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2014 On the eigenvalues of Aharonov–Bohm operators with varying poles
Virginie Bonnaillie-Noël, Benedetta Noris, Manon Nys, Susanna Terracini
Anal. PDE 7(6): 1365-1395 (2014). DOI: 10.2140/apde.2014.7.1365

Abstract

We consider a magnetic operator of Aharonov–Bohm type with Dirichlet boundary conditions in a planar domain. We analyze the behavior of its eigenvalues as the singular pole moves in the domain. For any value of the circulation we prove that the k-th magnetic eigenvalue converges to the k-th eigenvalue of the Laplacian as the pole approaches the boundary. We show that the magnetic eigenvalues depend in a smooth way on the position of the pole, as long as they remain simple. In case of half-integer circulation, we show that the rate of convergence depends on the number of nodal lines of the corresponding magnetic eigenfunction. In addition, we provide several numerical simulations both on the circular sector and on the square, which find a perfect theoretical justification within our main results, together with the ones by the first author and Helffer in Exp. Math. 20:3 (2011), 304–322.

Citation

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Virginie Bonnaillie-Noël. Benedetta Noris. Manon Nys. Susanna Terracini. "On the eigenvalues of Aharonov–Bohm operators with varying poles." Anal. PDE 7 (6) 1365 - 1395, 2014. https://doi.org/10.2140/apde.2014.7.1365

Information

Received: 3 October 2013; Revised: 22 February 2014; Accepted: 1 April 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1301.35069
MathSciNet: MR3270167
Digital Object Identifier: 10.2140/apde.2014.7.1365

Subjects:
Primary: 35J10 , 35J75 , 35P20 , 35Q40 , 35Q60

Keywords: Eigenvalues , magnetic Schrödinger operators , nodal domains

Rights: Copyright © 2014 Mathematical Sciences Publishers

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