Yannick Sire – Analysis & PDE Seminar

Title: Spectral analysis of Schrodinger operators and applications

Abstract: I will report on recent results about quasimode, eigenfunction and spectral projection bounds for Schrodinger operators, $H_V=-\Delta_g+V(x)$, on compact Riemannian manifolds $(M,g)$ of dimension $n\ge2$ with critically singular potentials V. Using the spectral projection bounds  we can prove a number of natural $L^p\to L^p$ spectral multiplier theorems under the assumption that $V\in L^{\frac{n}2}(M)\cap {\mathcal K}(M)$. We will draw also several applications to some PDEs. I will also mention some results when one considers some special Levy operators instead of the Laplace-Beltrami in H_V.