Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 1 (2014), 245-266.

The Aharonov–Bohm effect in spectral asymptotics of the magnetic Schrödinger operator

Gregory Eskin and James Ralston

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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.

Article information

Source
Anal. PDE, Volume 7, Number 1 (2014), 245-266.

Dates
Received: 25 February 2013
Revised: 30 October 2013
Accepted: 27 November 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731473

Digital Object Identifier
doi:10.2140/apde.2014.7.245

Mathematical Reviews number (MathSciNet)
MR3219506

Zentralblatt MATH identifier
1293.35186

Subjects
Primary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 35S30: Fourier integral operators 81S99: None of the above, but in this section

Keywords
Aharonov–Bohm effect magnetic Schrödinger operator wave trace

Citation

Eskin, Gregory; Ralston, James. The Aharonov–Bohm effect in spectral asymptotics of the magnetic Schrödinger operator. Anal. PDE 7 (2014), no. 1, 245--266. doi:10.2140/apde.2014.7.245. https://projecteuclid.org/euclid.apde/1513731473


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