Closed orbits and spatial density oscillations in the circular billiard

We present a case study for the semiclassical calculation of the oscillations in the particle and kinetic-energy densities for the two-dimensional circular billiard. For this system, we can give a complete classification of all closed periodic and non-periodic orbits. We discuss their bifurcations under variation of the starting point r and derive analytical expressions for their properties such as actions, stability determinants, momentum mismatches and Morse indices. We present semiclassical calculations of the spatial density oscillations using a recently developed closed-orbit theory (Roccia and Brack 2008 Phys. Rev. Lett. 100 200408), employing standard uniform approximations from perturbation and bifurcation theory, and test the convergence of the closed-orbit sum.