Eigenvalues variations for Aharonov-Bohm operators

Abstract

We study how the eigenvalues of a magnetic Schrödinger operator of Aharonov-Bohm type depend on the singularities of its magnetic potential. We consider a magnetic potential dened everywhere in R 2 except at a nite number of singularities, so that the associated magnetic eld is zero. On a xed planar domain, we dene the corresponding magnetic Hamiltonian with Dirichlet boundary conditions, and study its eigenvalues as functions of the singularities. We prove that these functions are continuous, and in some cases even analytic. We sketch the connection of this eigenvalue problem to the problem of nding spectral minimal partitions of the domain. IntroductionAharonov-Bohm operators have been introduced in [1] as models of Schrödinger operators with a localized magnetic eld. In addition to their physical relevance, it has been shown in [8] that these operators appear in the theory of spectral minimal partitions (see [9] for a denition of the latter object). In [4,3,5], the eigenvalues and eigenfunctions of an Aharonov-Bohm operator with Dirichlet boundary condition have thus been studied numerically to nd minimal partitions. One of the aim of the present work is to support and generalize some observations made in these papers.We dene Aharonov-Bohm operators as follows. Let ω be an open and connected set in R 2 . As usual, we denote by C ∞ c (ω) the set of smooth functions compactly supported in ω. Generally speaking, let us consider a magnetic potential, that is to say a vector eld A ∈ C ∞ (ω, R 2 ). We dene the sesquilinear form s A byand the norm · A by ) . In the rest of the paper, N is an integer, X = (X 1 , . . . , X N ) an N -tuple of points in R 2 , with X i = X j for i = j (at least for now), andWe dene the Aharonov-Bohm potential associated with X and α as the vector eldLet us point out that the X i 's can be in R 2 \ Ω, and in particular in ∂Ω. TheAharonov-Bohm operator associated with X and α is the magnetic Hamiltonianon Ω X . We denote it by −∆ X α , and the associated form domain and quadratic form by Q X α and q X α respectively. Along the paper, we make frequent references to the Dirichlet realization of the Laplacian on Ω , and to the sequence of its eigenvalues. We denote them by −∆Let us conclude by a few remarks of a more physical nature. To an AharonovBohm potential is associated a magnetic eldwhich is a measure. If B i is a small disk centered at X i , such that X j / ∈ B i for j = i ,The coecient α i can therefore be called the normalized (magnetic) ux at X i .Let nally note that for any closed loop γ inwhere ind γ (X i ) is the winding number of γ around X i . We use in this paper the following characterization of the form domain Q X α , which follows from Hardy-type inequalities proved in [12,2,14]. Proposition 1.1. Let H X α be dened aswith the natural scalar product. Then H X α is a Hilbert space compactly embedded in L 2 (Ω). Furthermore, there exists a continuous mapping(Ω) belongs to Q X α if, and only if, u ∈ H X α and γ 0 u = 0 .As a consequence of the compact embedding, the sp...

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