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

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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.

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Acknowledgments

The authors have been partially supported by the Gruppo Nazionale per l’ Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), 2014 INdAM-GNAMPA research project “Stabilità spettrale e analisi asintotica per problemi singolarmente perturbati”, and by the project ERC Advanced Grant 2013 ’ with Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”. The authors would like to thank Susanna Terracini for her encouragement and for fruitful discussions.

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Correspondence to Veronica Felli.

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Communicated by A. Malchiodi.

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Abatangelo, L., Felli, V. Sharp asymptotic estimates for eigenvalues of Aharonov-Bohm operators with varying poles. Calc. Var. 54, 3857–3903 (2015). https://doi.org/10.1007/s00526-015-0924-0

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