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

[摘要] We prove that a Hamiltonian $pin C^infty(T^*{f R}^n)$ is locally integrable near anon-degenerate criticalpoint $ho_0$ of the energy, provided that the fundamental matrixat $ho_0$ 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 formnear a hyperbolic fixed point of a canonical transformation.Then we investigate the complex case, showing that when $p$ is holomorphic near $ho_0in T^*{f C}^n$, then $e p$ becomes integrable in the complex domain forreal times, while the Birkhoff series and the Birkhoff transformsmay not converge, {em i.e.,} $p$ may not be integrable. These normal formsalso hold in the semi-classical frame.

[学科分类] 数学（综合）