Prove that the binding energy of the ground state for the quasispin model in Problem 31.3 is linear in the number of pairs of particles $N$ for small $N$.

Data from Problem 31.3

Show that for a Hamiltonian of the form

\[ H=-G \sum_{m, m^{\prime}>0} a_{m^{\prime}}^{\dagger} a_{-m^{\prime}}^{\dagger} a_{-m} a_{m} \]

the energy eigenvalues for the quasispin model of Problem 31.1 are given by

\[ E=G\left(S(S+1)+\frac{1}{4}(N-\Omega)^{2}+\frac{1}{2}(N-\Omega)\right) \]

where $N$ is half the particle number, $\Omega=\frac{1}{2}(2 j+1)$ is half the shell degeneracy, and $S(S+1)$ is the eigenvalue of the operator

\[ S^{2}=\frac{1}{2} \sum_{m>0}\left(s_{+}^{(m)} s_{-}^{(m)}+s_{-}^{(m)} s_{+}^{(m)}+s_{0}^{(m)} s_{0}^{(m)}\right) \]

corresponding to the total quasispin.