Vol. 7, No. 6, 2014

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ISSN: 1948-206X (e-only)
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On the eigenvalues of Aharonov–Bohm operators with varying poles

Virginie Bonnaillie-Noël, Benedetta Noris, Manon Nys and Susanna Terracini

Vol. 7 (2014), No. 6, 1365–1395
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.

Keywords
magnetic Schrödinger operators, eigenvalues, nodal domains
Mathematical Subject Classification 2010
Primary: 35J10, 35P20, 35Q40, 35Q60, 35J75
Milestones
Received: 3 October 2013
Revised: 22 February 2014
Accepted: 1 April 2014
Published: 18 October 2014
Authors
Virginie Bonnaillie-Noël
IRMAR, ENS Rennes
Université de Rennes 1
CNRS, UEB
av. Robert Schuman
35170 Bruz
France
Benedetta Noris
Laboratoire de Mathématiques
Université de Versailles-St Quentin
45 avenue des Etats-Unis
78035 Versailles
France
Manon Nys
Département de Mathématiques
Université Libre de Bruxelles (ULB)
Boulevard du triomphe
B-1050 Bruxelles
Belgium
Dipartimento di Matematica e Applicazioni
Università degli Studi di Milano-Bicocca
via Bicocca degli Arcimboldi 8
20126 Milano
Italy
Susanna Terracini
Dipartimento di Matematica “Giuseppe Peano”
Università di Torino
Via Carlo Alberto 10
20123 Torino
Italy