Uniform resolvent estimates for Schrödinger operator with an inverse-square potential

Abstract

We study the uniform resolvent estimates for Schrödinger 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 ${\mathcal{L}}_V$.

Comments are welcome.To appear in Journal of Functional Analysis