# Kinetic equation for finite systems of fermions with pairing

## TL;DR

This paper derives a kinetic equation for finite fermionic systems with pairing, addressing particle-number conservation issues in the Hartree-Fock-Bogoliubov framework and analyzing the collective response and excitation spectrum.

## Contribution

It introduces a method to restore particle-number conservation in the Wigner-transformed HFB equations and derives explicit expressions for eigenfrequencies and response functions.

## Key findings

- Restoration of particle-number conservation eliminates spurious contributions.
- Semiclassical spectrum exhibits an excitation gap of 2Δ.
- Derived response functions agree with quantum results.

## Abstract

The solutions of the Wigner-transformed time-dependent Hartree--Fock--Bogoliubov equations are studied in the constant-$\Delta$ approximation. This approximation is known to violate particle-number conservation. As a consequence, the density fluctuation and the longitudinal response function given by this approximation contain spurious contributions. A simple prescription for restoring both local and global particle-number conservation is proposed. Explicit expressions for the eigenfrequencies of the correlated systems and for the density response function are derived and it is shown that the semiclassical analogous of the quantum single--particle spectrum has an excitation gap of $2\Delta$, in agreement with the quantum result. The collective response is studied for a simplified form of the residual interaction.