Birkhoff Normal Forms in Semi-classical Inverse Problems

The purpose of this note is to apply the recent results on semi-classical trace formulæ [17], and on quantum Birkhoff normal forms for semi-classical Fourier Operators [12] to inverse problems. We show how the classical Birkhoff normal form can be recovered from semi-classical spectral invariants. In fact the full quantum Birkhoff normal form of the quantum Hamiltonian near a closed orbit, and infinitesimally with respect to the energy can be recovered. This generalizes recent results of Guillemin [7] and Zelditch [19],[20], [21] obtained in the high energy setting (a special case of semi-classical asymptotics). We will illustrate the results in a new setting to which they apply. Let P (h) be a semi-classical Schrödinger operator: