Characterization of sharp global Gaussian estimates for Schr\"odinger heat kernels

TL;DR

This paper characterizes when the fundamental solution of the Schrödinger equation with potential V has sharp Gaussian bounds globally in space and time, providing specific conditions for V.

Contribution

It offers a new characterization for V ≤ 0 and a sufficient condition for general V regarding sharp Gaussian estimates of Schrödinger heat kernels.

Findings

Characterization for V ≤ 0 cases.

Sufficient condition for general V.

Conditions ensure sharp Gaussian bounds in space and time.

Abstract

We investigate when the fundamental solution of the Schr\"odinger equation $\partial_t=\Delta+V$ posseses sharp Gaussian bounds global in space and time. We give a characterization for $V\leq 0$ and a sufficient condition for general $V$.