{"paper":{"title":"Degenerate elliptic operators in one dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Adam Sikora, Derek W. Robinson","submitted_at":"2009-09-03T04:01:19Z","abstract_excerpt":"Let $H$ be the symmetric second-order differential operator on $L_2(\\Ri)$ with domain $C_c^\\infty(\\Ri)$ and action $H\\varphi=-(c \\varphi')'$ where $ c\\in W^{1,2}_{\\rm loc}(\\Ri)$ is a real function which is strictly positive on $\\Ri\\backslash\\{0\\}$ but with $c(0)=0$. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $\\nu=\\nu_+\\vee\\nu_-$ where $\\nu_\\pm(x)=\\pm\\int^{\\pm 1}_{\\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $\\nu\\not\\in L_2(0,1)$ and a unique submarkovian extension if and only if $\\nu\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.0567","kind":"arxiv","version":1},"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"}