Christopher Sogge

Title: Quasimode estimates involving critically singular potentials

Abstract: We prove eigenfunction and quasimode estimates on compact Riemannian manifolds for Schrödinger operators, $H_V=-\Delta_g+V$ involving critically singular potentials $V$ which we assume to be in $L^{n/2}$ and/or the Kato class ${\mathcal K}$. Our proof is based on modifying the oscillatory integral/resolvent approach that was used to study the case where $V \equiv 0$ using recently developed techniques by many authors to study variable coefficient analogs of the uniform Sobolev estimates of Kenig, Ruiz and the speaker. Using the quasimode estimates we are able to obtain Strichartz estimates for wave equations. We are also able to prove corresponding results for Schrödinger operators in ${\mathbb R}^n$ and can obtain a natural generalization of the Stein-Tomas restriction theorem involving potentials with small $L^{n/2}({\mathbb R}^n)$ norms.

This is joint work with M. Blair and Y. Sire.