Periodic orbit quantization of bound chaotic systems

The authors present periodic orbit quantizations of the hyperbola billiard and the x2y2 potential. These two systems may be considered as belonging to the one-parameter family of potentials (x2y2)1a/. The quantum states are determined by means of the zeros of an expanded and truncated Selberg zeta function. The symmetries of the problem are considered and the Selberg zeta function is factorized into the irreducible representations of the symmetry group. The thus calculated eigenenergies are in good agreement with quantum mechanical calculations and converge when the number of terms in the expansion is increased. The results strongly indicate that the trace formula provides individual quantum eigenstates for chaotic systems.