Sharp and plain estimates for Schr\"odinger perturbation of Gaussian kernel

TL;DR

This paper examines conditions under which the fundamental solution of the Schrödinger equation with potential V exhibits sharp Gaussian estimates, especially focusing on potentials with fixed sign and those in the Kato class.

Contribution

It provides new insights into when sharp versus plain Gaussian estimates hold for Schrödinger perturbations of the Gaussian kernel, especially for fixed sign and Kato class potentials.

Findings

Sharp Gaussian estimates depend on the sign and class of the potential V.

Certain conditions are identified under which local in time sharp estimates are valid.

Results clarify the relationship between potential properties and Gaussian estimate types.

Abstract

We investigate whether a fundamental solution of the Schr\"odinger equation $\partial_t u =(\Delta +V)\, u$ has local in time sharp Gaussian estimates. We compare that class with the class of $V$ for which local in time plain Gaussian estimates hold. We concentrate on $V$ that have fixed sign and we present certain conclusions for $V$ in the Kato class.