Monodromy in perturbed Kepler systems: Hydrogen atom in crossed fields

¹ Mathematics Institute, University of Utrecht - Budapestlaan 6 3508 TA Utrecht, The Netherlands
² Université du Littoral - BP 5526, 59379 Dunkerque Cedex, France

Received: 21 December 1998
Accepted: 21 April 1999

Abstract

We demonstrate that an integrable approximation to the hydrogen atom in orthogonal electric and magnetic fields has monodromy, a fundamental dynamical property that makes a global definition of action-angle variables and of quantum numbers impossible. When the field strengths are sufficiently small, we find our integrable approximation using a two step normalization procedure. One of dynamically invariant sets of the resulting integrable system is a doubly pinched torus whose existence proves the presence of monodromy.