Recent probabilistic results on covariant Schrödinger operators on infinite weighted graphs

Abstract

We review recent probabilistic results on covariant Schrodinger operators on vector bundles over (possibly locally infinite) weighted graphs, and explain applications like semiclassical limits. We also clarify the relationship between these results and their formal analogues on smooth (possibly noncompact) Riemannian manifolds. 1 A review of covariant Schrodinger operators on smooth Riemannian manifolds Let us start by taking a look at covariant Schrodinger-type operators on Riemannian manifolds: Assume that E →M is a smooth finite dimensional Hermitian vector bundle over a possibly noncompact smooth Riemannian manifold M without boundary, with a Hermitian covariant derivative ∇ on E, which means that ∇ is a complex linear map ∇ : ΓC∞(M,E) −→ ΩC∞(M,E), such that for all Ψ1,Ψ2 ∈ ΓC∞(M,E) one has: d(Ψ1,Ψ2) = (∇Ψ1,Ψ2) + (Ψ1,∇Ψ2). (1) Then the symmetric nonnegative sesquilinear form