Improved resolvent bounds for radial potentials. II

TL;DR

This paper establishes improved semiclassical resolvent estimates for Schrödinger operators with radial potentials in higher dimensions, demonstrating how decay rates of potentials influence the bounds.

Contribution

It provides new resolvent bounds for radial potentials with specific decay rates, extending previous results and offering sharper estimates in the semiclassical regime.

Findings

Resolvent bounds are exponential in inverse semiclassical parameter h.

Stronger decay of potentials yields better resolvent estimates.

Results apply to potentials with polynomial and exponential decay.

Abstract

We prove semiclassical resolvent estimates for the Schr{\"o}dinger operator in R d , d $\ge$ 3, with real-valued radial potentials V $\in$ L $\infty$ (R d). We show that if V (x) = O x --$\delta$ with $\delta$ > 4, then the resolvent bound is of the form exp Ch -- $\delta$ $\delta$--1 log(h --1) 1 $\delta$--1 with some constant C > 0. If V (x) = O e -- C x $\alpha$ with C, $\alpha$ > 0, we get better resolvent bounds of the form exp Ch --1 log(h --1