Probabilistic Representation and Fall-Off of Bound States of Relativistic Schr\"odinger Operators with Spin 1/2

TL;DR

This paper derives a Feynman-Kac type formula for relativistic Schrödinger operators with spin 1/2, unbounded vector potentials, and magnetic fields with zeros, and analyzes the spatial decay of bound states.

Contribution

It introduces a novel probabilistic representation for these operators and establishes decay properties of bound states under various potential conditions.

Findings

Derived a Feynman-Kac formula for complex relativistic operators

Established energy comparison inequalities for bound states

Proved spatial decay of bound states for different potential types

Abstract

A Feynman-Kac type formula of relativistic Schr\"odinger operators with unbounded vector potential and spin 1/2 is given in terms of a three-component process consisting of Brownian motion, a Poisson process and a subordinator. This formula is obtained for unbounded magnetic fields and magnetic field with zeros. From this formula an energy comparison inequality is derived. Spatial decay of bound states is established separately for growing and decaying potentials by using martingale methods.