Trace formulas for Dirac operators with applications to resonances

Cheng, Bobby Ho Yene (2020) Trace formulas for Dirac operators with applications to resonances. Doctoral thesis (PhD), University of Sussex.

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Abstract

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-Kreĭn 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.

Item Type: Thesis (Doctoral)
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QC Physics > QC0170 Atomic physics. Constitution and properties of matter Including molecular physics, relativity, quantum theory, and solid state physics > QC0174.12 Quantum theory. Quantum mechanics
Depositing User: Library Cataloguing
Date Deposited: 29 Apr 2020 13:23
Last Modified: 01 Jun 2021 08:50
URI: http://sro.sussex.ac.uk/id/eprint/91095

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