Title

Spectral Theory For Discrete Laplacians

Keywords

34B45; 46E22; 47L30; 54E70; 60J10; 81S30; Absolutely continuous; Discrete Laplacians; Electrical network; Graph Laplacian; Infinite graphs; Multiplicity tables; Operators in Hilbert space; Rank-one perturbations; Semicircle laws; Spectral measures; Spectral representation; Spectrum

Abstract

Abstract.: We give the spectral representation for a class of selfadjoint discrete graph Laplacians Δ, with Δ depending on a chosen graph G and a conductance function c defined on the edges of G. We show that the spectral representations for Δ fall in two model classes, (1) tree-graphs with N-adic branching laws, and (2) lattice graphs. We show that the spectral theory of the first class may be computed with the use of rank-one perturbations of the real part of the unilateral shift, while the second is analogously built up with the use of the bilateral shift. We further analyze the effect on spectra of the conductance function c: How the spectral representation of Δ depends on c. Using ΔG, we introduce a resistance metric, and we show that it embeds isometrically into an energy Hilbert space. We introduce an associated random walk and we calculate return probabilities, and a path counting number. © 2009 Birkhäuser Verlag Basel/Switzerland.

Publication Date

4-1-2010

Publication Title

Complex Analysis and Operator Theory

Volume

4

Issue

1

Number of Pages

1-38

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s11785-008-0098-2

Socpus ID

77952105651 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/77952105651

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