Dispersive estimates for massive Dirac operators in dimension two

TL;DR

This paper investigates decay rates of solutions to the massive two-dimensional Dirac operator with electric potential, establishing decay bounds under various spectral conditions and highlighting the impact of resonances.

Contribution

It provides new dispersive estimates for the 2D massive Dirac operator, including cases with threshold resonances and regular thresholds, extending previous understanding.

Findings

Decay rate of t^{-1} in L^1 to L^∞ when thresholds are regular

Faster decay rate of t^{-1} (log t)^{-2} with spatial weights in certain cases

Presence of s-wave resonances affects decay bounds

Abstract

We study the massive two dimensional Dirac operator with an electric potential. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if the threshold energies are regular. We also show these bounds hold in the presence of s-wave resonances at the threshold. We further show that, if the threshold energies are regular that a faster decay rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of logarithmic spatial weights. The free Dirac equation does not satisfy this bound due to the s-wave resonances at the threshold energies.