Heat kernels of non-local Schr\"odinger operators with Kato potentials

TL;DR

This paper develops a comprehensive framework for analyzing heat kernels of non-local Schr"odinger operators with Kato potentials, covering both heavy- and light-tailed Lévy measures, and provides sharp estimates and regularity results.

Contribution

It introduces a general approach to construct and estimate heat kernels for non-local Schr"odinger operators with Kato potentials, encompassing a wide class of Lévy measures.

Findings

Established a relative-Kato bound for semigroups and potentials.

Constructed heat kernels with sharp estimates.

Proved regularity of heat kernels under stronger conditions.

Abstract

We study heat kernels of Schr\"odinger operators whose kinetic terms are non-local operators built for sufficiently regular symmetric L\'evy measures with radial decreasing profiles and potentials belong to Kato class. Our setting is fairly general and novel -- it allows us to treat both heavy- and light-tailed L\'evy measures in a joint framework. We establish a certain relative-Kato bound for the corresponding semigroups and potentials. This enables us to apply a general perturbation technique to construct the heat kernels and give sharp estimates of them. Assuming that the L\'evy measure and the potential satisfy a little stronger conditions, we additionally obtain the regularity of the heat kernels. Finally, we discuss the applications to the smoothing properties of the corresponding semigroups. Our results cover many important examples of non-local operators, including fractional and quasi-relativistic Schr\"odinger operators.