Feynman path integrals for magnetic Schrödinger operators on infinite weighted graphs

Abstract

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