  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.
