2011/03/29 by Palle E. T. Jørgensen, Palle E. T. Jorgensen, Jorgensen, Palle E. T. +2
Computer Science · Mathematics · Physics and Astronomy · #37A30 #46B22 #46E22 #47B15 #47B25 #47B32 #60J10. Secondary: 47B39 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Primary: 05C50 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Topological Materials and Phenomena #math-ph #math.FA #math.MP #math.SP #msc:05C50 #msc:37A30 #msc:46B22 #msc:46E22 #msc:47B15 #msc:47B25 #msc:47B32 #msc:47B39 #msc:60J10.
paper · pdf · doi:10.48550/arxiv.1103.5792
24 pages, 0 figures. Length reduced per referee recommendations
openalex publication_date 2011/03/29 · arxiv created 2012/03/06 · arxiv updated 2012/03/07 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
Let (G,c) be an infinite network, and let E be the canonical energy form. Let Δ2 be the Laplace operator with dense domain in ℓ2(G) and let ΔE be the Laplace operator with dense domain in the Hilbert space HE of finite energy functions on G. It is known that Δ2 is essentially self-adjoint, but that ΔE is not. In this paper, we characterize the Friedrichs extension of ΔE in terms of Δ2 and show that the spectral measures of the two operators are mutually absolutely continuous with Radon-Nikodym derivative λ (the spectral parameter), in the complement of λ=0. We also give applications to the effective resistance on (G,c). For transient networks, the Dirac measure at λ= 0 contributes to the spectral resolution of the Friedrichs extension of ΔE but not to that of the self-adjoint ℓ2 Laplacian.