Functional theory of the occupied spectral density

We address the problem of interacting electrons in an external potential by introducing the occupied spectral density $ρ(\mathbf{r},ω)$ as fundamental variable. First, we formulate the problem using an embedding framework, and prove a one-to-one correspondence between a $ρ(\mathbf{r},ω)$ and the local dynamical external potential $v_{\text{ext}}(\mathbf{r},ω)$ that embeds the interacting electrons into an open quantum system. Then, we use the Klein functional to ($i$) define a universal functional of $ρ(\mathbf{r},ω)$, ($ii$) introduce a variational principle for the total energy as a functional of $ρ(\mathbf{r},ω)$, and ($iii$) formulate a non-interacting mapping of spectral self-consistent equations suitable for numerical applications. At variance with time-dependent density-functional theory, this formulation aims at studying charged excitations and electronic spectra -- including electronic correlations -- with a functional theory; An explicit and formally correct description of all electronic levels could also lead to more accurate approximations for the total energy.