The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates

Abstract

We investigate $L^1\to L^{\infty }$ dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\frac{1}{2}}$ decay rate, which may be improved to $t^{-\frac{1}{2}-\gamma }$ for any $0\le \gamma <\frac{3}{2}$ at the cost of spatial weights. We classify the structure of threshold obstructions as being composed of a two dimensional space of p-wave resonances and a finite dimensional space of eigenfunctions at zero energy. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate except for a finite-rank piece. While in the case of a threshold eigenvalue only, the natural decay rate is preserved. In both cases we show that the decay rate may be improved at the cost of spatial weights.

arXiv comment42 pages, submitted. Added an improved version of the high energy result in Theorem 1.2, and its proof in Section 8

References in corpus (10)

• Wave packets in Honeycomb Structures and Two-Dimensional Dirac Equations
• Dispersive estimates for Schroedinger operators in dimension two
• The cubic Dirac equation: Small initial data in $H^{\frac{1}{2}}({\mathbb{R}}^2)$
• Stable directions for small nonlinear Dirac standing waves
• Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy
• A weighted dispersive estimate for Schrödinger operators in dimension two
• The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions
• A weighted estimate for two dimensional Schrodinger, matrix schrodinger and wave equations with resonance of first kind at zero energy
• Dispersive estimates for massive Dirac operators in dimension two
• Time decay estimates for the wave equation with potential in dimension two

Cited by in corpus (2)

• On the one dimensional Dirac equation with potential
• On Absence of Threshold Resonances for Schrodinger and Dirac Operators