What question did this study set out to answer?

To derive a Feynman-Kac type formula for relativistic Schrödinger operators with spin 1/2 and establish spatial decay of bound states.

April 1, 2026Open Access

Probabilistic representation and fall-off of bound states of relativistic Schrödinger operators with spin 1/2

Key Points

To derive a Feynman-Kac type formula for relativistic Schrödinger operators with spin 1/2 and establish spatial decay of bound states.
Derived Feynman-Kac formula involving Brownian motion, Poisson processes, and subordinators.
Analyzed unbounded magnetic fields and evaluated cases with zeros.
Utilized martingale methods to study spatial decay of bound states for varying potentials.
Energy comparison inequality is derived from the formulated Feynman-Kac formula.
Spatial decay of bound states is established for both growing and decaying potentials.

Abstract

A Feynman-Kac type formula for relativistic Schrödinger operators with unbounded vector potential and spin 1=2 is given in terms of a three-component process consisting of a Brownian motion, a Poisson process and a subordinator. This formula is obtained for unbounded magnetic fields and magnetic fields with zeros. From this formula an energy comparison inequality is derived. Spatial decay of bound states is established separately for growing and for decaying potentials by using martingale methods. © 2013 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

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Cite This Study

廣島 et al. (Tue,) studied this question.

synapsesocial.com/papers/69cd7b475652765b073a91db https://doi.org/https://doi.org/10.24517/00042078

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