Semiclassical calculation of bound states in a multidimensional system. Use of Poincaré’s surface of section

Abstract

A method utilizing integration along invariant curves on Poincaré’s surfaces of section is described for semiclassical calculation of eigenvalues. The systems treated are dynamically nonseparable and are quasiperiodic. Use is also made of procedures developed in the previous paper of this series. The calculated eigenvalues for an anharmonically coupled pair of oscillators agree well with the exact quantum values. They also agree with the previous semiclassical calculations in this laboratory, which instead used integrations along the caustics. The present paper increases the number of systems capable of being treated. Using numerical counter examples for nondegenerate systems, it is also shown that an alternative view in the literature, which assumes that periodic trajectories alone suffice, leads to wrong results for the individual eigenvalues.