{"paper":{"title":"Spectral results for free random variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"If the epsilon-derivative of the log-potential admits a real analytic extension through zero, then lambda lies outside the spectrum of a.","cross_cats":["math-ph","math.MP","math.PR"],"primary_cat":"math.OA","authors_text":"Brian C. Hall, Ching-Wei Ho","submitted_at":"2025-10-03T15:52:14Z","abstract_excerpt":"Let $(\\mathcal{A},\\mathrm{tr})$ be a von Neumann algebra with a faithful, normal trace $\\mathrm{tr}:\\mathcal{A}\\rightarrow\\mathbb{C}.$ For each $a\\in\\mathcal{A},$ define \\[ S(\\lambda,\\varepsilon)=\\mathrm{tr}[\\log((a-\\lambda)^{\\ast}(a-\\lambda )+\\varepsilon)],\\quad\\lambda\\in\\mathbb{C},~\\varepsilon>0, \\] so that the limit as $\\varepsilon\\rightarrow0^{+}$ of $S$ is the log potential of the Brown measure of $a.$ Suppose that for a fixed $\\lambda\\in\\mathbb{C},$ the function \\[ \\varepsilon\\mapsto\\frac{\\partial S}{\\partial\\varepsilon}(\\lambda ,\\varepsilon)=\\mathrm{tr}[((a-\\lambda)^{\\ast}(a-\\lambda)+\\v"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Suppose that for a fixed λ∈ℂ, the function ε↦∂S/∂ε(λ,ε)=tr[((a−λ)∗(a−λ)+ε)−1] admits a real analytic extension to a neighborhood of 0 in ℝ. Then λ is outside the spectrum of a.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The implication from real-analytic extendability of the ε-derivative at zero to λ lying in the resolvent set relies on the specific functional-analytic properties of the trace and the definition of the Brown measure via the limit of S as ε→0+ (abstract, first paragraph).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A criterion linking real-analytic extendability of the ε-derivative of the log-potential to λ being outside the spectrum is established and used to show spectrum equals Brown-measure support for circular, elliptic, and free multiplicative Brownian motion elements.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If the epsilon-derivative of the log-potential admits a real analytic extension through zero, then lambda lies outside the spectrum of a.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3c3dc3adcbbf12a45a8e8e815ceb16034ec9dfe37c3f88e3b709173199ad152d"},"source":{"id":"2510.03382","kind":"arxiv","version":5},"verdict":{"id":"5b3d44c3-eb67-4b2a-91b1-31956ca9c97e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T10:57:16.369272Z","strongest_claim":"Suppose that for a fixed λ∈ℂ, the function ε↦∂S/∂ε(λ,ε)=tr[((a−λ)∗(a−λ)+ε)−1] admits a real analytic extension to a neighborhood of 0 in ℝ. Then λ is outside the spectrum of a.","one_line_summary":"A criterion linking real-analytic extendability of the ε-derivative of the log-potential to λ being outside the spectrum is established and used to show spectrum equals Brown-measure support for circular, elliptic, and free multiplicative Brownian motion elements.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The implication from real-analytic extendability of the ε-derivative at zero to λ lying in the resolvent set relies on the specific functional-analytic properties of the trace and the definition of the Brown measure via the limit of S as ε→0+ (abstract, first paragraph).","pith_extraction_headline":"If the epsilon-derivative of the log-potential admits a real analytic extension through zero, then lambda lies outside the spectrum of a."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.03382/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"49c4873d961dd61b8e20e068194f052cf1192992a5ad6d21199e68d7bcd7c8af"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}