Periodic Orbit Theory Revisited in the Anisotropic Kepler Problem

The Gutzwiller's trace formula for the anisotropic Kepler problem is Fourier transformed with a convenient variable $u=1/\sqrt{-2E}$ which takes care of the scaling property of the AKP action $S(E)$. Proper symmetrization procedure (Gutzwiller's prescription) is taken by the introduction of half-orbits that close under symmetry transformations so that the two dimensional semi-classical formulas match correctly the quantum subsectors $m^\pi=0^+$ and $m^\pi=0^-$. Response functions constructed from half orbits in the POT side are explicitly given. In particular the response function $g_X$ from $X$-symmetric half orbit has an amplitude where the root of the monodromy determinant is inverse hyperbolic. The resultant {\it weighted densities of periodic orbits $D^{m=0}_e(\phi)$ and $D^{m=0}_o(\phi)$} from both quantum subsectors show peaks at the actions of the periodic orbits with correct peak heights corresponding to the Lyapunov exponents of them. The formulation takes care of the cut off of the energy levels, and the agreement between the $D(\phi)$s of QM and POT sides is observed independent for the choice of the cut off. The systematics appeared in the densities of the periodic orbits is explained in terms of features of periodic orbits. It is shown that from quantum energy levels one can extract the information of AKP periodic orbits, even the Lyapunov exponents-- the success of inverse quantum chaology in AKP.