Schrödinger Operators With $A_\infty$ Potentials

We study the heat kernel $p(x,y,t)$ associated to the real Schrödinger operator $H = -Δ+ V$ 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.

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