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

TL;DR

This paper develops integral representations for eigenfunctions of quantum Toda chains using QISM, revealing structural similarities between open and periodic cases and deriving new formulas related to harmonic analysis.

Contribution

It constructs integral eigenfunction representations for both open and periodic quantum Toda chains within the QISM framework, unifying their structure and deriving new Mellin-Barnes and Harish-Chandra formulas.

Findings

Eigenfunctions expressed as generalized Fourier transforms.

Recurrent relations lead to Mellin-Barnes type representations.

Derived Gindikin-Karpelevich formula for $GL(N,\RR)$.

Abstract

The integral representations for the eigenfunctions of $N$ particle quantum open and periodic Toda chains are constructed in the framework of Quantum Inverse Scattering Method (QISM). 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 to the Baxter equations of the second and first order respectively. In the latter case this leads to recurrent relations which result to representation of the Mellin-Barnes type for solutions of an open chain. As byproduct, we obtain the Gindikin-Karpelevich formula for the Harish-Chandra function in the case of $GL(N,\RR)$ group.