Decay estimates for Schr\"odinger heat semigroup with inverse square potential in Lorentz spaces

TL;DR

This paper derives precise decay estimates for the Schrödinger heat semigroup with inverse square potential in Lorentz spaces, enhancing understanding of its temporal behavior and operator norms.

Contribution

It provides sharp decay estimates for the heat semigroup and its gradient in Lorentz spaces for Schrödinger operators with inverse square potentials.

Findings

Sharp decay estimates for $e^{-tH}$ in Lorentz spaces

Decay estimates for $ abla e^{-tH}$ in Lorentz spaces

Improved understanding of the operator norms over time

Abstract

Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is an inverse square potential. In this paper we obtain sharp decay estimates of the operator norms of $e^{-tH}$ and $\nabla e^{-tH}$ in Lorentz spaces.