Subcriticality of subordinated Schr\"{o}dinger operators and their application to wave equations

TL;DR

This paper characterizes the criticality levels of subordinated Schrödinger operators using probabilistic methods and explores how subcriticality relates to bounded solutions of associated wave equations.

Contribution

It introduces a probabilistic framework for classifying Schrödinger operators and links subcriticality to wave equation solution boundedness.

Findings

Probabilistic criteria for criticality, subcriticality, supercriticality.

Connection between subcriticality and uniform boundedness of wave solutions.

New insights into the behavior of subordinated Schrödinger operators.

Abstract

We provide a probabilistic characterization of criticality, subcriticality, and supercriticality for subordinated Schr\"{o}dinger operators. We also investigate the relationship between the subcriticality of these operators and the uniform boundedness of solutions to the associated wave equation.