# Quasimode, eigenfunction and spectral projection bounds for Schr\"odinger operators on manifolds with critically singular potentials

## TL;DR

This paper extends spectral and eigenfunction bounds for Schrödinger operators on compact manifolds to critically singular potentials, enabling new $L^p$ spectral multiplier and Strichartz estimates.

## Contribution

It introduces methods to handle critically singular potentials in spectral bounds and extends classical estimates to these more general cases.

## Key findings

- Established spectral projection bounds for singular potentials.
- Derived $L^p$ spectral multiplier theorems under new conditions.
- Proved Strichartz estimates for Schrödinger equations with singular potentials.

## Abstract

We obtain quasimode, eigenfunction and spectral projection bounds for Schr\"odinger operators, $H_V=-\Delta_g+V(x)$, on compact Riemannian manifolds $(M,g)$ of dimension $n\ge2$, which extend the results of the third author~\cite{sogge88} corresponding to the case where $V\equiv 0$. We are able to handle critically singular potentials and consequently assume that $V\in L^{\tfrac{n}2}(M)$ and/or $V\in {\mathcal K}(M)$ (the Kato class). Our techniques involve combining arguments for proving quasimode/resolvent estimates for the case where $V\equiv 0$ that go back to the third author \cite{sogge88} as well as ones which arose in the work of Kenig, Ruiz and this author~\cite{KRS} in the study of "uniform Sobolev estimates" in ${\mathbb R}^n$. We also use techniques from more recent developments of several authors concerning variations on the latter theme in the setting of compact manifolds. Using the spectral projection bounds we can prove a number of natural $L^p\to L^p$ spectral multiplier theorems under the assumption that $V\in L^{\frac{n}2}(M)\cap {\mathcal K}(M)$. Moreover, we can also obtain natural analogs of the original Strichartz estimates~\cite{Strichartz77} for solutions of $(\partial_t^2-\Delta +V)u=0$. We also are able to obtain analogous results in ${\mathbb R}^n$ and state some global problems that seem related to works on absence of embedded eigenvalues for Schr\"odinger operators in ${\mathbb R}^n$ (e.g., \cite{IonescuJerison}, \cite{JK}, \cite{KenigNar}, \cite{KochTaEV} and \cite{iRodS}.)