Weprove eigenfunction and quasimode estimates on compact Riemannian manifolds for Schr\”odingeroperators, $H_V=-\Delta_g+V$ involving critically singular potentials $V$ which weassume tobe in $L^{n/2}$ and/or the Kato class ${\mathcal K}$.  Our proof is basedon modifying the oscillatory integral/resolvent approachthat was used to study the case where $V \equiv 0$ using recently developedtechniques by many authorsto study variable coefficient analogs of the uniform Sobolev estimates ofKenig, Ruiz and the speaker.

Usingthe quasimode estimates we are able to obtain Strichartz estimates for waveequations.  We are alsoable to prove corresponding results for Schr\”odinger operators in ${\mathbbR}^n$ and can obtain anatural generalization of the Stein-Tomas restriction theoreminvolving potentials with small $L^{n/2}({\mathbb R}^n)$ norms.

Thisis joint work with M. Blair and Y. Sire.