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K. Hornberger,
Spectral Properties of Magnetic Edge States,
Dissertation der Fakultät für Physik der Ludwig-Maximilians-Universität München, 2001


We study the spectral properties of magnetic edge states, which exist in the interior and exterior spectra of magnetic quantum billiards. To quantize the billiards, the boundary integral method is extended to the magnetic problem and to general boundary conditions. By virtue of an analytical regularization of the (hyper-)singular integral operators, we obtain for the first time precise quantum spectra even in the extreme semiclassical regime. The insight gained into the structure of the spectral determinant enables us to derive the semiclassical trace formula for magnetic billiards from first principles. We propose a spectral measure, which quantifies the intuitive notion of edge states. This density of edge states allows to analyse the interior and exterior spectra statistically, and to describe them semiclassically. We find strong, non-trivial cross-correlations between the interior and exterior spectra. These correlations are based on a duality of the corresponding classical dynamics. Our analytical results are confirmed by extensive numerical studies.

(187 pages, 52 figures)  [ps 7.0MByte, pdf 11.7MByte, LMU]