Gaussian Estimates for Heat Kernels of Higher Order Schrödinger Operators with Potentials in Generalized Schechter Classes

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 Schrödinger operator $\mathcal{L}:=P(D)+V$ possesses a heat kernel that satisfies the Gaussian upper bound and the Hölder 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 $\mathcal{L}$.

52 pages, Submitted