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Example case: pairing Hamiltonian

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 .