Two-sided heat kernel estimates for Schr\"{o}dinger operators with unbounded potentials

TL;DR

This paper derives sharp, global in space and time heat kernel bounds for Schr"odinger operators with unbounded potentials growing at infinity, using probabilistic methods, applicable even when the semigroup lacks ultracontractivity.

Contribution

It provides the first sharp two-sided heat kernel estimates for Schr"odinger operators with unbounded potentials under broad growth conditions, extending previous results.

Findings

Established global two-sided heat kernel bounds for unbounded potentials.

Derived Green's function estimates corresponding to the Schr"odinger operators.

Demonstrated bounds even when the semigroup is not intrinsically ultracontractive.

Abstract

Consider the Schr\"odinger operator $ \mathcal L^V=-\Delta+V $ on $\R^d$, where $V:\R^d\to [0,\infty)$ is a nonnegative and locally bounded potential on $\R^d$ so that for all $x\in \R^d$ with $|x|\ge 1$, $c_1g(|x|)\le V(x)\le c_2g(|x|)$ with some constants $c_1,c_2>0$ and a nondecreasing and strictly positive function $g:[0,\infty)\to [1,+\infty)$ that satisfies $g(2r)\le c_0 g(r)$ for all $r>0$ and $\lim_{r\to \infty} g(r)=\infty.$ We establish global in time and qualitatively sharp bounds for the heat kernel of the associated Schr\"{o}dinger semigroup by the probabilistic method. In particular, we can present global in space and time two-sided bounds of heat kernel even when the Schr\"{o}dinger semigroup is not intrinsically ultracontractive. Furthermore, two-sided estimates for the corresponding Green's functions are also obtained.