False singularities in partial sums over closed orbits

Abstract

The contribution to the spectral density from all the repetitions of a single classical closed orbit can have singularities that are not eigenenergies of the system. Only by summing over all different topologies of orbits do these false singularities get replaced by 8 peaks at the true eigenenergies. We illustrate this with an example where the expansion and sum over repetitions are exact rather than asymptotic. In studies of the energy spectra of quantum systems much use has been made of the semiclassical expansion of the density of states as a sum over the closed orbits of the corresponding classical system (Gutzwiller 1967, 1969, 1970, 1971, 1978, Balian and Bloch 1972, Berry and Tabor 1976, Berry 1983). It is known that this series has peculiar convergence properties and that all the periodic orbits must be included if there is to be any hope of getting the correct structure of any one singularity. In this letter we consider a case for which the expansion is exact (i.e. there are no asymptotic approxima- tions), non-trivial (i.e. there are infinitely many topologically different primitive periodic orbits) and for which the sum over repetitions of a given primitive orbit can be performed analytically. The system is the billiard on a torus: a spinless particle moving freely in a rectangular cell with periodic boundary conditions. This is integrable and has eigenenergies given, in suitable units, by