Semiclassical resolvent bounds for compactly supported radial potentials

Abstract

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2\Delta +V(|x|)-E$ in dimension $n\ge 2$, where $h,E>0$, and $V:[0,\infty )\to \mathbb{R}$ is $L^{\infty }$ and compactly supported. The weighted resolvent norm grows no faster than $\exp (C{h}^{-1})$, while an exterior weighted norm grows $\sim h^{-1}$. We introduce a new method based on the Mellin transform to handle the two-dimensional case.