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Limiting Absorption Principle and Strichartz Estimates for Dirac Operators in Two and Higher Dimensions

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Abstract

In this paper we consider Dirac operators in \({\mathbb{R}^n}\), \({n \ge 2}\), with a potential V. Under mild decay and continuity assumptions on V and some spectral assumptions on the operator, we prove a limiting absorption principle for the resolvent, which implies a family of Strichartz estimates for the linear Dirac equation. For large potentials the dynamical estimates are not an immediate corollary of the free case since the resolvent of the free Dirac operator does not decay in operator norm on weighted L2 spaces as the frequency goes to infinity.

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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, University of Cincinnati, Cincinnati, OH, 45221, USA

    Michael Goldberg

  3. 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. Michael Goldberg
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  3. William R. Green
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Corresponding author

Correspondence to William R. Green.

Additional information

Communicated by W. Schlag

The first author was partially supported by NSF Grant DMS-1501041. The second author is supported by Simons Foundation Grant 281057. The third author is supported by Simons Foundation Grant 511825 and acknowledges the support of a Rose-Hulman summer professional development Grant.

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Burak Erdoğan, M., Goldberg, M. & Green, W.R. Limiting Absorption Principle and Strichartz Estimates for Dirac Operators in Two and Higher Dimensions. Commun. Math. Phys. 367, 241–263 (2019). https://doi.org/10.1007/s00220-018-3231-8

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  • DOI: https://doi.org/10.1007/s00220-018-3231-8

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