Journal article
Unbounded graph-Laplacians in energy space, and their extensions
Journal of Applied Mathematics and Computing, Vol.39(1), pp.155-187
06/2012
DOI: 10.1007/s12190-011-0518-8
Abstract
Our purpose is to develop computational tools for determining spectra for operators associated with infinite weighted graphs. While there is a substantial literature concerning graph-Laplacians on infinite networks, much less developed is the distinction between the operator theory for the ℓ 2 space of the set V of vertices vs the case when the Hilbert space is defined by an energy form. A network is a triple (V,E,c) where V is a (typically countable infinite) set of vertices in a graph, with E denoting the set of edges. The function c is defined on E. It is given at the outset, symmetric and positive on E. We introduce a graph-Laplacian Δ, and an energy Hilbert space $\mathcal{H}_{E}$ (both depending on c). While it is known that Δ is essentially selfadjoint on its natural domain in ℓ 2(V), its realization in $\mathcal{H}_{E}$ is not. We give a characterization of the Friedrichs extension of the $\mathcal{H}_{E}$ -Laplacian, and prove a formula for its computation. We obtain several corollaries regarding the diagonalization of infinite matrices. To every weighted finite-interaction countable infinite graph there is a naturally associated infinite banded matrix. With the use of the Friedrichs spectral resolution, we obtain a diagonalization formula for this family of infinite matrices. With examples we give concrete illustrations of both spectral types, and spectral multiplicities.
Details
- Title: Subtitle
- Unbounded graph-Laplacians in energy space, and their extensions
- Creators
- Palle Jorgensen - Department of Mathematics University of Iowa Iowa City IA 52242-1419 USA
- Resource Type
- Journal article
- Publication Details
- Journal of Applied Mathematics and Computing, Vol.39(1), pp.155-187
- Publisher
- Springer-Verlag; Berlin/Heidelberg
- DOI
- 10.1007/s12190-011-0518-8
- ISSN
- 1598-5865
- eISSN
- 1865-2085
- Language
- English
- Date published
- 06/2012
- Academic Unit
- Mathematics
- Record Identifier
- 9983985924302771
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