Spectral Theory for Discrete Laplacians

    Authors

    D. E. Dutkay;P. E. T. Jorgensen

    Comments

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    Abbreviated Journal Title

    Complex Anal. Oper. Theory

    Keywords

    Operators in Hilbert space; discrete Laplacians; infinite graphs; spectral representation; spectral measures; multiplicity tables; semicircle laws; rank-one perturbations; spectrum; absolutely; continuous; graph Laplacian; electrical network; RANK-ONE PERTURBATIONS; SINGULAR CONTINUOUS-SPECTRUM; CATALAN NUMBERS; HILBERT-SPACE; OPERATORS; DIMENSIONS; TREES; SUBSPACES; H-2-CLASS; MATRICES; Mathematics, Applied; Mathematics

    Abstract

    We give the spectral representation for a class of selfadjoint discrete graph Laplacians Delta, with Delta depending on a chosen graph G and a conductance function c defined on the edges of G. We show that the spectral representations for Delta 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 Delta depends on c. Using Delta(G), 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.

    Journal Title

    Complex Analysis and Operator Theory

    Volume

    4

    Issue/Number

    1

    Publication Date

    1-1-2010

    Document Type

    Article

    Language

    English

    First Page

    1

    Last Page

    38

    WOS Identifier

    WOS:000277137200001

    ISSN

    1661-8254

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