Factorized Weight Functions vs. Factorized Scattering

Abstract:

Starting from an extensive class of factorized weight functions W(p) on the N-dimensional torus ?, we construct an orthonormal base of symmetric N-variable polynomials for L ² _s (?,W(p)dp) via lexicographic ordering of the monomial symmetric functions (free boson states) and the Gram-Schmidt procedure. We show that the dominant asymptotics of these polynomials is factorized. As a corollary, we obtain a large class of quantum integrable soliton systems on the symmetric subspace of l ²(ℤ^N). The class of weight functions contains in particular the weight function yielding Macdonald polynomials. For that special case, the quantum soliton system can be viewed as the dual relativistic Calogero–Sutherland system.