Semiclassical extension of Levinson's theorem to a particle interacting with a harmonic oscillator

Abstract

A semiclassical extension of Levinson's theorem to a model three-body system is presented. The system chosen is a particle interacting with a harmonic oscillator for collinear configurations. The theorem predicts the number N of three-body bound states that a potential energy surface can support, and this number is calculated for a set of surfaces for this model collinear system. The calculation involves the extrapolation of the S matrix to zero collision energy, where S matrix elements needed in the extrapolation are found in the classical limit. For comparison, N is also calculated in a purely classical fashion from the multidimensional phase space volume integral. The results from the semiclassical and purely classical calculations are in fairly good agreement.