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

The appearance of avoided crossings among energy levels as a system paramet er is varied is signaled by the presence of square-root branch points in th e complex parameter-plane. Even hidden crossings, which are so gradual as t o 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 a nd give rise to avoided crossings in energy. Either may be present in an ac tual quantum-mechanical problem. This poses special problems in perturbatio n 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. De termining which analytic structure is present from the perturbation series is illustrated here with the example of a dimensional perturbation treatmen t of the diamagnetic hydrogen problem. The branch point trajectories for th is system in the complex plane of the perturbation parameter delta (related to the magnetic quantum number and the dimensionality) as the magnetic fie ld strength is varied are also examined. It is shown how the trajectories o f the two branch-point pairs as the magnetic field strength varies are a na tural consequence of the particular analytic structure the energy manifests in the complex delta-plane. There is no need to invoke any additional anal ytic structures as a function of the field strength parameter. (C) 2000 Ame rican Institute of Physics. [S0022-2488(99)00612-X].