Sharp Gaussian upper bounds for Schr\"odinger semigroups on the half-line

TL;DR

This paper extends sharp Gaussian upper bounds for Schr"odinger semigroups from the whole space to the half-line, employing new weighted ultracontractivity techniques under Kato class conditions.

Contribution

It introduces a novel approach using weighted ultracontractivity to establish Gaussian bounds for Schr"odinger semigroups on the half-line, generalizing previous results.

Findings

Established sharp Gaussian upper bounds on the half-line

Developed a new weighted ultracontractivity technique

Extended Kato class conditions to half-line setting

Abstract

In 1998, V. Liskevich and Y. Semenov showed sharp Gaussian upper bounds for Schr\"odinger semigroups on $\mathbb R^3$ with potentials satisfying a global Kato class condition. Using similar basic ideas we show sharp Gaussian upper bounds for Schr\"odinger semigroups on the half-line, also assuming a suitable global Kato class condition. Our proof strategy includes a new technique of weighted ultracontractivity estimates.