Symplectic inverse spectral theory for pseudodifferential operators

TL;DR

This paper demonstrates that, under generic conditions, the semiclassical spectrum of a one-dimensional pseudodifferential operator uniquely encodes the symplectic geometry and Hamiltonian dynamics of its classical system.

Contribution

It establishes a symplectic inverse spectral theory linking the semiclassical spectrum to the classical phase space geometry.

Findings

Spectrum modulo O(h^2) determines symplectic geometry

Spectrum encodes Hamiltonian dynamics

Results hold under generic assumptions

Abstract

We prove, under some generic assumptions, that the semiclassical spectrum modulo O(h^2) of a one dimensional pseudodifferential operator completely determines the symplectic geometry of the underlying classical system. In particular, the spectrum determines the hamiltonian dynamics of the principal symbol.