Applications of classical periodic orbit theory to circular billiards with small scattering centers

Abstract

Mesoscopic systems like metallic clusters in three dimensions as well as quantum dots in two dimensions have raised a particular interest in the semiclassical interpretation of shell effects. We use periodic orbit theory to calculate the density of states for two-dimensional circular billiards. When a singular magnetic flux line is added at the center of the disk, we show that the Aharonov-Bohm (AB) effect manifests itself through the cancellation of periodic orbits and the appearance of a new signal in the Fourier transform of the quantum density of states. The same effects are also found analytically for a two-dimensional harmonic oscillator potential with flux line. Finally, we show that a homogeneous magnetic field B perpendicular to the disk plane leads to B-periodic oscillations of the level density, which recently have also been observed experimentally in two-dimensional circular quantum dots.