Semiclassical resolvent bounds for compactly supported radial potentials

TL;DR

This paper establishes weighted resolvent bounds for semiclassical Schrödinger operators with compactly supported radial potentials, introducing a new Mellin transform method for the two-dimensional case.

Contribution

It provides novel resolvent estimates for radial potentials and introduces a Mellin transform technique for the 2D case.

Findings

Weighted resolvent norm grows at most exponentially with inverse semiclassical parameter

Exterior weighted norm scales linearly with inverse semiclassical parameter

New Mellin transform method effectively handles the two-dimensional case

Abstract

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schr\"odinger 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 $\exp(Ch^{-1})$, 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.