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

Abstract

We investigate semi-classical properties of Maupertuis-Jacobi correspondence for families of 2-D Hamiltonians $(H_{\lambda }(x,\xi ),{\mathcal{H}}_{\lambda }(x,\xi ))$, when ${\mathcal{H}}_{\lambda }(x,\xi )$ is the perturbation of a completely integrable Hamiltonian $\tilde{\mathcal{H}}$ veriying some isoenergetic non-degeneracy conditions. Assuming ${\hat{H}}_{\lambda }$ has only discrete spectrum near $E$, and the energy surface $\{{\tilde{\mathcal{H}}}_0=\mathcal{E}\}$ is separated by some pairwise disjoint Lagrangian tori, we show that most of eigenvalues for ${\hat{H}}_{\lambda }$ near $E$ are asymptotically degenerate as $h\to 0$. 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.

Conference Days of Diffraction 2012, St Petersburg