Vol. 7, No. 1, 2014

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ISSN: 1948-206X (e-only)
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The Aharonov–Bohm effect in spectral asymptotics of the magnetic Schrödinger operator

Gregory Eskin and James Ralston

Vol. 7 (2014), No. 1, 245–266
Abstract

We show that in the absence of a magnetic field the spectrum of the magnetic Schrödinger operator in an annulus depends on the cosine of the flux associated with the magnetic potential. This result follows from an analysis of a singularity in the “wave trace” for this Schrödinger operator, and hence shows that even in the absence of a magnetic field the magnetic potential can change the asymptotics of the Schrödinger spectrum; that is, the Aharonov–Bohm effect takes place. We also study the Aharonov–Bohm effect for the magnetic Schrödinger operator on a torus.

In memory of Hans Duistermaat

Keywords
Aharonov–Bohm effect, magnetic Schrödinger operator, wave trace
Mathematical Subject Classification 2010
Primary: 35P20, 35S30, 81S99
Milestones
Received: 25 February 2013
Revised: 30 October 2013
Accepted: 27 November 2013
Published: 7 May 2014
Authors
Gregory Eskin
Mathematics Department
University of California, Los Angeles
Los Angeles, CA 90095-1555
United States
James Ralston
Mathematics Department
University of California, Los Angeles
Los Angeles, CA 90095-1555
United States