On multiple eigenvalues for Aharonov–Bohm operators in planar domains

Abstract

Abstract. We study multiple eigenvalues of a magnetic Aharonov-Bohm operator with Dirichlet boundary conditions in a planar domain. In particular, we study the structure of the set of the couples position of the pole-circulation which keep fixed the multiplicity of a double eigenvalue of the operator with the pole at the origin and half-integer circulation. We provide sufficient conditions for which this set is made of an isolated point. The result confirms and validates a lot of numerical simulations available in preexisting literature.

Multiple Aharonov–Bohm eigenvalues: The case of the first eigenvalue on the disk

“…In particular, we are going to focus our attention on the first eigenvalue to problem (1.5) and to study its multiplicity as the pole is moving from the origin around the disk. One can prove that this situation fulfills the assumptions of [6,Theorem 1.6], so that we know that the origin is locally the only point where the first eigenvalue is double. The main result of the paper is then the following Theorem 1.1.…”

“…For what concerns us, the disk gives us the opportunity to begin to tackle the interesting question about how rare multiple eigenvalues are with respect to the position of the pole globally in the domain. This is a first contribution to carry on the analysis started in [6]. On the other hand, the present paper is not dealing directly with the aforementioned conjecture, but it presents arguments which may be useful towards it.…”

“…(2.1) Thus, it results (see [16,17,6] for deeper explanations) that 2A a is gauge equivalent to 0, as 2A a = −ie −iθa ∇e iθa = ∇θ a . We introduce the following antilinear and antiunitary operator…”

“…The paper [6] started the study of multiple eigenvalues of this operator with respect both to the position of the pole a ∈ Ω and the circulation α ∈ (0, 1). It shows that multiple eigenvalues in general occur, even if under suitable assumptions they are very rare locally with respect to the two parameters.…”

“…We will show that in fact they are really different by means of their Taylor expansion's first terms. The first derivatives of the two branches at the origin can be computed in terms of the corresponding eigenfunctions' asymptotic expansions in the spirit of [6]. This is the content of Section 3.…”