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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 thisarticle, 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 theH\"older regularity when $V$ belongs to this new class. The Davies--Gaffneyestimates for the associated semigroup and their local versions are also given.These results pave the way for many further studies on the analysis of$\mathcal{L}$.

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