Exactly Solvable Pairing Model Using an Extension of Richardson-gaudin Approach

The notion of pairing plays a central role in the BCS theory of superconductivity. In 1963, R.W. Richardson showed that exact energy eigenvalues and eigenstates of the BCS pairing Hamiltonian can be computed if one can solve a given set of highly coupled nonlinear equations. The limit of these equations in which the BCS coupling constant is very large were also obtained by Gaudin in 1976. Gaudin‘s tool was the algebraic Bethe ansatz method. Gaudin’s Hamiltonians are the constants of motion of BCS Hamiltonian in the large coupling constant limit. In this paper, we present a new class of exactly solvable boson pairing models using the technique pioneered by Richardson and Gaudin. The technique is outlined in the next section and in Section 3 we introduce the new class of exactly solvable models together with all energy eigenvalues and first few energy eigenstates. Richardson-Gaudin technique is based on an equation of which three solutions are known. To each solution there is a corresponding set of mutually commuting Hamiltonians and a related Lie algebra. In Section 4 we present another solution