The Problem of Two Fixed Centers: Bifurcations, Actions, Monodromy

A comprehensive analysis of the Euler-Jacobi problem of motion in the field of two fixed attracting centers is given, first classically and then quantum mechanically in semiclassical approximation. The system was originally studied in the context of celestial mechanics but, starting with Pauli’s dissertation, became a model for one-electron molecules such as H2 (symmetric case of equal centers) or HHe 2+ (asymmetric case of different centers). The present paper deals with arbitrary relative strength of the two centers and considers separately the planar and the three-dimensional problems. All versions represent nontrivial examples of integrable dynamics and are studied here from the unifying point of view of the energy momentum mapping from phase space to the space of integration constants. The interesting objects are the critical values of this mapping, i. e., its bifurcation diagram, and their pre-images which organize the foliation of phase space into Liouville-Arnold tori. The classical analysis culminates in the explicit derivation of the action variable representation of iso-energetic surfaces. The attempt to identify a system of global actions, smoothly dependent on the integration constants wherever these are non-critical, leads to the detection of monodromy of a special kind which is here described for the first time. The classical monodromy has its counterpart in the quantum version of the two-center problem where it prevents the assignments of unique quantum numbers even though the system is separable. PACS: 05.45.-a Nonlinear dynamics and nonlinear dynamical systems 03.65.Sq Semiclassical theories and applications 31.10.+z Theory of electronic structure, electronic transitions, and chemical binding