Closed orbits and the regular bound spectrum

Abstract

The energy levels of systems whose classical motion is multiply periodic are accurately given by the quantum conditions of Einstein, Brillouin & Keller (E. B. K.). We transform the E. B. K. conditions into a representation of the spectrum in terms of a ‘topological sum’ involving only the closed classical orbits; the theory applies equally to separable and non-separable systems; stability parameters are not involved. Significant contributions come from complex closed orbits which however have real constants of the motion. Clustering of levels on different scales is demonstrated by smoothing the spectrum using the formal device, due to Balian & Bloch, of adding a variable imaginary part to the energy. The topological sum is shown to agree very well with exactly-computed spectra for circular and spherical potential wells with repulsive core.