The Semi Classical Maupertuis-Jacobi Correspondance for Quasi-Periodic Hamiltonian Flows

TL;DR

This paper extends the Maupertuis-Jacobi correspondence to semi-classical Hamiltonian flows, enabling the construction of quasi-modes in integrable or invariant torus settings, with applications to water-wave theory.

Contribution

It introduces a semi-classical extension of the Maupertuis-Jacobi correspondence for Hamiltonians, allowing quasi-mode construction in integrable and invariant torus cases.

Findings

Constructs quasi-modes for semi-classical Hamiltonians at specific energies.

Applies to water-wave theory, identifying trapped modes from Liouville metrics.

Extends classical correspondence to semi-classical and quantum regimes.

Abstract

We extend to the semi-classical setting the Maupertuis-Jacobi correspondance for a pair of hamiltonians $(H(x,hD_x), {\cal H}(x,hD_x)$. If ${\cal H}(p,x)$ is completely integrable, or has merely has invariant diohantine torus $\Lambda$ in energy surface ${\cal E}$, then we can construct a family of quasi-modes for $H(x,hD_x)$ at the corresponding energy $E$. This applies in particular to the theory of water-waves in shallow water, and determines trapped modes by an island, from the knowledge of Liouville metrics.