Daniel Elton Lancaster University

Spectral reduction and zero modes of the Weyl–Dirac operator

Abstract:

For a given magnetic potential A one can define the Weyl–Dirac operator σ·(-i∇-A)$ on ℝ³. The question of whether or not this operator possess a zero-energy L² eigenfunction, or zero mode, is quite subtle. We review some known results, including recent work on a conjecture for asymptotics in the strong field (or semi-classical) regime. In particular, we look at cases where the magnetic field is parallel to a conformal Killing field; such a symmetry can be exploited to reduce spectral questions to the consideration of a family of two dimensional Dirac-type problems. A complication arises as the basic two dimensional object involves a quotient by a typically non-proper action. Nevertheless, estimates for the existence of zero modes of the original Dirac operator can be obtained.