Self-adjoint and Markovian extensions of infinite quantum graphs

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.

to appear in J. London Math. Soc.; 47 pages, 2 figures

References in corpus (6)

• Spectral Estimates for Infinite Quantum Graphs
• Quantum graphs on radially symmetric antitrees
• Analysis of Dirichlet forms on graphs
• Canonical measures on metric graphs and a Kazhdan's theorem
• On the Decomposition of the Laplacian on Metric Graphs
• Zero Measure and Singular Continuous Spectra for Quantum Graphs

Cited by in corpus (3)

• Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra
• A note on the Gaffney Laplacian on infinite metric graphs
• Laplacians on infinite graphs: discrete vs continuous