Path-generating function of the diamagnetic Kepler problem

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Abstract
In the framework of Gutzwiller’s theory for nonintegrable Hamiltonian systems the diagonal part of the Green’s function is expressed as a sum over classical trajectories which are closed in configuration space. We can distinguish two types of recurrent trajectories, those which are in the neighborhood of a periodic orbit and those whose lengths vanish as the allotted time continuously approaches zero. For the hydrogen atom in a uniform magnetic field we identify both types of orbits in the path-generating function, i.e, the Fourier transform of the Green’s function with respect to 1/ħ. Projections of the Green’s function onto coherent states eliminate the contributions from recurrent but nonperiodic orbits. The corresponding phase-space representation of the path-generating function shows a strong localization along the classical periodic orbits.

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