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

par Abatangelo, Laura;Felli, Veronica;Noris, Benedetta;Nys, Manon
Référence Journal of functional analysis, 273, 7, page (2428-2487)
Publication Publié, 2017-10

Article révisé par les pairs

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Résumé :

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.