Normalization and global analysis of perturbations of the hydrogen atom

Abstract

The hydrogen atom perturbed by sufficiently small homogeneous static electric and magnetic fields of arbitrary mutual alignment is a specific perturbation of the Kepler system with three degrees of freedom and three parameters. Normalization of the Keplerian symmetry reveals that the parameter space is stratified into resonant zones of systems, each zone with an internal dynamical stratification of its own ͑Efstathiou, Sadovskií, and Zhilinskií, 2007, Proc. R. Soc. London, Ser. A 463, 1771͒. Based on the fully integrable approximation, the bundle of invariant tori of individual systems within zones is characterized globally and the qualitative dynamical stratification is uncovered. The techniques involved in this analysis are illustrated with the example of the 1:1 resonance zone ͑near orthogonal fields͒ whose structure is known at present. Applications in the corresponding quantum system are also described. 2148 References 2148 2101 K. Efstathiou and D. A. Sadovskií: Normalization and global analysis of …

Spin-1 microcondensate in a magnetic field

“…[10], see also Ref. [11]). Finally Section IV is devoted to a very different approach, in which the quantum Hamiltonian is solved directly using the Bethe ansatz.…”

Hamiltonian tools for the analysis of optical polarization control

“…This role is similar to the one played by the separatrix in purely one-dimensional systems [30]. We remark that the existence of these singular structures had been essentially ignored in the physics literature until their recent introduction in the domain of atomic and molecular systems [29,31,32]. The aim of the present article is to render these new mathematical tools accessible to a broad audience in the context of nonlinear optics.…”

“…It can be shown that the variables x 0 ;x 1 ;x 2 ;x 3 constitute a basis for the set of polynomial functions commuting with K [29]. In particular, since fH;Kg 0, one can write the Hamiltonian as…”

“…The aim of the present article is to render these new mathematical tools accessible to a broad audience in the context of nonlinear optics. In this respect, this paper can also be viewed as a pedagogical introduction to this geometric approach of Hamiltonian singularities, which may subsequently h2elp the interested reader to enter into a more specialized literature of Hamiltonian dynamics [28,29,33].…”

“…However, little has been done in order to provide a theoretical description of the phenomenon of polarization attraction. Our aim in this article is to make a step in this direction by introducing a new set of tools that find their origin in recently developed mathematical techniques, in particular for the study of Hamiltonian singularities [28,29]. We have successfully used this mathematical approach in Refs.…”