Integral representation for the eigenfunctions of quantum periodic Toda chain

TL;DR

This paper develops an integral representation for the eigenfunctions of the quantum periodic Toda chain with N particles, combining Gutzwiller's ideas, R-matrix methods, and Whittaker functions, and reproduces known results for small N.

Contribution

It introduces a novel integral representation for the eigenfunctions of the quantum periodic Toda chain for arbitrary N, unifying multiple approaches and interpreting the Sklyanin measure via Whittaker functions.

Findings

Reproduces Gutzwiller's results for N=2,3,4

Calculates Sklyanin's scalar product from the Plancherel formula

Provides a natural interpretation of the Sklyanin measure

Abstract

Integral representation for the eigenfunctions of quantum periodic Toda chain is constructed for N-particle case. The multiple integral is calculated using the Cauchy residue formula. This gives the representation which reproduces the particular results obtained by Gutzwiller for N=2,3 and 4-particle chain. Our method to solve the problem combines the ideas of Gutzwiller and R-matrix approach of Sklyanin with the classical results in the theory of the Whittaker functions. In particular, we calculate Sklyanin's invariant scalar product from the Plancherel formula for the Whittaker functions derived by Semenov-Tian-Shansky thus obtaining the natural interpretation of the Sklyanin measure in terms of the Harish-Chandra function.