Semiclassical Transition from an Ellipticalto an Oval

Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when h is small in comparison to relevant actions or action diierences in the corresponding classical system. In many situations, however, action diierences can be arbitrarily small and then uniform approximations are more appropriate. In the present paper we examine diierent uniform approximations for describing the spectra of integrable systems and systems with mixed phase space. This is done on the example of two billiard systems, an elliptical billiard and a deformation of it, an oval billiard. We derive a trace formula for the ellipse which involves a uniform approximation for the Maslov phases near the separatrix, and a uniform approximation for tori of periodic orbits close to a bifurcation. We then examine how the trace formula is modiied when the ellipse is deformed into an oval. This involves uniform approximations for the break-up of tori and uniform approximations for bifurcations of periodic orbits. Relations between diierent uniform approximations are discussed. PACS numbers: 03.65.Ge Solutions of wave equations: bound states. 03.65.Sq Semiclassical theories and applications. 05.45.+b Theory and models of chaotic systems.