Exact solution of the Goryachev-Chaplygin problem in quantum mechanics

Abstract

The quantum mechanical analogue of the classical integrable system, originally founded by Goryachev and Chaplygin (1900), is considered in detail. The problem is formulated in terms of the Euclid E(3) group. The Euler-Poisson equations of motion and their integrals are derived. The determination of the spectra of the integrals of motion is equivalent to construction of the special basis of representation of E(3). The eigenvalue problem admits separation in new variables, which are closely connected with two boson creation-annihilation operators. They are the same as in the Majorana representation of the Lorentz group. In the new variables the quantum mechanical equations of motion look similar to the classical ones. The constants of motion are determined by the spectral problems for the two Jacobi-type tridiagonal infinite matrices. Some numerical results are given.