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arXiv · 1807.00219

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

Burak Erdogan·Michael Goldberg·William R. Green
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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 $t^{-\frac12}$ decay rate, which may be improved to $t^{-\frac12-\gamma}$ for any $0\leq \gamma<\frac{3}{2}$ 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.

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Burak Erdogan, Michael Goldberg, William R. Green. 2018-06-30. The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates. https://arxiv.org/abs/1807.00219

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