AKP energy levels by a simple shooting scheme for a periodic orbit

We revisit the periodic orbit theory for the anisotropic Kepler problem, which is an important playground for quantum chaos. In order to explore the periodic orbit, Gutzwiller devised an iteration scheme which computes the Fourier coefficients of the orbit iteratively. Here we note, in a nutshell, that all one needs is the primary periodic orbit. We propose an alternative scheme taking into account the symmetry of the target trajectory and the scaling property of the AKP equation of motion. We show that a simple shooting scheme gives the final periodic orbit almost immediately.