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An application of Nekhoroshev theory to the study of the perturbed hydrogen atom. (English) Zbl 1337.37042

In the paper under review the Keplerian or n-shell approximation to the hydrogen atom in the presence of weak static electric and magnetic fields is considered. At the classical level, this is a Hamiltonian system with the phase space \(S^2 \times S^2\). Its principal order Hamiltonian \(H_0\) was known already to W. Pauli jr. [Z. Phys. 36, 336–363 (1926; JFM 52.0978.02)]. \(H_0\) defines an isochronous system with a linear flow on \(S^2 \times S^2\) 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. The article under review studies 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. This analysis is based on Nekhoroshev theory – a branch of the Hamiltonian perturbation theory that complements the Kolmogorov-Arnold-Moser theorem. The answer depends on the a priori choice of the maximal order \(N\) of resonances that are 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.

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
70H08 Nearly integrable Hamiltonian systems, KAM theory
70F05 Two-body problems
81V45 Atomic physics
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)

Citations:

JFM 52.0978.02
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References:

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