Level statistics for continuous energy spectra with application to the hydrogen atom in crossed electric and magnetic fields.

Abstract

The statistical analysis of energy levels, a powerful tool in the study of quantum systems, is applicable to discrete spectra. Here we propose an approach to carry level statistics over to continuous energy spectra, 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 nearest-neighbor spacing (NNS), usually associated with quantum chaos and regularity, can be carried over to systems with a purely continuous energy spectrum. The approach is demonstrated for the hydrogen atom in perpendicular electric and magnetic fields. This system has a purely continuous energy spectrum from [minus][infinity] to [infinity]. Depending on the field parameters, we find for the NNS a Poisson or a Wigner distribution, or a transitional behavior. We also outline how to determine physically relevant resonances in our approach by a stabilization method.