A Canonical Gauge for Computing of Eigenpairs of the Magnetic Schr\"odinger Operator

TL;DR

This paper introduces a canonical magnetic gauge for the magnetic Schrödinger operator, enabling more efficient and stable numerical computation of eigenpairs by transforming the problem into a less oscillatory form.

Contribution

The paper proposes a novel canonical gauge computed via a Poisson problem, improving numerical approximation of eigenpairs for the magnetic Schrödinger operator.

Findings

More accurate eigenpair computation with the canonical gauge

Enhanced numerical stability and efficiency

Eigenvectors become less oscillatory

Abstract

We consider the eigenvalue problem for the magnetic Schr\"odinger operator and take advantage of a property called gauge invariance to transform the given problem into an equivalent problem that is more amenable to numerical approximation. More specifically, we propose a canonical magnetic gauge that can be computed by solving a Poisson problem, that yields a new operator having the same spectrum but eigenvectors that are less oscillatory. Extensive numerical tests demonstrate that accurate computation of eigenpairs can be done more efficiently and stably with the canonical magnetic gauge.