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 Λ 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.