Periodic Orbit Quantization beyond the Semiclassical Theory

A quantum generalization of the semiclassical theory of Gutzwiller is given. The new formulation leads to systematic orbit-by-orbit inclusion of higher $\ensuremath{\Elzxh}$ contributions to the spectral determinant. We apply the theory to billiard systems, and compare the periodic orbit quantization including the first $\ensuremath{\Elzxh}$ contribution to the exact quantum mechanical results.