## The problem of two fixed centres: bifurcations, actions, monodromy

preprint

posted on 21.07.2005 by Holger Waalkens, Holger R. Dullin, Peter H. Richter#### preprint

Preprints are manuscripts made publicly available before they have been submitted for formal peer review and publication. They might contain new research findings or data. Preprints can be a draft or final version of an author's research but must not have been accepted for publication at the time of submission.

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
H+
2 (symmetric case of equal centers) or HHe2+ (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.

### History

#### School

- Science

#### Department

- Mathematical Sciences