Gaussian Estimates for Heat Kernels of Higher Order Schr\"odinger Operators with Potentials in Generalized Schechter Classes

TL;DR

This paper establishes Gaussian upper bounds and regularity properties for heat kernels of higher order Schrödinger operators with potentials in a newly defined generalized Schechter class, advancing the analysis of such operators.

Contribution

The authors introduce a new generalized Schechter class for potentials and prove heat kernel bounds and regularity for higher order Schrödinger operators with potentials in this class.

Findings

Heat kernel satisfies Gaussian upper bounds.

H"older regularity of the heat kernel is established.

Davies--Gaffney estimates are proved for the semigroup.

Abstract

Let $m\in\mathbb N$, $P(D):=\sum_{|\alpha|=2m}(-1)^m a_\alpha D^\alpha$ be a $2m$-order homogeneous elliptic operator with real constant coefficients on $\mathbb{R}^n$, and $V$ a measurable function on $\mathbb{R}^n$. In this article, the authors introduce a new generalized Schechter class concerning $V$ and show that the higher order Schr\"odinger operator $\mathcal{L}:=P(D)+V$ possesses a heat kernel that satisfies the Gaussian upper bound and the H\"older regularity when $V$ belongs to this new class. The Davies--Gaffney estimates for the associated semigroup and their local versions are also given. These results pave the way for many further studies on the analysis of $\mathcal{L}$.