---
_id: '14694'
abstract:
- lang: eng
  text: We study the unique solution m of the Dyson equation \( -m(z)^{-1} = z\1 -
    a + S[m(z)] \) on a von Neumann algebra A with the constraint Imm≥0. Here, z lies
    in the complex upper half-plane, a is a self-adjoint element of A and S is a positivity-preserving
    linear operator on A. We show that m is the Stieltjes transform of a compactly
    supported A-valued measure on R. Under suitable assumptions, we establish that
    this measure has a uniformly 1/3-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 m 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.
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Johannes
  full_name: Alt, Johannes
  id: 36D3D8B6-F248-11E8-B48F-1D18A9856A87
  last_name: Alt
- first_name: László
  full_name: Erdös, László
  id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
  last_name: Erdös
  orcid: 0000-0001-5366-9603
- first_name: Torben H
  full_name: Krüger, Torben H
  id: 3020C786-F248-11E8-B48F-1D18A9856A87
  last_name: Krüger
  orcid: 0000-0002-4821-3297
citation:
  ama: 'Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral
    bands, edges and cusps. <i>Documenta Mathematica</i>. 2020;25:1421-1539. doi:<a
    href="https://doi.org/10.4171/dm/780">10.4171/dm/780</a>'
  apa: 'Alt, J., Erdös, L., &#38; Krüger, T. H. (2020). The Dyson equation with linear
    self-energy: Spectral bands, edges and cusps. <i>Documenta Mathematica</i>. EMS
    Press. <a href="https://doi.org/10.4171/dm/780">https://doi.org/10.4171/dm/780</a>'
  chicago: 'Alt, Johannes, László Erdös, and Torben H Krüger. “The Dyson Equation
    with Linear Self-Energy: Spectral Bands, Edges and Cusps.” <i>Documenta Mathematica</i>.
    EMS Press, 2020. <a href="https://doi.org/10.4171/dm/780">https://doi.org/10.4171/dm/780</a>.'
  ieee: 'J. Alt, L. Erdös, and T. H. Krüger, “The Dyson equation with linear self-energy:
    Spectral bands, edges and cusps,” <i>Documenta Mathematica</i>, vol. 25. EMS Press,
    pp. 1421–1539, 2020.'
  ista: 'Alt J, Erdös L, Krüger TH. 2020. The Dyson equation with linear self-energy:
    Spectral bands, edges and cusps. Documenta Mathematica. 25, 1421–1539.'
  mla: 'Alt, Johannes, et al. “The Dyson Equation with Linear Self-Energy: Spectral
    Bands, Edges and Cusps.” <i>Documenta Mathematica</i>, vol. 25, EMS Press, 2020,
    pp. 1421–539, doi:<a href="https://doi.org/10.4171/dm/780">10.4171/dm/780</a>.'
  short: J. Alt, L. Erdös, T.H. Krüger, Documenta Mathematica 25 (2020) 1421–1539.
corr_author: '1'
date_created: 2023-12-18T10:37:43Z
date_published: 2020-09-01T00:00:00Z
date_updated: 2025-04-15T08:05:00Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.4171/dm/780
external_id:
  arxiv:
  - '1804.07752'
file:
- access_level: open_access
  checksum: 12aacc1d63b852ff9a51c1f6b218d4a6
  content_type: application/pdf
  creator: dernst
  date_created: 2023-12-18T10:42:32Z
  date_updated: 2023-12-18T10:42:32Z
  file_id: '14695'
  file_name: 2020_DocumentaMathematica_Alt.pdf
  file_size: 1374708
  relation: main_file
  success: 1
file_date_updated: 2023-12-18T10:42:32Z
has_accepted_license: '1'
intvolume: '        25'
keyword:
- General Mathematics
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: 1421-1539
publication: Documenta Mathematica
publication_identifier:
  eissn:
  - 1431-0643
  issn:
  - 1431-0635
publication_status: published
publisher: EMS Press
quality_controlled: '1'
related_material:
  record:
  - id: '6183'
    relation: earlier_version
    status: public
status: public
title: 'The Dyson equation with linear self-energy: Spectral bands, edges and cusps'
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2020'
...