Sharp Gaussian upper bounds for Schrödinger semigroups on the half-line

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