Feynman path integrals for magnetic Schr\"odinger operators on infinite weighted graphs

TL;DR

This paper establishes a Feynman path integral formula for magnetic Schr"odinger operators on infinite graphs, leading to new estimates that control the unitary group uniformly across potentials.

Contribution

It introduces a Feynman path integral representation for magnetic Schr"odinger operators on infinite graphs and derives a novel Kato-Simon estimate for the associated unitary group.

Findings

Path integral formula for magnetic Schr"odinger operators on infinite graphs

A new Kato-Simon estimate controlling the unitary group

Uniform bounds in terms of the graph's weighted degree

Abstract

We prove a Feynman path integral formula for the unitary group $ \exp(-itL_{v,\theta})$, $t\geq 0$, associated with a discrete magnetic Schr\"odinger operator $L_{v,\theta}$ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate $$ |\exp(-itL_{v,\theta})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y), $$ which controls the unitary group uniformly in the potentials in terms of a Schr\"odinger semigroup, where the potential $\mathrm{deg}$ is the weighted degree function of the graph.