Semiclassical Quantization of the Diamagnetic Hydrogen Atom at the Field-free Ionization Threshold

Semiclassical quantization methods have been successfully applied to the hydrogen atom in a magnetic field at low excitation energies where the classical dynamics is near integrable or weakly chaotic, and at high energies far above the field-free ionization threshold where the classical dynamics is completely hyperbolic. However, the energy region around E = 0 has so far resisted any attempts for semiclassical quantization. We present an efficient multi-shooting algorithm for the periodic orbit search and a novel quantization technique based on harmonic inversion with constraints which allow, for the first time, the semiclassical quantization of the diamagnetic hydrogen atom in the very challenging energy region.