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

Abstract In 1998, V. Liskevich and Y. Semenov proved sharp Gaussian upper bounds for Schrödinger semigroups on $$\mathbb {R}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> 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.