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
_λ
(x; ξ), H
_λ
(x; ξ)), when H
_λ
(x; ξ) is the perturbation of a completely integrable Hamiltonian ̃“ verifying some isoenergetic non-degeneracy conditions. Assuming ̂H
_λ
has only discrete spectrum near E, and the energy surface {̃H̃ = ε} is separated by some pairwise disjoint Lagrangian tori, we show that most of eigenvalues for ̂H
_λ
near E are asymptotically degenerate as h→0. 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.