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We investigate semi-classical properties of Maupertuis-Jacobi correspondencein 2-D for families of Hamiltonians $(H_\lambda(x,\xi), {\calH}_\lambda(x,\xi))$, when ${\cal H}_\lambda(x,\xi)$ is the perturbation ofcompletely integrable Hamiltonian $\widetilde{\cal H}$ veriying someisoenergetic non-degeneracy conditions. Assuming the Weyl $h$-PDO $H^w_\lambda$has only discrete spectrum near $E$, and the energy surface $\{\widetilde{\calH}={\cal E}\}$ is separated by some pairwise disjoint lagrangian tori, we showthat most of eigenvalues for $\hat H_\lambda$ near $E$ are asymptoticallydegenerate as $h\to0$. This applies in particular for the determination oftrapped modes by an island, in the linear theory of water-waves. We alsoconsider quasi-modes localized near rational tori. Finally, we discuss breakingof Maupertuis-Jacobi correspondence on the equator of Katok sphere.

The semiclassical Maupertuis-Jacobi correspondence for quasi-periodic Hamiltonian flows: stable and unstable spectra