Restriction of Schr\"odinger eigenfunctions to submanifolds

TL;DR

This paper establishes sharp restriction estimates for Schr"odinger eigenfunctions with singular potentials on compact manifolds, extending previous results to all codimensions and including negatively curved spaces and flat tori.

Contribution

It introduces a refined perturbative method that handles submanifolds of all codimensions for Schr"odinger eigenfunctions with critically singular potentials.

Findings

Sharp restriction estimates for eigenfunctions with singular potentials

Extension of uniform $L^2$ restriction estimates to singular potentials on flat tori

Improved estimates on negatively curved manifolds

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.