The semi-classical Maupertuis-Jacobi correspondence for quasi-periodic Hamiltonian flows with applications to linear water waves theory

We extend to the semi-classical setting the Maupertuis-Jacobi correspondence for a pair of Hamiltonians (H(x, hD(x)), H(x, hD(x))). If H(x, p) is completely integrable, or has merely an invariant Diophantine torus. in energy surface {H = E}, then we can construct a family of quasi-modes for H(x, hD(x)) at the corresponding energy E. This applies in particular to the linear theory of water waves, and determines trapped modes by an island, from the knowledge of Liouville metrics.