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Essential self-adjointness of the graph-Laplacian
Journal article   Peer reviewed

Essential self-adjointness of the graph-Laplacian

Palle E.T Jorgensen
Journal of mathematical physics, Vol.49(7), pp.073510-073510-33
2008
DOI: 10.1063/1.2953684
url
https://arxiv.org/pdf/0802.0133View
Open Access

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 Δ 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 Δ is automatically essential self-adjoint. By this we mean that Δ is defined on the dense subspace 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 Δ is self-adjoint 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 Δ = Δ 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.

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