The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps

  • Johannes Alt

    Section of Mathematics, University of Geneva, 2-4 Rue du Lièvre, CH-1211 Genève 4, Switzerland
  • László Erdős

    IST Austria, Am Campus 1, A-3400 Klosterneuburg, Austria
  • Torben Krüger

    Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 København, Denmark
The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps cover
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We study the unique solution mm of the Dyson equation m(z)1=z\1a+S[m(z)]-m(z)^{-1} = z\1 - a + S[m(z)] on a von Neumann algebra A\mathcal{A} with the constraint Imm0\text{Im}\, m\ge 0. Here, zz lies in the complex upper half-plane, aa is a self-adjoint element of A\mathcal{A} and SS is a positivity-preserving linear operator on A\mathcal{A}. We show that mm is the Stieltjes transform of a compactly supported A\mathcal{A}-valued measure on R\mathbb{R}. Under suitable assumptions, we establish that this measure has a uniformly 1/31/3-Hölder continuous density with respect to the Lebesgue measure, which is supported on finitely many intervals, called bands. In fact, the density is analytic inside the bands with a square-root growth at the edges and internal cubic root cusps whenever the gap between two bands vanishes. The shape of these singularities is universal and no other singularity may occur. We give a precise asymptotic description of mm near the singular points. These asymptotics generalize the analysis at the regular edges given in the companion paper on the Tracy-Widom universality for the edge eigenvalue statistics for correlated random matrices [the first author et al., Ann. Probab. 48, No. 2, 963--1001 (2020; Zbl 1434.60017)] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner-type matrices [G. Cipolloni et al., Pure Appl. Anal. 1, No. 4, 615--707 (2019; Zbl 07142203); the second author et al., Commun. Math. Phys. 378, No. 2, 1203--1278 (2020; Zbl 07236118)]. We also extend the finite dimensional band mass formula from [the first author et al., loc. cit.] to the von Neumann algebra setting by showing that the spectral mass of the bands is topologically rigid under deformations and we conclude that these masses are quantized in some important cases.

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Johannes Alt, László Erdős, Torben Krüger, The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps. Doc. Math. 25 (2020), pp. 1421–1539

DOI 10.4171/DM/780