$L^q$ Estimates on the Restriction of Schr\"odinger Eigenfunctions with singular potentials

Abstract

We consider eigenfunction estimates in $L^p$ for Schr\"odinger operators, $H_V=-\Delta_g+V(x)$, on compact Riemannian manifolds $(M, g)$. Eigenfunction estimates over the full manifolds were already obtained by Sogge \cite{Sogge1988concerning} for $V\equiv 0$ and the first author, Sire, and Sogge \cite{BlairSireSogge2021Quasimode}, and the first author, Huang, Sire, and Sogge \cite{BlairHuangSireSogge2022UniformSobolev} for critically singular potentials $V$. For the corresponding restriction estimates for submanifolds, the case $V\equiv 0$ was considered in Burq, G\'erard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and Hu \cite{Hu2009lp}. In this article, we will handle eigenfunction restriction estimates for some submanifolds $\Sigma$ on compact Riemannian manifolds $(M, g)$ with $n:=\dim M\geq 2$, where $V$ is a singular potential.