  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
$\mathcal{L}$.
