Representation theory and quantum inverse scattering method: the open Toda chain and the hyperbolic Sutherland model

Abstract

Using the representation theory of Gl(N), we express the wave function of the GL(N,ℝ) Toda chain, which was recently obtained by two of us by the quantum inverse scattering method, in terms of multiple integrals. The main tool is our generalization of the Gelfand-Zetlin method to the case of infinite-dimensional representations of Gl(N). The interpretation of this generalized construction in terms of the coadjoint orbits is given and the connection with the Yangian Gl(N) is discussed. We also give the hyperbolic Sutherland model eigenfunctions expressed in terms of integrals in the Gelfand-Zetlin representation. Using the example of the open Toda chain, we discuss the connection between the quantum inverse scattering method and representation theory.