Restriction of Schr\"odinger eigenfunctions to submanifolds

Abstract

For Schr\"odinger operators $H_V=-\Delta_g+V$ with critically singular potentials $V$ on compact manifolds, we prove sharp estimates for the restriction of eigenfunctions to submanifolds. Our method refines the perturbative argument by Blair-Sire-Sogge and enables us to deal with submanifolds of all codimensions. As applications, we obtain improved estimates on negatively curved manifolds and flat tori. In particular, we extend the uniform $L^2$ restriction estimates on flat tori by Bourgain-Rudnick to singular potentials.