Equations in dual variables for Whittaker functions

TL;DR

This paper derives new difference equations in the spectral variables for Whittaker functions related to SL(N), expanding their known differential equation properties by connecting open and closed Toda chain frameworks.

Contribution

It introduces a novel set of difference equations in the spectral variables for Whittaker functions, linking open and closed Toda chain models.

Findings

Whittaker functions satisfy new difference equations in spectral variables

Established relations between open and closed Toda chains

Extended understanding of Whittaker functions' properties

Abstract

It is known that the Whittaker functions $w(q,\lambda)$ associated to the group SL(N) are eigenfunctions of the Hamiltonians of the open Toda chain, hence satisfy a set of differential equations in the Toda variables $q_i$. Using the expression of the $q_i$ for the closed Toda chain in terms of Sklyanin variables $\lambda_i$, and the known relations between the open and the closed Toda chains, we show that Whittaker functions also satisfy a set of new difference equations in $\lambda_i$.