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

TL;DR

This paper extends eigenfunction restriction estimates to Schr"odinger operators with singular potentials on compact Riemannian manifolds, building on prior work for smooth and critically singular potentials, and addresses submanifold restrictions.

Contribution

It provides new eigenfunction restriction estimates for Schr"odinger operators with singular potentials on submanifolds of compact Riemannian manifolds.

Findings

Established restriction estimates for eigenfunctions with singular potentials.

Extended previous results from smooth and critically singular potentials.

Analyzed eigenfunction behavior on submanifolds 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.