SEMICLASSICAL QUANTIZATION OF MULTIDIMENSIONAL SYSTEMS

The solution of a stationary k-dimensional Schrodinger equation H-PSI = E-PSI in the semiclassical limit HBAR --> 0 is reduced to a discrete (k - 1)-dimensional quantum map psi = T-psi where the integral kernel (the matrix) T is built through classical trajectories corresponding to the classical Poincare map of the given problem. High-excited energy eigenvalues obey the quantization condition zeta(s)(E) = 0 where the function zeta(s)(E) = det(1 - T) coincides with the Selberg zeta function defined as the product over primitive per-iodic orbits. Different properties of the constructed Poincare map are discussed, in particular the Riemann-Siegel relation for the dynamical zeta function.