Approximation of Dirac Operators with Delta-shell Potentials in the Norm Resolvent Sense
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Beschreibung
The present thesis is devoted to the approximation of Dirac operators with δ-shell potentials supported on the boundary of a two- or three-dimensional C^2-domain. These singular potentials are used as idealized replacements for potentials which are strongly localized in a neighbourhood of the support of the δ-shell potential and they often simplify the spectral analysis. To justify the usage of such potentials it is essential to prove that Dirac operators with δ-shell potentials can be approximated by Dirac operators with strongly localized potentials in a way which transfers the spectral properties. The most important contribution of this thesis is the establishment of conditions for the convergence of Dirac operators with strongly localized potentials in the norm resolvent sense. This type of convergence implies that the spectrum of the Dirac operator with δ-shell potential can be completely characterized by the spectra of the approximating operators and vice versa. In the special case of electrostatic and Lorentz scalar δ-shell potentials an explicit convergence condition is provided. Furthermore, counterexamples which imply the sharpness of this condition are also presented.
Autor*in
Christian Stelzer
Themen in »Approximation of Dirac Operators with Delta-shell Potentials in the Norm Resolvent Sense«
Dirac operators delta-shell potentials strongly localized potentials norm resolvent convergence convergence conditions by parts formula boundary element methods FMM adaptivity parallelization HPC