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

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