# Quasi-low-dimensional electron gas with one populated band as a testing ground for time-dependent density-functional theory

## TL;DR

This paper analytically solves the time-dependent density-functional theory for a quasi-low-dimensional electron gas with one filled band, providing insights into many-body interactions and excitation spectra.

## Contribution

It presents an explicit nonlocal exchange potential solution for TDDFT in quasi-low-dimensional systems with one populated band, advancing theoretical understanding.

## Key findings

- Calculated interband excitation spectra of Q2DEG.
- Identified cancellations between Hartree and exchange kernels at low density.
- Compared TDDFT and Kohn-Sham transition insights.

## Abstract

We find the analytical solution to the time-dependent density-functional theory (TDDFT) problem for the quasi-low-dimensional (2D and 1D) electron gas (QLDEG) with only one band filled in the direction perpendicular to the system extent. The theory is developed at the level of TD exact exchange and yields the exchange potential as an explicit nonlocal operator of the spin-density. The dressed interband (image states) excitation spectra of the Q2DEG are calculated, while the comparison with the Kohn-Sham (KS) transitions provides the insight into the qualitative and quantitative role of the many-body interactions. Important cancellations between the Hartree $f_H$ and the exchange $f_x$ kernels are found in the low-density limit, shedding light on the interrelations between the KS and many-body excitations.