The semi-classical Maupertuis-Jacobi correspondence: stable and unstable spectra

We investigate semi-classical properties of Maupertuis-Jacobi correspondence for families of 2-D Hamiltonians $(H_\lambda(x,\xi), {\cal H}_\lambda(x,\xi))$, when ${\cal H}_\lambda(x,\xi)$ is the perturbation of a completely integrable Hamiltonian $\widetilde{\cal H}$ veriying some isoenergetic non-degeneracy conditions. Assuming $\hat H_\lambda$ has only discrete spectrum near $E$, and the energy surface $\{\widetilde{\cal H}_0={\cal E}\}$ is separated by some pairwise disjoint Lagrangian tori, we show that most of eigenvalues for $\hat H_\lambda$ near $E$ are asymptotically degenerate as $h\to0$. This applies in particular for the determination of trapped modes by an island, in the linear theory of water-waves. We also consider quasi-modes localized near rational tori. Finally, we discuss breaking of Maupertuis-Jacobi correspondence on the equator of Katok sphere.