, locally in Ω ˙ , a regular curve, exceptpossibly at a finite number of singular points { X j } j { 1 , , n } .

  2. For j { 1 , , n } , in the neighborhood of X j , 𝒩(u) consists of an even number of regular half-curves meeting at Xj with equal angles (so that Xj can be seen as a cross-point).

  3. In the neighborhood of a + (resp. a - ), 𝒩 ( u ) consists of an odd number of regular half-curves meeting at a + (resp. a - ) with equal angles (in particular this means that a + and a - are always contained in 𝒩 ( u ) ).

Proof.

The proof is essentially contained in [20, Theorem 1.5] (see also [5]);for the sake of completeness we present a sketch of it.Let the eigenfunction u be associated with the eigenvalue λ, so that Hu=λu.Let x0 be a point in Ω˙.For ε>0, we denote by D(x0,ε) the open disk {x:|x-x0|<ε}.Let us show that we can find ε>0 small enough and a local gauge transformationφ:D(x0,ε) such that𝐀*=𝐀-iφφ=0 andu*=φu is a real-valued function in D(x0,ε).Indeed, let us define, as before, a gauge function ψ such that 2𝐀=-iψψ.For ε>0 small enough, we can define a smooth function φ:D(x0,ε) such that ψ¯(x)=(φ(x))2 for all xD(x0,ε), by taking

φ ( x ) = exp ( - i 2 arg ( ψ ( x ) ) ) ,

with arg a determination of the argument in ψ(D(x0,ε)).A direct computation shows that for xD(x0,ε),

i φ ( x ) φ ( x ) = i 2 ψ ¯ ( x ) ψ ¯ ( x ) = 𝐀 ( x ) .

The gauge transformation on D(x0,ε) associated with φ therefore sends 𝐀 to 0.Furthermore, since u is K-real, we have ψu¯=φ2u¯=u in D(x0,ε), and therefore φu¯=φu.The real-valued function v=φu satisfies -Δv=λv,and, since |φ|1 on D(x0,ε),we have that 𝒩(v)D(x0,ε)=𝒩(u)D(x0,ε).Parts (i) and (ii) of Theorem 3.6 then follow from classical results on the nodal set of Laplacian eigenfunctions (see, for instance,[15, Theorem 2.1] and [20, Theorem 4.2]).

To prove part (iii) of Theorem 3.6, we use the regularity result of [20] for the Dirichlet problem associated with a one-pole Aharonov–Bohm operator.Indeed, let ε>0 be small enough so that D=D(a+,ε)Ω and a-D.By this choice of ε, 𝐀a-=f on D, with f asmooth function, so that the domain D and the magnetic potential𝐀, restricted to D, satisfy the hypotheses of[20, Theorem 1.5]. The function u is a solution of the Dirichlet problem

{ ( i + 𝐀 ) 2 u - λ u = 0 in D , u = γ on D ,

with γ=u|DW1,(D). A direct application of [20, Theorem 1.5] gives property (iii) around a+.We can obtain property (iii) around a- by exchanging the role of a+ and a-.∎

3.2 Reduction to the Dirichlet–Laplacian

Our aim in this subsection is to show that, as the two poles of the operator (1.2) coalesce into a point at which uN does not vanish, λNa is equal to the N-th eigenvalue of the Laplacian in Ω with a small subset concentrating at 0 removed.

Theorem 3.7

Let us assume that there exists N1 such that the N-theigenvalue λN of the Dirichlet Laplacian in Ω issimple. Let uN be a L2(Ω)-normalized eigenfunction associatedλN and assume that uN(0)0.Then, for all a>0 sufficiently small, there exists a compact connected set KaΩ such that

λ N a = λ N ( Ω K a ) ,

and Ka concentrates around 0 as a0+, i.e., for any ε>0, there exists δ>0 such that if a<δ, then KaDε.

We will divide the proof into two lemmas.

Lemma 3.8

Let R>0 be such that DR¯Ω anduN(x)0 for all xDR¯. Let r(0,R).We denote by Cr,R the closed ring

C r , R = { x 2 : r | x | R } .

There exists δ>0 such that if 0<a<δ and u is a magnetic-real eigenfunction associated with λNa, then u does not vanish inCr,R.

Proof.

Let us assume, by contradiction, that there exists a sequence an0+ such that for all n1, λNan admits an eigenfunction φn whichvanishes somewhere in Cr,R.Let us denotes by Xn a zero of φn in Cr,R.

According to [18, Section III], we can assume, up to extraction and a suitable normalization of φn, that φnuN in L2(Ω).Since H is a uniformly regular elliptic operator in a neighborhood of Cr,R, φn converges to uN uniformly on Cr,R.Furthermore, up to one additional extraction, we can assume that XnXCr,R.This implies that uN(X)=0, contradicting the fact that uN(x)0 for all xDR¯.∎

Lemma 3.9

For all R>0 such that DR¯Ω and uN(x)0 for all xDR¯,there exists δ>0 such that if 0<a<δ and uNa is a magnetic-real eigenfunction associated with λNa,then N(uNa)DR consists in a single regular curve connecting a- and a+.

Proof.

By the continuity of (a-,a+)λNa (see [18]), we have that

(3.3) Λ = max a [ 0 , R ] λ N a ( 0 , + ) .

Let us choose r(0,R) such that

(3.4) r < λ 1 ( D 1 ) Λ ,

where λ1(D1) is the 1-st eigenvalue of the Laplacian in the unit disk D1.According to Lemma 3.8, there exists δ(r)>0 such that if a<δ(r), then any eigenfunction associated to λNa does not vanish in the closed ring Cr,R.

Let us assume that 0<a<δ(r) and a<r, and let uNa be an eigenfunction associated with λNa.The proof relies on a topological analysis of 𝒩:=𝒩(uNa)DR, inspired by previous work on nodal sets and minimal partitions (see [8, Section 6] and references therein).Lemma 3.8 implies that 𝒩 is compactly included in Dr.Theorem 3.6 implies that 𝒩 consists of a finite number of regular arcs connecting a finite number of singular points.In other words, 𝒩 is a regular planar graph.Let us denote by V the set of vertices of 𝒩, by b1 the number of its connected components and by μ the number of its faces.By face, we mean a connected component of 2𝒩.There is always one unbounded face, so μ1.Furthermore, for all wV, we denote by ν(w) the degree of the vertex w, that is to say the number of half-curves ending at w.Let us note that, according to Theorem 3.6, both a- and a+ belong to V and have an odd degree, and any other vertex can only have an even degree.These quantities are related through Euler’s formula for planar graphs:

(3.5) μ = b 1 + w V ( ν ( w ) 2 - 1 ) + 1 .

For this classical formula, see, for instance, [6, Theorems 1.1 and 9.5].Note that this reference treats the case of a connected graph.The generalization used here is easily obtained by linking the b1 connected components of the graph with b1-1 edges, in order to go back to the connected case.

Let us show by contradiction that μ=1.If μ2, there exists a bounded face of the graph 𝒩, which is a nodal domain of uNa entirely contained in Dr.Let us call it ω.We denote by λk(ω,a-,a+) the k-th eigenvalue of the operator (i+𝐀a-,a+)2 in ω, with homogeneous Dirichlet boundary condition on ω.Since ω is a nodal domain, for some k(a){0} depending on a, we have that

λ N a = λ k ( a ) ( ω , a - , a + ) λ 1 ( ω , a - , a + ) .

By the diamagnetic inequality,

λ 1 ( ω , a - , a + ) λ 1 ( ω ) ,

where λ1(ω) is the 1-st eigenvalue of the Dirichlet Laplacian in ω.By domain monotonicity,

λ 1 ( ω ) λ 1 ( D r ) = λ 1 ( D 1 ) r 2 .

Hence, we obtain

r λ 1 ( D 1 ) λ N a ,

thus contradicting (3.4).We conclude that μ=1.

Going back to Euler’s formula (3.5), we obtain

(3.6) w V ( ν ( w ) 2 - 1 ) = - b 1 - 1 .

According to Theorem 3.6, we have ν(w)2-1-12 if w{a-,a+}, and ν(w)2-11 if wV{a-,a+}.Inequality (3.6) can therefore be satisfied only if V={a-,a+} and ν(a-)=ν(a+)=1, that is to say if 𝒩 is a regular arc connecting a- and a+.∎

We are now in position to prove Theorem 3.7.

Proof of Theorem 3.7.

From Lemma 3.9, it follows that for a sufficiently small, there exists a curve Ka in 𝒩(uNa) connecting a- and a+ and (in view of Lemma 3.8) concentrating at 0, where uNa is a magnetic-real eigenfunction associated with λNa.

Let us write Ωa=ΩKa.Since Ka is contained in 𝒩(uNa), we have that there exists k(a){0} (depending on a) such that

(3.7) λ N a = λ k ( a ) ( Ω a , a - , a + ) ,

where λk(a)(Ωa,a-,a+) denotes the k(a)-th eigenvalue of Ha-,a+Ωa.

Let us consider a closed path γ in Ωa.By the definition of Ωa, γ does not meet Ka, which means that Ka is contained in a connected component of 2γ. Since the function XIndγ(X) is constant on all connected components of 2γ, we have that Indγ(a-)=Indγ(a+).According to (3.1), this implies that

1 2 π γ 𝐀 a - , a + 𝑑 𝐬 = 0 .

In view of Lemma 3.3, we conclude that 𝐀a-,a+ is gauge equivalent to 0 in Ωa, and hence Proposition 3.2 ensures that

(3.8) λ k ( a ) ( Ω a , a - , a + ) = λ k ( a ) ( Ω a ) .

Combining (3.7) and (3.8), we obtain

(3.9) λ N a = λ k ( a ) ( Ω a ) .

We observe that ak(a) stays bounded as a0+.Indeed if, by contradiction, k(an)+ along some sequence an0+, by (3.9), we should have

λ N a n = λ k ( a n ) ( Ω a n ) λ k ( a n ) ( Ω ) + ,

thus contradicting (3.3).

Then, for any sequence an0+, there exists a subsequence anj such that k(anj)k for some k.Since k(a) is integer-valued we have that necessarily k(anj)=k{0} for j sufficiently large.Hence, (3.9) yields λNanj=λk(ΩKanj).It is well known (see, e.g., [11, Theorem 1.2]) that λk(ΩKanj)λk(Ω) as j+; hence, taking into account (1.3), we conclude that k=N.Moreover, since the limit of k(anj) does not depend on the subsequence and ak(a) isinteger-valued, we conclude that k(a)=N for all a sufficientlysmall, so that (3.9) becomes

λ N a = λ N ( Ω a ) ,

and the proof is complete.∎

4 Proof of Theorem 1.2

We are in position to complete the proof of Theorem 1.2.

Proof of Theorem 1.2.

For a>0 small, let KaΩ be as in Theorem 3.7.We denote as

d a := diam K a

the diameter of Ka.From Theorem 1.3, it follows that

λ N ( Ω K a ) - λ N = u N 2 ( 0 ) 2 π | log d a | + o ( 1 | log d a | ) as a 0 + .

Hence, in view of Theorem 3.7,

(4.1) λ N a - λ N = u N 2 ( 0 ) 2 π | log d a | + o ( 1 | log d a | ) as a 0 + .

From (4.1) and Proposition 2.3, it follows that for every τ(0,1),

1 | log d a | ( 1 + o ( 1 ) ) 1 ( 1 - τ ) | log a | ( 1 + o ( 1 ) ) ,

and then

(4.2) | log a | | log d a | 1 1 - τ ( 1 + o ( 1 ) ) as a 0 + .

On the other hand, since a-,a+Ka, we have that da2a,so that |loga||logda|+log2 and

(4.3) | log a | | log d a | 1 + O ( 1 | log d a | ) = 1 + o ( 1 ) , as a 0 + .

Combining (4.2) and (4.3), we conclude that

1 lim inf a 0 + | log a | | log d a | lim sup a 0 + | log a | | log d a | 1 1 - τ

for every τ(0,1), and then, letting τ0+, weobtain that

(4.4) lim a 0 + | log a | | log d a | = 1 .

The conclusion then follows from (4.1) and (4.4).∎


Dedicated to Professor Ireneo Peral on the occasion of his 70th birthday



Communicated by Antonio Ambrosetti and David Arcoya


Award Identifier / Grant number: 339958

Award Identifier / Grant number: 201274FYK7_008

Funding statement: The authors have been partially supported by the project ERC Advanced Grant 2013 n. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems –COMPAT”, funded by the European Research Council.V. Felli is partially supported by PRIN-2012-grant n. 201274FYK7_008: “Variational and perturbative aspects of nonlinear differential problems”, funded by the Ministero dell’Istruzione, dell’Università e della Ricerca.

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Received: 2016-12-14
Accepted: 2017-01-31
Published Online: 2017-03-18
Published in Print: 2017-05-01

© 2017 by De Gruyter

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