Essential selfadjointness of the graph-Laplacian

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 $l^2$. 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.

50 pages with TOC and figures

References in corpus (3)

• Iterated function systems, representations, and Hilbert space
• Harmonic analysis of iterated function systems with overlap
• Commuting self-adjoint extensions of symmetric operators defined from the partial derivatives

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