{"paper":{"title":"Self-adjoint extensions of network Laplacians and applications to resistance metrics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.SP","authors_text":"Erin P. J. Pearse, Palle E. T. Jorgensen","submitted_at":"2011-03-29T23:17:48Z","abstract_excerpt":"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 absolu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.5792","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}