Semiclassical resolvent estimates for short-range L ∞ potentials. II

Abstract

We prove semiclassical resolvent estimates for real-valued potentials $\mathit{V}\in {\mathit{L}}^{\infty }\mathrm{(}{\mathbb{R}}^{\mathit{n}}\mathrm{)}$ , $\mathit{n}⩾3$ , of the form $\mathit{V}={\mathit{V}}_{\mathit{L}}+{\mathit{V}}_{\mathit{S}}$ , where ${\mathit{V}}_{\mathit{L}}$ is a long-range potential which is ${\mathit{C}}^1$ with respect to the radial variable, while ${\mathit{V}}_{\mathit{S}}$ is a short-range potential satisfying ${\mathit{V}}_{\mathit{S}}\mathrm{(}\mathit{x}\mathrm{)}=\mathcal{O}\mathrm{(}{⟨\mathit{x}⟩}^{-\mathit{\delta }}\mathrm{)}$ with $\mathit{\delta }\mathrm{>}1$ .