Semi-classical Integrability, Hyperbolic Flows and the Birkhoff Normal Form

Rouleux, Michel

Canadian journal of mathematics, Tome 56 (2004) no. 5, pp. 1034-1067

Résumé

We prove that a Hamiltonian $p\in {{C}^{\infty }}({{T}^{*}}{{\mathbf{R}}^{n}})$ is locally integrable near a non-degenerate critical point ${{\rho }_{0}}$ of the energy, provided that the fundamental matrix at ${{\rho }_{0}}$ has rationally independent eigenvalues, none purely imaginary. This is done by using Birkhoff normal forms, which turn out to be convergent in the ${{C}^{\infty }}$ sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the complex case, showing that when $p$ is holomorphic near ${{\rho }_{0}}\in {{T}^{*}}{{\mathbf{C}}^{n}},$ then Re $p$ becomes integrable in the complex domain for real times, while the Birkhoff series and the Birkhoff transforms may not converge, i.e., $p$ may not be integrable. These normal forms also hold in the semi-classical frame.

Détail

DOI : 10.4153/CJM-2004-047-6

Mots-clés : 35S, 37J10, 70H08

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