Application of Bogomolny's transfer operator to a circular harmonic oscillator plus potential

Bogomolny's transfer operator has been used to find an analytical solution for the semiclassical energy eigenvalues of a simple two-dimensional integrable system. The system studied consists of a particle moving in an isotropic harmonic oscillator potential plus a potential. The classical trajectories are used to construct the transfer matrix, and an expression is derived for the eigenvalues of this matrix as a function of the energy. These eigenvalue curves yield the semiclassical energy eigenvalues for the quantum system, which turn out to be exactly the same as the results obtained by solving the Schrödinger equation. Some insight into this unexpected agreement is provided by considering an exact transfer operator. We show that when this operator is expanded in powers of Planck's constant, the leading term in the expansion is Bogomolny's transfer operator.