Because of the SU(2) algebra, we know that the eigenvalues of \( \hat{P}^2 \) must be of the form \( p(p+1) \), with \( p \) either integer or half-integer, and the eigenvalues of \( \hat{P}_z \) are \( m_p \) with \( p \geq | m_p| \), with \( m_p \) also integer or half-integer.

But because \( \hat{P}_z = (1/2)(\hat{N}-\Omega) \), we know that for \( N \) particles the value \( m_p = (N-\Omega)/2 \). Furthermore, the values of \( m_p \) range from \( -\Omega/2 \) (for \( N=0 \)) to \( +\Omega/2 \) (for \( N=2\Omega \), with all states filled).

We deduce the maximal \( p = \Omega/2 \) and for a given \( n \) the values range of \( p \) range from \( |N-\Omega|/2 \) to \( \Omega/2 \) in steps of 1 (for an even number of particles)

Following Racah we introduce the notation \( p = (\Omega - v)/2 \) where \( v = 0, 2, 4,..., \Omega - |N-\Omega| \) With this it is easy to deduce that the eigenvalues of the pairing Hamiltonian are $$ -G(N-v)(2\Omega +2-N-v)/4 $$ This also works for \( N \) odd, with \( v= 1,3,5, \dots \).