Recovering the Hamiltonian from spectral data

TL;DR

This paper demonstrates that spectral data related to periodic trajectories and localized observables can determine the full Taylor expansion of a quantum Hamiltonian near those trajectories, including special cases like the bottom of a well.

Contribution

It establishes a method to recover the full Taylor expansion of a quantum Hamiltonian from spectral data associated with periodic trajectories and localized observables.

Findings

Spectral data determines the Hamiltonian's Taylor expansion near a trajectory.

The method applies to both general and Schrödinger operators at the bottom of a well.

The results connect spectral contributions to the Gutzwiller formula with Hamiltonian reconstruction.

Abstract

We show that the contributions to the Gutzwiller formula with observable associated to the iterates of a given elliptic nondegenerate periodic trajectory $\gamma$ and to certain families of observables localized near $\gamma$ determine the quantum Hamiltonian in a formal neighborhood of the trajectory $\gamma$, that is the full Taylor expansion of its total symbol near $\gamma$. We also treat the "bottom of a well" case both for general and Schr\"odinger operators.