{"paper":{"title":"The Dyson equation with linear self-energy: spectral bands, edges and cusps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP","math.PR"],"primary_cat":"math.OA","authors_text":"Johannes Alt, Laszlo Erdos, Torben Kr\\\"uger","submitted_at":"2018-04-20T17:54:30Z","abstract_excerpt":"We study the unique solution $m$ of the Dyson equation \\[ -m(z)^{-1} = z - a + S[m(z)] \\] on a von Neumann algebra $\\mathcal{A}$ with the constraint $\\mathrm{Im}\\,m\\geq 0$. Here, $z$ lies in the complex upper half-plane, $a$ is a self-adjoint element of $\\mathcal{A}$ and $S$ is a positivity-preserving linear operator on $\\mathcal{A}$. We show that $m$ is the Stieltjes transform of a compactly supported $\\mathcal{A}$-valued measure on $\\mathbb{R}$. Under suitable assumptions, we establish that this measure has a uniformly $1/3$-H\\\"{o}lder continuous density with respect to the Lebesgue measure,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.07752","kind":"arxiv","version":5},"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"}