Spectral Properties of Magnetic Edge States

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.