Evolution of the $N$-particle state in the BCS model

We obtain an exact scheme describing the dynamics of the $N$-particle state in the BCS model that is similar to the Gel'fand–Yaglom procedure for finding the transition amplitude of the harmonic oscillator with a time-dependent frequency. We find the many-particle spectrum and the eigenfunctions of the BCS Hamiltonian in the limit as the system volume tends to infinity.