Eigenvalues of the Nuclear Pairing Hamiltonian

A new technique is developed in this paper for solving Hamiltonians of the form $H={Σ}{}{k>0}{{ε}}ₖ({{a}ₖ}⁺{a}ₖ+{{a}_{{-}k}}⁺{a}_{{-}k}){-}{Σ}{}{k,l>0}G(k,l){{a}ₖ}⁺{{a}_{{-}k}}⁺{a}_{{-}l}{a}ₗ,$ where $G(k, l)$ may have the forms: $G(k, l)=G$ (independent of $k$ and $l$), $G(k, l)={G}ₖ{G}ₗ$, or $G(k, l)=G(k, l)$ (nonseparable in $k$ and $l$). The eigenvalues obtained with this technique are compared with the exact solutions of Kerman, Lawson, and Macfarlane. The agreement with the exact solutions is fairly good.