  We study the heat kernel $p(x,y,t)$ associated to the real Schr\"odinger
operator $H = -\Delta + V$ on $L^2(\mathbb{R}^n)$, $n \geq 1$. Our main result
is a pointwise upper bound on $p$ when the potential $V \in A_\infty$. In the
case that $V\in RH_\infty$, we also prove a lower bound. Additionally, we
compute $p$ explicitly when $V$ is a quadratic polynomial.
