Schrödinger Operators With $A_{\infty }$ Potentials

Abstract

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

arXiv comment15 pages. Comments welcome!