Monodromy in Perturbed Kepler Systems: Hydrogen Atom in Crossed Elds

{ We demonstrate that an integrable approximation to the hydrogen atom in orthogonal electric and magnetic elds has monodromy, a fundamental dynamical property that makes a global deenition of action-angle variables and of quantum numbers impossible. When the eld strengths are suuciently small, we nd 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. Introduction. { In 1980 Duistermaat 1] introduced the concept of monodromy in the study of two degree of freedom integrable Hamiltonian systems. Since that time monodromy has been analyzed in several integrable systems of classical mechanics 2]. Monodromy describes the global twisting of a family of invariant 2-tori parameterized by a circle of regular values of the energy momentum map of the integrable system. Its presence is signaled by the existence of a singular ber of the energy momentum map which is topologically a \pinched torus" 3]. Loosely speaking, if an integrable system has monodromy, then it is impossible to label the tori in a unique way by values of the actions. Since invariant tori are at the foundation of semiclassical EBK quantization of integrable systems, monodromy should manifest itself in the corresponding quantum systems 4, 5, 6]. Because monodromy is quite common in classical integrable systems of two degrees of freedom, it should have many important physical implications in quantum mechanics. In this note we report the results of the analysis of the geometry of an integrable approximation to the problem of the hydrogen atom in crossed magnetic and electric elds. (This fundamental physical system is an atomic analog 7] of a perturbed Keplerian system.) We show that its geometry is nontrivial and has monodromy.