Semiclassical quantization of multidimensional systems

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Abstract
The solution of a stationary k-dimensional Schrodinger equation H Psi =E Psi in the semiclassical limit h(cross) to 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 periodic orbits. Different properties of the constructed Poincare map are discussed, in particular the Riemann-Siegel relation for the dynamical zeta function.

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