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The Dirac Equation in Two Dimensions: Dispersive Estimates and Classification of Threshold Obstructions

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    M. Burak Erdoğan

  2. Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, IN, 47803, USA

    William R. Green

Authors
  1. M. Burak Erdoğan
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  2. William R. Green
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Corresponding author

Correspondence to M. Burak Erdoğan.

Additional information

Communicated by W. Schlag

The first author was partially supported by the NSF Grant DMS-1501041.

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Erdoğan, M.B., Green, W.R. The Dirac Equation in Two Dimensions: Dispersive Estimates and Classification of Threshold Obstructions. Commun. Math. Phys. 352, 719–757 (2017). https://doi.org/10.1007/s00220-016-2811-8

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  • DOI: https://doi.org/10.1007/s00220-016-2811-8

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