Periodic orbit theory revisited in the anisotropic Kepler problem

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