Asymptotic behaviour of energy levels of a quantum free particle in an elliptic sector

Abstract

We study quantum solution for a free particle in a domain bounded by an ellipse and arc(s) of confocal hyperbola(s). We found asymptotic behaviour of energy levels as focal distance tends to zero and show how it is related to the energy levels of limiting wedge billiard. Classical billiard system in the considered domains is integrable due to existence of an additional conserved quantity. There is a corresponding quantum counterpart, and we calculate its eigenvalues.