Heat kernels and Green functions for fractional Schr\"{o}dinger operators with confining potentials

TL;DR

This paper provides sharp two-sided estimates for the heat kernel and Green function of fractional Schrödinger operators with confining potentials that grow at infinity, using probabilistic and analytic methods.

Contribution

It introduces new sharp bounds for these operators with potentials having the doubling property, combining probabilistic and analytic techniques.

Findings

Sharp two-sided estimates for heat kernel and Green function.

Results applicable to potentials with the doubling property.

Methods based on strong Markov property and Feynman–Kac formula.

Abstract

We give two-sided, global (in all variables) estimates of the heat kernel and the Green function of the fractional Schr\"odinger operator with a non-negative and locally bounded potential $V$ such that $V(x) \to \infty$ as $|x| \to \infty$. We assume that $V$ is comparable to a radial profile with the doubling property. Our bounds are sharp with respect to spatial variables and qualitatively sharp with respect to time. The methods we use combine probabilistic and analytic arguments. They are based on the strong Markov property and the Feynman--Kac formula.