LEVEL STATISTICS FOR CONTINUOUS ENERGY-SPECTRA WITH APPLICATION TO THE HYDROGEN-ATOM IN CROSSED ELECTRIC AND MAGNETIC-FIELDS

The statistical analysis of energy levels, a powerful tool in the stud y of quantum systems, is applicable to discrete spectra. Here we propo se an approach to carry level statistics over to continuous energy spe ctra, paradoxical as this may sound at first. The approach proceeds in three steps, first a discretization of the spectrum by cutoffs, then a statistical analysis of the resulting discrete spectra, and finally a determination of the limit distributions as the cutoffs are removed. In this way the notions of Wigner and Poisson distributions for neare st-neighbor spacing (NNS), usually associated with quantum chaos and r egularity, can be carried over to systems with a purely continuous ene rgy spectrum. The approach is demonstrated for the hydrogen atom in pe rpendicular electric and magnetic fields. This system has a purely con tinuous energy spectrum from -infinity to infinity. Depending on the f ield parameters, we find for the NNS a Poisson or a Wigner distributio n or a transitional behavior. We also outline how to determine physica lly relevant resonances in our approach by a stabilization method.