Bulk behaviour of ground states for relativistic Schr\"odinger operators with compactly supported potentials

TL;DR

This paper develops a probabilistic framework to analyze the ground states of relativistic Schrödinger operators with compactly supported potentials, providing explicit estimates and insights into their behavior inside and outside the potential well.

Contribution

It introduces a novel probabilistic representation of ground states for non-local Schrödinger operators, enabling detailed analysis of their spatial behavior and regularity.

Findings

Derived explicit estimates for the first exit and entrance times of stable processes.

Established local rates of ground states inside and outside the potential well.

Showed the ground state regularity depends on the fractional power of the operator.

Abstract

We propose a probabilistic representation of the ground states of massive and massless Schr\"{o}dinger operators with a potential well in which the behaviour inside the well is described in terms of the moment generating function of the first exit time from the well, and the outside behaviour in terms of the Laplace transform of the first entrance time into the well. This allows an analysis of their behaviour at short to mid-range from the origin. In a first part we derive precise estimates on these two functionals for stable and relativistic stable processes. Next, by combining scaling properties and heat kernel estimates, we derive explicit local rates of the ground states of the given family of non-local Schr\"{o}dinger operators both inside and outside the well. We also show how this approach extends to fully supported decaying potentials. By an analysis close-by to the edge of the potential well, we furthermore show that the ground state changes regularity, which depends qualitatively on the fractional power of the non-local operator.