On the Leading Term of the Eigenvalue Variation for Aharonov--Bohm Operators with a Moving Pole

Abstract

Abstract. We study the behavior of eigenvalues for magnetic Aharonov-Bohm operators with half-integer circulation and Dirichlet boundary conditions in a planar domain. We analyse the leading term in the Taylor expansion of the eigenvalue function as the pole moves in the interior of the domain, proving that it is a harmonic homogeneous polynomial and detecting its exact coefficients.

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

“…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].…”

“…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). (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).…”

“…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.…”

“…Theorem C. 16. 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).…”

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

“…We refer to papers [12,13,18,17,19,20,21,22,23] for details on the deep relation between behavior of eigenfunctions, their nodal domains, and spectral minimal partitions. Related to this, the investigation carried out in [2,3,4,14,26,29] highlighted a strong connection between nodal properties of eigenfunctions and asymptotic expansion of the function which maps the position of the pole a in the domain to eigenvalues of the operator (i∇ + A a ) 2 (see also [1,Section 3] for a brief overview).…”