Maximal Accretive Extensions of Schrödinger Operators on Vector Bundles over Infinite Graphs

Abstract

Given a Hermitian vector bundle over an infinite weighted graph, we define the Laplacian associated to a unitary connection on this bundle and study a perturbation of this Laplacian by an operator-valued potential. We give a sufficient condition for the resulting Schrödinger operator to serve as the generator of a strongly continuous contraction semigroup in the corresponding ℓ p -space. Additionally, in the context of ℓ 2 -space, we study the essential self-adjointness of the corresponding Schrödinger operator.

Domination of quadratic forms

“…In our setup we essentially follow the works [12], [13] for graphs and Dirichlet forms over discrete spaces and the article [16] for vector bundles over graphs and magnetic Schrödinger operators. Further discussion can be found in these references.…”

Semiclassical limits of quantum partition functions on infinite graphs

“…Proof of Lemma 2.3. a) By Cauchy-Schwarz we get that (7) implies the second inclusion in (8), and Green's formula (cf. Lemma 3.1 in [16]) shows that the first inclusion in (8) implies (9). It remains to prove that (7) implies the first inclusion in (8).…”

“…We refer the reader to [16] for problems concerning the explicit domain of definition of H Φ,V , and essential-selfadjointness questions related with H Φ,V .…”

Recent probabilistic results on covariant Schrödinger operators on infinite weighted graphs

“…We fix a Kato decomposable potential V on F → X, where we refer the reader to [9] for essential self-adjointness properties of H Φ,V . Let us now prepare the ingredients for the Feynman-Kac formula: As Q is a regular Dirichlet form on a nice space, we can associate a reversible strong right-Markoff process to it.…”