The Goryachev-Chaplygin gyrostat in quantum mechanics

Abstract

The authors investigate the quantum mechanical analogue of the classical integrable system named in the title. The Goryachev-Chaplygin (GC) gyrostat is a generalisation of the GC top, where the Coriolis interaction is taken into account. The problem is formulated in terms of the Euclid E(3) group. The integrals of motion are derived. The separation of variables is based on the connection between the degenerated representation of the E(3) group and the special representation of the SO(3,2) group. Some numerical results on the spectra of energy and separation constant are presented. The strong field limit for the integrals of motion is considered in detail, as well as a correlation diagram connecting the states in the limits of weak and strong fields. It appears that the dependence of energy and separation constant on the strength of the Coriolis interaction has the zone behaviour.