On the spectrum of the magnetic Dirac operator

TL;DR

This paper investigates the mathematical properties of the magnetic Dirac operator, providing eigenvalue estimates and explicit spectrum calculations on certain manifolds, advancing understanding of its fundamental characteristics.

Contribution

It offers new eigenvalue bounds and explicit spectral computations for the magnetic Dirac operator on specific geometric settings.

Findings

Eigenvalue estimates for the magnetic Dirac operator on closed manifolds

Explicit spectrum calculations on flat torus and 3-sphere

Identification of spectral properties depending on magnetic field choices

Abstract

The magnetic Dirac operator describes the relativistic motion of a charged particle in a magnetic field. Although this operator got a lot of attention in physics many of its fundamental mathematical properties remain unexplored and this article is a first step towards filling this gap. To this end we provide a number of eigenvalue estimates for the magnetic Dirac operator on closed Riemannian manifolds and explicitly compute its spectrum for specific choices of the magnetic field on the flat torus and on the three-dimensional round sphere.