A generalized Sham-Schl\"uter equation: the link between nonlocality and frequency dependence

We present an in principle exact approach to construct effective potentials and kernels for the calculation of electronic spectra. In particular, the potential that yields the spectral function needed to describe photoemission turns out to be dynamical but local and real. Using explicit examples we illustrate how the nonlocality of the physical self-energy is converted into a frequency dependence of the effective potential. We also show that our approach leads to a very short derivation of a kernel that gives a very good description of absorption or energy-loss spectra of a wide range of materials.