Recovering the Hamiltonian from spectral data

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.