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

TL;DR

This paper investigates decay rates of Schr"odinger heat semigroup operators with inverse square potentials in Lorentz spaces, providing sharp estimates and characterizing the Laplacian through decay behavior.

Contribution

It establishes precise upper and lower decay estimates for the Schr"odinger heat semigroup with inverse square potential in Lorentz spaces, including sharp decay characterizations.

Findings

Derived upper and lower decay bounds for the semigroup operator norms.

Identified sharp decay estimates for various Lorentz space mappings.

Characterized the Laplacian based on decay properties of the semigroup.

Abstract

Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is a radially symmetric inverse square potential. Let $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$ be the operator norm of $\nabla^\alpha e^{-tH}$ from the Lorentz space $L^{p,\sigma}({\bf R}^N)$ to $L^{q,\theta}({\bf R}^N)$, where $\alpha\in\{0,1,2,\dots\}$. We establish both of upper and lower decay estimates of $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$ and study sharp decay estimates of $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$. Furthermore, we characterize the Laplace operator $-\Delta$ from the view point of the decay of $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$.