Branch-point structure and the energy level characterization of avoided crossings

Abstract

The appearance of avoided crossings among energy levels as a system parameter is varied is signaled by the presence of square-root branch points in the complex parameter-plane. Even hidden crossings, which are so gradual as to be difficult to resolve experimentally, can be uncovered by the knowledge of the locations of these branch points. As shown in this paper, there are two different analytic structures that feature square-root branch points and give rise to avoided crossings in energy. Either may be present in an actual quantum-mechanical problem. This poses special problems in perturbation theory since the analytic structure of the energy is not readily apparent from the perturbation series, and yet the analytic structure must be known beforehand if the perturbation series is to be summed to high accuracy. Determining which analytic structure is present from the perturbation series is illustrated here with the example of a dimensional perturbation treatment of the diamagnetic hydrogen problem. The branch point trajectories for this system in the complex plane of the perturbation parameter δ (related to the magnetic quantum number and the dimensionality) as the magnetic field strength is varied are also examined. It is shown how the trajectories of the two branch-point pairs as the magnetic field strength varies are a natural consequence of the particular analytic structure the energy manifests in the complex δ-plane. There is no need to invoke any additional analytic structures as a function of the field strength parameter.