The Dyson equation with linear self-energy: Spectral bands, edges and cusps

preprint The Dyson equation with linear self-energy: Spectral bands, edges and cusps draft Johannes Alt author 36D3D8B6-F248-11E8-B48F-1D18A9856A87 László Erdös author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603 Torben H Krüger author 3020C786-F248-11E8-B48F-1D18A9856A870000-0002-4821-3297 LaEr department Random matrices, universality and disordered quantum systems project We study the unique solution $m$ of the Dyson equation \[ -m(z)^{-1} = z - a + S[m(z)] \] on a von Neumann algebra $\mathcal{A}$ with the constraint $\mathrm{Im}\,m\geq 0$. Here, $z$ lies in the complex upper half-plane, $a$ is a self-adjoint element of $\mathcal{A}$ and $S$ is a positivity-preserving linear operator on $\mathcal{A}$. We show that $m$ is the Stieltjes transform of a compactly supported $\mathcal{A}$-valued measure on $\mathbb{R}$. Under suitable assumptions, we establish that this measure has a uniformly $1/3$-H\"{o}lder 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 [arXiv:1804.07744] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner-type matrices [arXiv:1809.03971,arXiv:1811.04055]. We also extend the finite dimensional band mass formula from [arXiv:1804.07744] 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. 2018 eng arXiv 1804.0775210.48550/arXiv.1804.07752 https://research-explorer.ista.ac.at/record/14694 https://research-explorer.ista.ac.at/record/149 Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. arXiv. doi:<a href="https://doi.org/10.48550/arXiv.1804.07752">10.48550/arXiv.1804.07752</a> J. Alt, L. Erdös, T.H. Krüger, ArXiv (n.d.). Alt, Johannes, et al. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” ArXiv, 1804.07752, doi:<a href="https://doi.org/10.48550/arXiv.1804.07752">10.48550/arXiv.1804.07752</a>. Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. arXiv, 1804.07752. Alt, J., Erdös, L., & Krüger, T. H. (n.d.). The Dyson equation with linear self-energy: Spectral bands, edges and cusps. arXiv. <a href="https://doi.org/10.48550/arXiv.1804.07752">https://doi.org/10.48550/arXiv.1804.07752</a> J. Alt, L. Erdös, and T. H. Krüger, “The Dyson equation with linear self-energy: Spectral bands, edges and cusps,” arXiv. . Alt, Johannes, László Erdös, and Torben H Krüger. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” ArXiv, n.d. <a href="https://doi.org/10.48550/arXiv.1804.07752">https://doi.org/10.48550/arXiv.1804.07752</a>. 61832019-03-28T09:20:06Z2026-04-08T14:11:36Z