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

Johannes Alt, Laszlo Erdos, Torben Krüger

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 $\mathrm{Im}m\ge 0$. Here, $…