Symmetry breaking and bifurcations in the periodic orbit theory. I - Elliptic billiard

We derive an analytical trace formula for the level density of two-dimensio nal elliptic billiards using an improved stationary phase method. The resul t is a continuous function of the deformation parameter (eccentricity) thro ugh all bifurcation points of the short diameter orbit and its repetitions, and possesses the correct limit of circular billiard at zero eccentricity. Away from the circular limit and the bifurcations, it reduces to the usual (extended) Gutzwiller trace formula, which for the leading-order families of periodic orbits is identical to the result of Berry and Tabor. We show t hat the circular disk limit of the diameter-orbit contribution is also reac hed through contributions from closed (periodic and non-periodic) orbits of the hyperbolic type with an even number of reflections from the boundary. We obtain the Maslov indices depending on deformation and energy in terms o f the phases of the complex error and Airy functions. We find enhancement o f the amplitudes near the common bifurcation points of short-diameter and h yperbolic orbits. The calculated semiclassical level densities and shell en ergies are in good agreement with the quantum mechanical ones.