Schr\"odinger Operators With $A_{\mathrm{\infty }}$ Potentials

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