Contractivity and ground state domination properties for non-local Schr\"odinger operators

TL;DR

This paper investigates the supercontractivity and hypercontractivity of semigroups derived from non-local Schrödinger operators linked to symmetric jump Lévy processes, establishing sharp conditions for these properties and revealing their equivalence in certain cases.

Contribution

It introduces refined contractivity concepts for non-local Schrödinger operators and provides sharp criteria for their validity, highlighting differences from classical Schrödinger operators.

Findings

Sharp necessary and sufficient conditions for contractivity properties.

Equivalence of supercontractivity, ultracontractivity, and hypercontractivity for certain potentials.

Contrast with classical Schrödinger operators where these properties differ.

Abstract

We study supercontractivity and hypercontractivity of Markov semigroups obtained via ground state transformation of non-local Schr\"odinger operators based on generators of symmetric jump-paring L\'evy processes with Kato-class confining potentials. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps, and the related operators include pseudo-differential operators of interest in mathematical physics. We refine these contractivity properties by the concept of $L^p$-ground state domination and its asymptotic version, and derive sharp necessary and sufficient conditions for their validity in terms of the behaviour of the L\'evy density and the potential at infinity. As a consequence, we obtain for a large subclass of confining potentials that, on the one hand, supercontractivity and ultracontractivity, on the other hand, hypercontractivity and asymptotic ultracontractivity of the transformed semigroup are equivalent properties. This is in stark contrast to classical Schr\"odinger operators, for which all these properties are known to be different.