Generalized Schr\"odinger semigroups on infinite graphs

TL;DR

This paper develops Feynman-Kac-type formulas for Schrödinger semigroups on infinite graphs with Hermitian vector bundles, enabling new analysis tools for quantum systems modeled on complex graph structures.

Contribution

It introduces a generalized framework for Schrödinger semigroups on infinite graphs using Hermitian bundles and connections, extending previous finite or simpler models.

Findings

Established Feynman-Kac representations for these semigroups

Derived applications for quantum systems on complex graphs

Extended analysis techniques to locally infinite graph structures

Abstract

With appropriate notions of Hermitian vector bundles and connections over weighted graphs which we allow to be locally infinite, we prove Feynman-Kac-type representations for the corresponding semigroups and derive several applications thereof.