Heat Kernel Estimates for Schr\"odinger Operators in the Domain Above a Bounded Lipschitz Function

TL;DR

This paper establishes precise upper and lower bounds for the heat kernel of Schrödinger operators with decaying potentials in domains defined by Lipschitz functions, advancing understanding of heat distribution in such geometries.

Contribution

It provides the first matching bounds for the Dirichlet heat kernel of Schrödinger operators in Lipschitz domains with rapidly decaying potentials.

Findings

Established explicit upper and lower bounds for the heat kernel.

Demonstrated the bounds' sharpness in Lipschitz domain settings.

Extended heat kernel estimates to Schrödinger operators with specific decay conditions.

Abstract

We give matching upper and lower bounds for the Dirichlet heat kernel of a Schr\"odinger operator $\Delta+W$ in the domain above the graph of a bounded Lipschitz function, in the case when $W$ decays away from the boundary faster than quadratically.