Integral representations for the eigenfunctions of quantum open and periodic Toda chains from the QISM formalism

Abstract

The integral representations for the eigenfunctions of N particle quantum open and periodic Toda chains are constructed within the framework of the quantum inverse scattering method. Both periodic and open N-particle solutions have essentially the same structure, being written as a generalized Fourier transform over the eigenfunctions of the N-1 particle open Toda chain with the kernels satisfying the Baxter equations of second and first order, respectively. In the latter case this leads to recurrent relations which result in a representation of Mellin-Burnes-type solutions of an open chain. As a byproduct, we obtain the Gindikin-Karpelevich formula for the Harish-Chandra function in the case of the {GL}(N,) group.