Majorization, 4G Theorem and Schr\"odinger perturbations

TL;DR

This paper develops methods to estimate Schr"odinger perturbations of transition densities using majorants and the 4G inequality, focusing on stable subordinators and inverse Gaussian processes, to better understand their behavior under singular potentials.

Contribution

It proves the 4G inequality for specific subordinators and discusses admissible potentials, providing new tools for analyzing Schr"odinger perturbations of transition densities.

Findings

Proved 4G inequality for 1/2-stable and inverse Gaussian subordinators.

Identified classes of admissible potentials for these processes.

Provided estimates for transition densities of Schr"odinger operators.

Abstract

Schr\"odinger perturbations of transition densities by singular potentials may fail to be comparable with the original transition density. For instance this is so for the transition density of a subordinator perturbed by any time-independent unbounded potential. In order to estimate such perturbations it is convenient to use an auxilary transition density as a majorant and the 4G inequality for the original transition density and the majorant. We prove the 4G inequality for the $1/2$-stable and inverse Gaussian subordinators, discuss the corresponding class of admissible potentials and indicate estimates for the resulting transition densities of Schr\"odinger operators. The connection of the transition densities to their generators is made via the weak-type notion of fundamental solution.