Self-adjoint extensions of network Laplacians and applications to resistance metrics

Abstract

Let $(G,c)$ be an infinite network, and let $\mathcal{E}$ be the canonical energy form. Let ${\Delta }_2$ be the Laplace operator with dense domain in ${\ell }^2(G)$ and let ${\Delta }_{\mathcal{E}}$ be the Laplace operator with dense domain in the Hilbert space ${\mathcal{H}}_{\mathcal{E}}$ of finite energy functions on $G$. It is known that ${\Delta }_2$ is essentially self-adjoint, but that ${\Delta }_{\mathcal{E}}$ is \emph{not}. In this paper, we characterize the Friedrichs extension of ${\Delta }_{\mathcal{E}}$ in terms of ${\Delta }_2$ and show that the spectral measures of the two operators are mutually absolutely continuous with Radon-Nikodym derivative $\lambda$ (the spectral parameter), in the complement of $\lambda =0$. We also give applications to the effective resistance on $(G,c)$. For transient networks, the Dirac measure at $\lambda =0$ contributes to the spectral resolution of the Friedrichs extension of ${\Delta }_{\mathcal{E}}$ but not to that of the self-adjoint ${\ell }^2$ Laplacian.

24 pages, 0 figures. Length reduced per referee recommendations

References in corpus (4)

• Stochastic Completeness of Graphs
• Operator theory of electrical resistance networks
• A discrete Gauss-Green identity for unbounded Laplace operators and transience of random walks
• Spectral Theory for Discrete Lapacians

Cited by in corpus (1)

• Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian