Heat kernel estimates for Schrödinger operators with supercritical killing potentials

Abstract

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

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