Gaussian estimates for heat kernels of higher order Schrödinger operators with potentials in generalized Schechter classes

Abstract

Let m ∈ N $m\in \mathbb {N}$ , P ( D ) : = ∑ | α | = 2 m ( − 1 ) m a α D α $P(D):=\sum _{|\alpha |=2m}(-1)^m a_\alpha D^\alpha$ be a 2 m $2m$ -order homogeneous elliptic operator with real constant coefficients on R n $\mathbb {R}^n$ , and V $V$ a real-valued measurable function on R n $\mathbb {R}^n$ . In this article, the authors introduce a new generalized Schechter class concerning V $V$ and show that the higher order Schrödinger operator L : = P ( D ) + V $\mathcal {L}:=P(D)+V$ possesses a heat kernel that satisfies the Gaussian upper bound and the Hölder regularity when V $V$ 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 L $\mathcal {L}$ .