Method for reconstructing the self-energy from the spectral function

A fundamental question about the nature of quantum materials such as high-${T}_{c}$ systems remains open to date: it is unclear whether they are (some variety of) Fermi liquids, or (some variety of) non Fermi liquids. A direct avenue to determine their nature is to study the (imaginary part of the) self-energy at low energies. Here we present a method to extract this low-$\ensuremath{\omega}$ self-energy from experimentally derived spectral functions. The method seems suited for implementation with high-quality angle-resolved photoemission data. It is based on a helpful theorem proposed here, which assures us that the method has minimal (or vanishing) error at the lowest energies. We provide numerical examples showing that a few popular model systems yield distinguishably different low-energy self-energies.