Uniform approximation for bifurcations of periodic orbits with high repetition numbers

Abstract

The semiclassical contribution of a periodic orbit to the quantum density of states diverges when the orbit bifurcates. In this case one has to apply approximations which are uniformly valid both in and a parameter which describes the distance to the bifurcation. The form of the approximation depends on the repetition number m of the orbit that bifurcates. In a two-dimensional system, the approximations are different for m = 1 up to m = 5, and for they have the same form as for m = 5. In this article, we consider the case which occurs first when an integrable system is perturbed. A uniform approximation for the contribution to the spectral density is derived, which in the limit of large reduces to a sum of semiclassical contributions of isolated periodic orbits.