If one assumes that energy levels are never half filled (always occupied by either 0 or 2 fermions), then the pairing model is equivalent to a system of \( N \) pairs of fermions that occupy \( P \) doubly-degenerate energy levels $$ \begin{align} H = 2\sum_{p} \delta_pA_p^{\dagger}A_p+\sum_{pq}g_{pq}A_p^{\dagger}A_q, \tag{7} \end{align} $$ where \( p \) and \( q \) sum from over the set \( \{1,...,p\} \) and $$ \begin{align*} A_p &= a_{p-}a_{p+} \\ A^{\dagger}_p &= a^{\dagger}_{p+}a^{\dagger}_{p-}, \end{align*} $$ are the fermionic pair creation and annihilation operators.