MONODROMY IN THE HYDROGEN ATOM IN CROSSED FIELDS

R. H. Cushman

Mathematics Institute, University of Utrecht,
Budapestlaan 6, 3508 TA Utrecht, The Netherlands 

D. A. Sadovski\'{\i}

Universit\'e du Littoral, Citadelle, BP 5526, 59379 Dunkerque Cedex, France

Received 8 June 1999; revised 1 February 2000; accepted 9 February 2000
Communicated by J.D. Meiss
Available online 25 May 2000.

       We show that the hydrogen atom in orthogonal electric and magnetic
       fields has a special property of certain integrable classical
       Hamiltonian systems known as monodromy. The strength of the fields is
       assumed to be small enough to validate the use of a truncated normal
       form Hsnf which is obtained from a two step normalization of the
       original system. We consider the level sets of Hsnf on the second
       reduced phase space.  For an open set of field parameters we show that
       there is a special dynamically invariant set which is a ``doubly pinched
       2-torus''. This implies that the integrable Hamiltonian Hsnf has
       monodromy.  Manifestation of monodromy in quantum mechanics is also
       discussed.

PACS: 03.20.+i Classical mechanics of discrete systems; 
      32.60+i  Zeeman and Stark effects

Keywords: Singular reduction; Monodromy; Energy-momentum map

Physica D: Nonlinear Phenomena
Volume 142, Issues 1-2 , 1 August 2000, Pages 166-196