#### Vol. 7, No. 6, 2014

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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