# Continuity equation for the many-electron spectral function

## TL;DR

This paper derives a continuity equation for the many-electron spectral function using a dynamical exchange-correlation framework, simplifying the calculation of spectral functions by focusing on the diagonal Green function and proposing a Kohn-Sham-like scheme.

## Contribution

It introduces a continuity equation approach for the many-electron spectral function and a Kohn-Sham-like scheme to simplify exchange-correlation potential approximations.

## Key findings

- Derived a continuity equation for the spectral function.
- Proposed a Kohn-Sham-like scheme for spectral function calculation.
- Provided a formal solution for the spectral function using approximate exchange-correlation fields.

## Abstract

Starting from the recently proposed dynamical exchange-correlation field framework, the equation of motion of the diagonal part of the many-electron Green function is derived, from which the spectral function can be obtained. The resulting equation of motion takes the form of the continuity equation of charge and current densities in electrodynamics with a source. An unknown quantity in this equation is the current density, corresponding to the kinetic energy. A procedure \`a la Kohn-Sham scheme is then proposed, in which the difference between the kinetic potential of the interacting system and the non-interacting Kohn-Sham system is shifted into the exchange-correlation field. The task of finding a good approximation for the exchange-correlation field should be greatly simplified since only the diagonal part is needed. A formal solution to the continuity equation provides an explicit expression for calculating the spectral function, given an approximate exchange-correlation field.