Heat kernel estimates for Schr\"odinger operators with supercritical killing potentials

In this paper, we study the Schr\"odinger operator $\Delta-V$, where $V$ is a supercritical non-negative potential belonging to a large class of functions containing functions of the form $b|x|^{-(2+2\beta)}$, $b, \beta>0$. We obtain two-sided estimates on the heat kernel $p(t, x, y)$ of $\Delta-V$, along with estimates for the corresponding Green function. Unlike the case of the fractional Schr\"odinger operator $-(-\Delta)^{\alpha/2}-V$, $\alpha\in (0, 2)$, 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.