Inverse spectral problems for Schr\"odinger and pseudo-differential operators

Starting from the semi-classical spectrum of Schr"odinger operators $-h^2\mathrm{\Delta }+V$ (on ${\mathbb{R}}^n$ or on a Riemannian manifold) it is possible to detect critical levels of the potential $V$. Via micro-local methods one can express spectral statistics in terms of different invariants: \begin{itemize} \item Geometry of energy surfaces (heat invariant like). \item Classical orbits (wave invariants). \item But also classical equilibria (new wave invariants). \end{itemize} Any critical point of $V$ with zero momentum is an equilibrium of the flow and generates many singularities in the semi-classical distribution of eigenvalues. Via sharp spectral estimates, this phenomena indicates the presence of a critical energy level and the information contained in this singularity allows to reconstruct partially the local shape of $V$. Several generalizations of this approach are also proposed. Keywords : Spectral analysis, P.D.E., Micro-local analysis; Schr"odinger operators; Inverse spectral problems.