The Semi Classical Maupertuis-Jacobi Correspondance for Quasi-Periodic Hamiltonian Flows

Abstract

We extend to the semi-classical setting the Maupertuis-Jacobi correspondance for a pair of hamiltonians $(H(x,h{D}_x),\mathcal{H}(x,h{D}_x)$. If $\mathcal{H}(p,x)$ is completely integrable, or has merely has invariant diohantine torus $\Lambda$ in energy surface $\mathcal{E}$, then we can construct a family of quasi-modes for $H(x,h{D}_x)$ at the corresponding energy $E$. This applies in particular to the theory of water-waves in shallow water, and determines trapped modes by an island, from the knowledge of Liouville metrics.

~40 pages