Integrability, hyperbolic flows and the Birkhoff normal form

34 pagesWe prove that a Hamiltonian $p\in C^\infty(T^*{\bf R}^n)$ is locally integrable near a non-degenerate critical point $\rho_0$ of the energy, provided that the fundamental matrix at $\rho_0$ has no purely imaginary eigenvalues. 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 almost holomorphic case

The Maupertuis-Jacobi principle for Hamiltonians of the form F(x, |p|) in two-dimensional stationary semiclassical problems

11 pages, 1 figureInternational audienceWe make use of the Maupertuis -- Jacobi correspondence, well known in Classical Mechanics, to simplify 2-D asymptotic formulas based on Maslov's canonical operator, when constructing Lagrangian manifolds invariant with respect to phase flows for Hamiltonians of the form $F(x,|p|)$. As examples we consider Hamiltonians coming from the Schr\"odinger equation, the 2-D Dirac equation for graphene and linear water wave theory

The semiclassical Maupertuis-Jacobi correspondence and applications to linear water wave theory

International audienc

Hyperbolic Hamiltonian flows and the semi-classical Poincaré map

International audienceWe consider semi-excited resonances created by a periodic orbit of hyperbolic type for Schrödinger type operators with a small "Planck constant". They are defined within an analytic framework based on the semi-classical quantization of Poincaré map in action-angle variables