Completeness of the Bethe Ansatz for an open $q$-boson system with integrable boundary interactions

Abstract

We employ a discrete integral-reflection representation of the double affine Hecke algebra of type $C^{\vee }C$ at the critical level q=1, to endow the open finite $q$-boson system with integrable boundary interactions at the lattice ends. It is shown that the Bethe Ansatz entails a complete basis of eigenfunctions for the commuting quantum integrals in terms of Macdonald's three-parameter hyperoctahedral Hall-Littlewood polynomials.

31 pages, LaTeX