Essential selfadjointness of the graph-Laplacian

TL;DR

This paper proves that the graph Laplacian operator on infinite graphs, relevant in various fields, is inherently selfadjoint and typically has continuous spectrum, with a framework for approximating its spectral data via finite graphs.

Contribution

It establishes the essential selfadjointness of the graph Laplacian with conductance functions and provides a method to approximate spectral data through finite graph sequences.

Findings

The Laplace operator is automatically essentially selfadjoint.

The spectrum of the operator is generically continuous.

Finite graph approximations converge to the spectral properties of the infinite graph.

Abstract

We study the operator theory associated with such infinite graphs $G$ as occur in electrical networks, in fractals, in statistical mechanics, and even in internet search engines. Our emphasis is on the determination of spectral data for a natural Laplace operator associated with the graph in question. This operator $\Delta$ will depend not only on $G$, but also on a prescribed positive real valued function $c$ defined on the edges in $G$. In electrical network models, this function $c$ will determine a conductance number for each edge. We show that the corresponding Laplace operator $\Delta$ is automatically essential selfadjoint. By this we mean that $\Delta$ is defined on the dense subspace $\mathcal{D}$ (of all the real valued functions on the set of vertices $G^{0}$ with finite support) in the Hilbert space $l^{2}% (G^{0})$. The conclusion is that the closure of the operator $\Delta$ is selfadjoint in $l^{2}(G^{0})$, and so in particular that it has a unique spectral resolution, determined by a projection valued measure on the Borel subsets of the infinite half-line. We prove that generically our graph Laplace operator $\Delta=\Delta_{c}$ will have continuous spectrum. For a given infinite graph $G$ with conductance function $c$, we set up a system of finite graphs with periodic boundary conditions such the finite spectra, for an ascending family of finite graphs, will have the Laplace operator for $G$ as its limit.