{"date_created":"2019-03-28T09:20:06Z","article_number":"1804.07752","date_published":"2018-04-20T00:00:00Z","title":"The Dyson equation with linear self-energy: Spectral bands, edges and cusps","external_id":{"arxiv":["1804.07752"]},"year":"2018","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"6183","date_updated":"2021-01-12T08:06:36Z","type":"preprint","department":[{"_id":"LaEr"}],"status":"public","month":"04","oa_version":"Preprint","language":[{"iso":"eng"}],"article_processing_charge":"No","publication_status":"submitted","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"149"}]},"author":[{"last_name":"Alt","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","full_name":"Alt, Johannes","first_name":"Johannes"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"Erdös, László"},{"full_name":"Krüger, Torben H","first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-4821-3297","last_name":"Krüger"}],"citation":{"mla":"Alt, Johannes, et al. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” *ArXiv*, 1804.07752.","ista":"Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. arXiv, 1804.07752.","ieee":"J. Alt, L. Erdös, and T. H. Krüger, “The Dyson equation with linear self-energy: Spectral bands, edges and cusps,” *arXiv*. .","chicago":"Alt, Johannes, László Erdös, and Torben H Krüger. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” *ArXiv*, n.d.","short":"J. Alt, L. Erdös, T.H. Krüger, ArXiv (n.d.).","apa":"Alt, J., Erdös, L., & Krüger, T. H. (n.d.). The Dyson equation with linear self-energy: Spectral bands, edges and cusps. *arXiv*.","ama":"Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. *arXiv*."},"publication":"arXiv","abstract":[{"text":"We study the unique solution $m$ of the Dyson equation \\[ -m(z)^{-1} = z - a\r\n+ S[m(z)] \\] on a von Neumann algebra $\\mathcal{A}$ with the constraint\r\n$\\mathrm{Im}\\,m\\geq 0$. Here, $z$ lies in the complex upper half-plane, $a$ is\r\na self-adjoint element of $\\mathcal{A}$ and $S$ is a positivity-preserving\r\nlinear operator on $\\mathcal{A}$. We show that $m$ is the Stieltjes transform\r\nof a compactly supported $\\mathcal{A}$-valued measure on $\\mathbb{R}$. Under\r\nsuitable assumptions, we establish that this measure has a uniformly\r\n$1/3$-H\\\"{o}lder continuous density with respect to the Lebesgue measure, which\r\nis supported on finitely many intervals, called bands. In fact, the density is\r\nanalytic inside the bands with a square-root growth at the edges and internal\r\ncubic root cusps whenever the gap between two bands vanishes. The shape of\r\nthese singularities is universal and no other singularity may occur. We give a\r\nprecise asymptotic description of $m$ near the singular points. These\r\nasymptotics generalize the analysis at the regular edges given in the companion\r\npaper on the Tracy-Widom universality for the edge eigenvalue statistics for\r\ncorrelated random matrices [arXiv:1804.07744] and they play a key role in the\r\nproof of the Pearcey universality at the cusp for Wigner-type matrices\r\n[arXiv:1809.03971,arXiv:1811.04055]. We also extend the finite dimensional band\r\nmass formula from [arXiv:1804.07744] to the von Neumann algebra setting by\r\nshowing that the spectral mass of the bands is topologically rigid under\r\ndeformations and we conclude that these masses are quantized in some important\r\ncases.","lang":"eng"}],"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1804.07752","open_access":"1"}],"day":"20"}