Sharp asymptotic estimates for eigenvalues of Aharonov-Bohm operators with varying poles

Abstract

We investigate the behavior of eigenvalues for a magnetic Aharonov-Bohm operator with half-integer circulation and Dirichlet boundary conditions in a planar domain. We provide sharp asymptotics for eigenvalues as the pole is moving in the interior of the domain, approaching a zero of an eigenfunction of the limiting problem along a nodal line. As a consequence, we verify theoretically some conjectures arising from numerical evidences in preexisting literature. The proof relies on an Almgren-type monotonicity argument for magnetic operators together with a sharp blow-up analysis.

Sharp convergence rate of eigenvalues in a domain with a shrinking tube

“…Moreover, from the simplicity assumption (4) and Fredholm Alternative, one can easily prove the following result (see e.g. [1] for details for a similar operator).…”

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

“…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.…”

“…The existence part is proved by taking Φ k = ψ k + w k . To prove uniqueness, one can argue by contradiction exploiting the Hardy Inequality (see [1,Proposition 4.3] for a detailed proof in a similar problem).…”

“…Proof. Once the negative sign of M (ε, u, λ) is established (Lemma C.5) the proof proceeds as in [1,Section 5].…”

“…Proof. We omit the proof which can be derived from the monotonicity result given in Lemma C.6 following the same argument as Lemma C.10; for details in an analogous problem we refer to [1,Theorem 5.9].…”