Elsevier

Journal of Functional Analysis

Volume 273, Issue 7, 1 October 2017, Pages 2428-2487
Journal of Functional Analysis

Sharp boundary behavior of eigenvalues for Aharonov–Bohm operators with varying poles

https://doi.org/10.1016/j.jfa.2017.06.023Get rights and content
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Abstract

In this paper, we investigate the behavior of the eigenvalues of a magnetic Aharonov–Bohm operator with half-integer circulation and Dirichlet boundary conditions in a bounded planar domain. We establish a sharp relation between the rate of convergence of the eigenvalues as the singular pole is approaching a boundary point and the number of nodal lines of the eigenfunction of the limiting problem, i.e. of the Dirichlet-Laplacian, ending at that point. The proof relies on the construction of a limit profile depending on the direction along which the pole is moving, and on an Almgren-type monotonicity argument for magnetic operators.

MSC

35P15
35J10
35J75
35B40
35B44

Keywords

Aharonov–Bohm operators
Almgren monotonicity formula
Spectral theory

Cited by (0)

The authors are partially supported by the project ERC Advanced Grant 2013 No. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems – COMPAT”. L. Abatangelo and V. Felli are partially supported by the 2015 INdAM-GNAMPA research project “Operatori di Schrödinger con potenziali elettromagnetici singolari: stabilità spettrale e stime di decadimento”. V. Felli is partially supported by PRIN-2012-grant “Variational and perturbative aspects of nonlinear differential problems”. B. Noris was supported by the projects MIS F.4508.14 (FNRS) & ARC AUWB-2012-12/17-ULB1-IAPAS.