Gaussian estimates with best constants for higher-order Schr\"odinger operators with Kato potentials

TL;DR

This paper derives optimal Gaussian bounds for the heat kernel of higher-order Schrödinger operators with variable coefficients and Kato potentials, providing precise exponential decay rates.

Contribution

It introduces sharp Gaussian estimates for heat kernels of higher-order elliptic Schrödinger operators with Kato potentials, including the best possible constants.

Findings

Established Gaussian bounds with sharp constants for heat kernels.

Extended estimates to operators with variable highest order coefficients.

Provided a framework for analyzing higher-order Schrödinger operators with Kato potentials.

Abstract

We establish Gaussian estimates on the heat kernel of a higher-order uniformly elliptic Schr\"odinger operator with variable highest order coefficients and with a Kato class potential. The estimates involve the sharp constant in the Gaussian exponent.