Remark on Spectral Rigidity for Magnetic Schr\"odinger Operators

TL;DR

This paper provides a straightforward proof of Guillemin's theorem, showing that the magnetic field on a torus can be uniquely determined from the spectrum of the magnetic Schr"odinger operator.

Contribution

It offers a simplified proof of a key spectral rigidity result for magnetic Schr"odinger operators on the torus, enhancing understanding of inverse spectral problems.

Findings

Magnetic field on the torus is spectrally determined.

Simplified proof of Guillemin's theorem.

Spectral data uniquely determines the magnetic field.

Abstract

We give a simple proof of Guillemin's theorem on the determination of the magnetic field on the torus by the spectrum of the corresponding Schr\"odinger operator.