The massless Dirac equation in two dimensions: zero-energy obstructions and dispersive estimates

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\leq \gamma<\frac{3}...

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Bibliographic Details
Published inJournal of spectral theory Vol. 11; no. 3; pp. 935 - 979
Main Authors Erdoğan, M. Burak, Goldberg, Michael, Green, William R.
Format Journal Article
LanguageEnglish
Published European Mathematical Society Publishing House 01.01.2021
Subjects
Dirac equation
Eigenvalues
Mathematical research
Resonance
United States
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Summary: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\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.
ISSN:1664-039X
1664-0403
DOI:10.4171/jst/362