On the spectrum of the magnetic Dirac operator

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.