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

We study the uniform resolvent estimates for Schrödinger operator with a Hardy-type singular potential. Let $\mathcal{L}_V=-Δ+V(x)$ where $Δ$ is the usual Laplacian on $\mathbb{R}^n$ and $V(x)=V_0(θ) r^{-2}$ where $r=|x|, θ=x/|x|$ and $V_0(θ)\in\mathcal{C}^1(\mathbb{S}^{n-1})$ is a real function such that the operator $-Δ_θ+V_0(θ)+(n-2)^2/4$ 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 $\mathcal{L}_V$.

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