Generalized Schrödinger Semigroups on Infinite Graphs

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.

A note on the surjectivity of operators on vector bundles over discrete spaces

“…As mentioned above, we prove Eidelheit's surjectivity criterion for finite hopping range operators on finite-dimensional vector bundles over infinite discrete set. It includes the result for linear cellular automata from [17] but can also be applied to magnetic Schrödinger operators on bundles, which were recently introduced in [6]. Our result for magnetic Schrödinger operators (Theorem 2.2) contains the aforementioned results for graph Laplacians and discrete Schrödinger operators (graph Laplacian plus real potential).…”

Semiclassical limits of quantum partition functions on infinite graphs

“…One of our main results, Theorem 2.8, states that (5) indeed is always true, whenever v is such that H + v/ is well-defined as a form sum for all > 0. Here, in contrast to the continuum setting, we even do not have to assume e −βv < ∞, in the sense that the limit in (5) exists anyway.…”

Recent probabilistic results on covariant Schrödinger operators on infinite weighted graphs

“…Generalizing the scalar magnetic situation from [6], the following Feynman-Kac formula, which is our discrete analogue of (3), has been proven in [7]:…”

“…Let τ n : Ω → [0, ∞], n ∈ N 0 , denote the n-th jump time of X, and let N(t) : Ω → N 0 ∪ {∞} be its number of jumps until t ≥ 0. Many path properties of this process have been derived in [6,7]. We shall need in the sequel that τ = sup n τ n and…”