# Uniform resolvent estimates for Schr\"odinger operator with an inverse-square potential

## TL;DR

This paper establishes new uniform resolvent and Sobolev estimates for Schr"odinger operators with inverse-square potentials, extending understanding of their spectral properties and resolvent behavior in weighted spaces.

## Contribution

It provides novel uniform resolvent and Sobolev estimates for Schr"odinger operators with Hardy-type inverse-square potentials, under specific positivity conditions.

## Key findings

- Derived new uniform weighted resolvent estimates
- Established uniform Sobolev estimates for the operator
- Extended spectral analysis for inverse-square potential operators

## Abstract

We study the uniform resolvent estimates for Schr\"odinger operator with a Hardy-type singular potential. Let $\mathcal{L}_V=-\Delta+V(x)$ where $\Delta$ is the usual Laplacian on $\mathbb{R}^n$ and $V(x)=V_0(\theta) r^{-2}$ where $r=|x|, \theta=x/|x|$ and $V_0(\theta)\in\mathcal{C}^1(\mathbb{S}^{n-1})$ is a real function such that the operator $-\Delta_\theta+V_0(\theta)+(n-2)^2/4$ is a strictly positive operator on $L^2(\mathbb{S}^{n-1})$. We prove some new uniform weighted resolvent estimates and also obtain some uniform Sobolev estimates associated with the operator $\mathcal{L}_V$.