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Trace formulas for Dirac operators with applications to resonances

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posted on 2023-06-09, 21:10 authored by Bobby Cheng
Motivated by their appearance in the physical sciences, scattering resonances of the three-dimensional Dirac operator perturbed by a real-valued, smooth, compactly supported, electric potential are studied. The potentials are 4×4 matrix-valued, multiplication operators. Under a prescribed mapping, the cut-off full resolvent is extended meromorphically from the physical half-plane to the whole complex plane. The poles that lie in the un-physical plane are defined as resonances for the perturbed Dirac operator. This thesis presents basic properties of the free and full Dirac resolvents and introduces the resonances that the latter creates. Particular attention is paid to the resonances appearing at the threshold points when the full resolvent is studied near these limits. The scattering matrix is analysed as a mapping between solutions of the Dirac eigenvalue problem and then used to establish the Birman-Krein formula, which relates the trace difference between functions of the full and free Dirac operators. In turn, a Poisson wave trace formula in the distributional sense is established via an upper bound counting function and factorization of the scattering matrix determinant. Both trace formulas are generalized such that resonances appearing at the threshold points are considered. Finally, under further restrictions on the potential, the existence of infinitely many Dirac resonances is proved as an application of our trace formulas.

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  • Published version

Pages

95.0

Department affiliated with

  • Mathematics Theses

Qualification level

  • doctoral

Qualification name

  • phd

Language

  • eng

Institution

University of Sussex

Full text available

  • Yes

Legacy Posted Date

2020-04-29

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