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arXiv · 1903.05040

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

Haruya Mizutani·Junyong Zhang·Jiqiang Zheng
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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 $\mathcal{L}_V$.

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Haruya Mizutani, Junyong Zhang, Jiqiang Zheng. 2019-03-12. Uniform resolvent estimates for Schr\"odinger operator with an inverse-square potential. https://doi.org/10.1016/j.jfa.2019.108350

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