Polynomial algebras and exact solutions of general quantum nonlinear optical models I: two-mode boson systems

Abstract

We introduce higher order polynomial deformations of A1 Lie algebra. We construct their unitary representations and the corresponding single-variable differential operator realizations. We then use the results to obtain exact (Bethe ansatz) solutions to a class of two-mode boson systems, including the Bose–Einstein condensate (BEC) models as special cases. Up to an overall factor, the eigenfunctions of the two-mode boson systems are given by polynomials whose roots are solutions of the associated Bethe ansatz equations. The corresponding eigenvalues are expressed in terms of these roots. We also establish the spectral equivalence between the BEC models and certain quasi-exactly solvable Schördinger potentials.