Maximal accretive extensions of Schr\"odinger operators on vector bundles over infinite graphs

TL;DR

This paper investigates the conditions under which Schr"odinger operators on vector bundles over infinite graphs are essentially self-adjoint and generate contraction semigroups, extending spectral theory in discrete geometric settings.

Contribution

It introduces a framework for analyzing maximal accretive extensions of Schr"odinger operators on vector bundles over infinite graphs, including criteria for self-adjointness and semigroup generation.

Findings

Provides sufficient conditions for essential self-adjointness.

Characterizes when Schr"odinger operators generate contraction semigroups.

Extends spectral analysis to vector bundle Laplacians on 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 the essential self-adjointness of a perturbation of this Laplacian by an operator-valued potential. Additionally, we give a sufficient condition for the resulting Schr\"odinger operator to serve as the generator of a strongly continuous contraction semigroup in the corresponding l^{p}-space.