A formally exact one-frequency-only Bethe-Salpeter-like equation. Similarities and differences between GW +BSE and self-consistent RPA

A formally exact Bethe-Salpeter-like equation for the linear-response function is introduced with a kernel which depends only on the one frequency of the applied field. This is in contrast with the standard Bethe-Salpeter equation (BSE) which involves multiple-frequency integrals over the kernel and response functions. From the one-frequency kernel, known approximations are straightforwardly recovered. However, the present formalism lends itself to more powerful approximations. This is demonstrated with the exact analytical solution of the Hubbard molecule. Similarities and differences of the $GW$+BSE approach with the self-consistent random-phase approximation (RPA) is also discussed.