Cheng, Bobby Ho Yene.pdf (813.36 kB)
Trace formulas for Dirac operators with applications to resonances
thesis
posted on 2023-06-09, 21:10 authored by Bobby ChengMotivated 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.
History
File Version
- Published version
Pages
95.0Department affiliated with
- Mathematics Theses
Qualification level
- doctoral
Qualification name
- phd
Language
- eng
Institution
University of SussexFull text available
- Yes
Legacy Posted Date
2020-04-29Usage metrics
Categories
No categories selectedKeywords
Licence
Exports
RefWorks
BibTeX
Ref. manager
Endnote
DataCite
NLM
DC