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

2012/06/23 by Dobrokhotov, Sergey, Rouleux, Michel
#FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1206.5409

Abstract

We investigate semi-classical properties of Maupertuis-Jacobi correspondence for families of 2-D Hamiltonians (H_λ(x,ξ), \cal H_λ(x,ξ)), when \cal H_λ(x,ξ) is the perturbation of a completely integrable Hamiltonian \widetilde\cal H veriying some isoenergetic non-degeneracy conditions. Assuming H_λ has only discrete spectrum near E, and the energy surface \\widetilde\cal H₀=\cal E\ 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.

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