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

TL;DR

This paper derives two-sided estimates for the heat kernel and Green function of Schr"odinger operators with supercritical non-negative potentials, revealing exponential decay near the origin, which advances understanding of such operators.

Contribution

It provides new two-sided heat kernel estimates for Schr"odinger operators with a broad class of supercritical potentials, including explicit decay behavior near the origin.

Findings

Heat kernel decays exponentially near the origin.

Established two-sided estimates for the Green function.

Extended analysis to a large class of supercritical potentials.

Abstract

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.