# Eigenvalue equation for interacting Green's functions

Studying the articles "Topological Hamiltonian as an exact tool for topological invariants" (https://arxiv.org/abs/1207.7341) and "Simplified Topological Invariants for Interacting Insulators" (https://arxiv.org/abs/1201.6431) I stumbled upon the following eigenvalue equation

\begin{align} G^{-1}(k,i\omega) |\alpha(k,i\omega)> &= \mu_\alpha(k,i\omega) |\alpha(k,i\omega)> \end{align}

where

\begin{align} G^{-1}(k,i\omega) &= i\omega - H_0(k) - \Sigma(k,i\omega) \end{align}

is the full Green's function of an interacting problem evaluated at the Matsubara frequencies $\omega_n = (2n+1)\pi/\beta$ (Probably a similar formular also holds for real frequencies, then just replace $i\omega \rightarrow \omega$). $H_0(k)$ is the Hamiltonian matrix of the noninteracting problem and $\Sigma(k,i\omega)$ the $\omega$ dependent self-energy that appears in Dyson's equation $G = G_0 + G_0 \Sigma G$ (I come from a condensed matter background and know Green's functions from that context.). On a mathematical level, $G^{-1}(k,i\omega) \in\$GL(N,C) is an invertible complex $N\times N$ matrix, and $\{|\alpha(k,i\omega)>\}$ a set of orthogonal eigenvectors of that matrix with eigenvalues $\mu_\alpha(k,i\omega)$.

The eigenvalue equation can also be written as

\begin{align} \left(i\omega - H_0(k) - \Sigma(k,i\omega)\right) |\alpha(k,i\omega)> &= \mu_\alpha(k,i\omega) |\alpha(k,i\omega)>\\ \left(H_0(k) + \Sigma(k,i\omega)\right) |\alpha(k,i\omega)> &= -(\mu_\alpha(k,i\omega)+i\omega) |\alpha(k,i\omega)> \end{align}

where the last line looks pretty similar to the time independent Schroedinger equation of a noninteracting system

\begin{align} H_0(k) |\alpha(k)> &= \epsilon(k)|\alpha(k)> \end{align}

with the big difference that the first equation depends on $\omega$ and the second does not. I want to understand that equation and have the following questions

• What is the meaning of an $\omega$-dependent eigenstate?
• What is the meaning of an $\omega$-dependent (energy?) eigenvalue?