An Application of Nekhoroshev Theory to the Study of the Perturbed Hydrogen Atom

We return to the Keplerian or n-shell approximation to the hydrogen atom in the presence of weak static electric and magnetic fields. At the classical level, this is a Hamiltonian system with the phase space S^2×S^2. Its principal order Hamiltonian H_0 was known already to Pauli in 1926. H_0 defines an isochronous system with a linear flow on S^2 × S^2 and with frequencies depending on the external fields. Small perturbations of H_0 due to higher order terms can be studied by further normalization, either resonant or nonresonant. We study the question, raised previously, of how to decide for given parameters of the fields what normalization should be used and with regard to which resonances. We base this analysis on the Nekhoroshev theory—a branch of the Hamiltonian perturbation theory that complements the Kolmogorov-Arnold-Moser theorem. Our answer depends on the a priori choice of the maximal order N of resonances that are going to be taken into account (a cutoff). For any given N, there is a decomposition of the parameter space into resonant and nonresonant zones, and a normal form with a remainder of order exp(−N) may be consistently constructed in each of such zones.