Heat kernel estimates for Schrödinger operators with supercritical killing potentials

In this paper, we study the Schrödinger operator $\mathrm{\Delta }-V$, where $V$ is a supercritical non-negative potential belonging to a large class of functions containing functions of the form $b\mathrm{\mid }x{\mathrm{\mid }}^{-(2+2\beta )}$, $b,\beta >0$. We obtain two-sided estimates on the heat kernel $p(t,x,y)$ of $\mathrm{\Delta }-V$, along with estimates for the corresponding Green function. Unlike the case of the fractional Schrödinger operator $-(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}-V$, $\alpha \in (0,2)$, with supercritical killing potential dealt with in [11], in the present case, the heat kernel $p(t,x,y)$ decays to 0 exponentially as $x$ or $y$ tends to the origin.