Self-adjoint and Markovian extensions of infinite quantum graphs

TL;DR

This paper explores the connection between graph boundaries and self-adjoint extensions of the Kirchhoff Laplacian on infinite metric graphs, introducing finite volume ends as a key concept for understanding Markovian extensions.

Contribution

It introduces the notion of finite volume ends for metric graphs and characterizes Markovian and self-adjoint extensions based on these ends, providing a geometric perspective.

Findings

Finite volume ends serve as the boundary for Markovian extensions.

The Gaffney Laplacian is self-adjoint if the graph has no finite volume ends.

Complete description of Markovian extensions when finitely many finite volume ends occur.

Abstract

We investigate the relationship between one of the classical notions of boundaries for infinite graphs, \emph{graph ends}, and self-adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph. We introduce the notion of \emph{finite volume} for ends of a metric graph and show that finite volume graph ends is the proper notion of a boundary for Markovian extensions of the Kirchhoff Laplacian. In contrast to manifolds and weighted graphs, this provides a transparent geometric characterization of the uniqueness of Markovian extensions, as well as of the self-adjointness of the Gaffney Laplacian -- the underlying metric graph does not have finite volume ends. If however finitely many finite volume ends occur (as is the case of edge graphs of normal, locally finite tessellations or Cayley graphs of amenable finitely generated groups), we provide a complete description of Markovian extensions upon introducing a suitable notion of traces of functions and normal derivatives on the set of graph ends.