The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions

Abstract

We investigate dispersive estimates for the two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a $t^{-1}$ decay rate as an operator from the Hardy space $H^1$ to $BMO$, the space of functions of bounded mean oscillation. This estimate, along with the $L^2$ conservation law allows one to deduce a family of Strichartz estimates. We classify the structure of threshold obstructions as being composed of s-wave resonances, p-wave resonances and eigenfunctions. We show that, as in the case of the Schrödinger evolution, the presence of a threshold s-wave resonance does not destroy the $t^{-1}$ decay rate. As a consequence of our analysis we obtain a limiting absorption principle in the neighborhood of the threshold, and show that there are only finitely many eigenvalues in the spectral gap.

Revised according to referee suggestions. Added several references and expanded the discussion of the spectral theory of the perturbed Dirac operator. 40 pages

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• On Absence of Threshold Resonances for Schrodinger and Dirac Operators
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• The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions