{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"8b77cc8492cd49943920f839899b40c528742cae86deb7ea123d02e6b2be41bc","created":"2026-10-03T07:18:01Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"72687d6fb8945064531c0ec8cada023158ea5eae6ab5e80ddec7b4690cf4992a","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"proof","assisted_by":[],"external_id":"cm.sign-problem-numerics.analytic-continuation-limits","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Most electron simulations work in imaginary time, and converting their output into measurable spectra is mathematically unstable: tiny noise allows wildly different answers. The best possible resolution for a given noise level, and a way to certify an answer, are not established.","posed_since":"","precise":"Given $G(\\tau) = \\operatorname{integral} \\mathrm{d}w\\, K(\\tau, w) A(w)$ with $K = \\operatorname{exp}(-\\tau w)/(1 + \\operatorname{exp}(-\\beta w))$ on $0 \\le \\tau \\le \\beta$ and Gaussian noise of size $\\sigma$, determine the optimal resolution of the spectral function $A(w)$ (minimal resolvable width and amplitude of a feature at frequency $w$) as a function of $\\sigma, \\beta \\text{ and } w$, and construct estimators (maximum entropy, stochastic, Nevanlinna, Bayesian) with certified error bounds that reach it. An answer is proven bounds plus an estimator attaining them on benchmarks.","problem_ref":null,"references":"","settled_by":"A theorem bounding achievable resolution from below and an estimator whose certified intervals match that bound on test spectra.","status_note":"Nevanlinna interpolation (Fei, Yeh and Gull 2021) enforces causality exactly but gives no certified bounds for noisy data.","title":"Resolution limits of analytic continuation from imaginary-time 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