We consider a magnetic Schrödinger operator with magnetic field concentrated at one point (the pole) of a domain and half integer circulation, and we focus on the behavior of Dirichlet eigenvalues as functions of the pole. Although the magnetic field vanishes almost everywhere, it is well known that it affects the operator at the spectral level (the Aharonov–Bohm effect, Phys Rev (2) 115:485–491, 1959). Moreover, the numerical computations performed in (Bonnaillie-Noël et al., Anal PDE 7(6):1365–1395, 2014; Noris and Terracini, Indiana Univ Math J 59(4):1361–1403, 2010) show a rather complex behavior of the eigenvalues as the pole varies in a planar domain. In this paper, in continuation of the analysis started in (Bonnaillie-Noël et al., Anal PDE 7(6):1365–1395, 2014; Noris and Terracini, Indiana Univ Math J 59(4):1361–1403, 2010), we analyze the relation between the variation of the eigenvalue and the nodal structure of the associated eigenfunctions. We deal with planar domains with Dirichlet boundary conditions and we focus on the case when the singular pole approaches the boundary of the domain: then, the operator loses its singular character and the k-th magnetic eigenvalue converges to that of the standard Laplacian. We can predict both the rate of convergence and whether the convergence happens from above or from below, in relation with the number of nodal lines of the k-th eigenfunction of the Laplacian. The proof relies on the variational characterization of eigenvalues, together with a detailed asymptotic analysis of the eigenfunctions, based on an Almgren-type frequency formula for magnetic eigenfunctions and on the blow-up technique.

On the Aharonov–Bohm Operators with Varying Poles: The Boundary Behavior of Eigenvalues

NYS, Manon;Susanna Terracini
2015-01-01

Abstract

We consider a magnetic Schrödinger operator with magnetic field concentrated at one point (the pole) of a domain and half integer circulation, and we focus on the behavior of Dirichlet eigenvalues as functions of the pole. Although the magnetic field vanishes almost everywhere, it is well known that it affects the operator at the spectral level (the Aharonov–Bohm effect, Phys Rev (2) 115:485–491, 1959). Moreover, the numerical computations performed in (Bonnaillie-Noël et al., Anal PDE 7(6):1365–1395, 2014; Noris and Terracini, Indiana Univ Math J 59(4):1361–1403, 2010) show a rather complex behavior of the eigenvalues as the pole varies in a planar domain. In this paper, in continuation of the analysis started in (Bonnaillie-Noël et al., Anal PDE 7(6):1365–1395, 2014; Noris and Terracini, Indiana Univ Math J 59(4):1361–1403, 2010), we analyze the relation between the variation of the eigenvalue and the nodal structure of the associated eigenfunctions. We deal with planar domains with Dirichlet boundary conditions and we focus on the case when the singular pole approaches the boundary of the domain: then, the operator loses its singular character and the k-th magnetic eigenvalue converges to that of the standard Laplacian. We can predict both the rate of convergence and whether the convergence happens from above or from below, in relation with the number of nodal lines of the k-th eigenfunction of the Laplacian. The proof relies on the variational characterization of eigenvalues, together with a detailed asymptotic analysis of the eigenfunctions, based on an Almgren-type frequency formula for magnetic eigenfunctions and on the blow-up technique.
2015
339
3
1101
1146
https://www.scopus.com/inward/record.uri?eid=2-s2.0-84939654947&partnerID=40&md5=5a375bd788a7c72be52c283783e6bc4d
http://arxiv.org/abs/1411.5244
Aharonov-Bohm operators, Almgren monotonicity formula, spectral theory.
Bendetta, Noris; Nys, Manon; Terracini, Susanna
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1651851
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