Semiclassical quantization of nonseparable systems: A new look at periodic orbit theory

Abstract

A modified version of Gutzwiller’s periodic orbit theory of semiclassical eigenvalues is presented which eliminates some of the principal shortcomings of the original result. In particular, for a nonseparable system with N degrees of freedom the new quantum condition characterizes the eigenvalues by N quantum numbers (rather than just one), and it also reduces to the correct result in the limit that the system is a separable set of harmonic oscillators (whereas the original quantum condition does not). This new periodic orbit quantum condition is seen to bear an interesting relation to Marcus’ recent theory of semiclassical eigenvalues which involves manifolds of quasiperiodic trajectories.