# Bispectral dual difference equations for the quantum Toda chain with boundary perturbations

## TL;DR

This paper establishes a bispectral duality for hyperoctahedral Whittaker functions, showing they satisfy a dual system of difference equations in the spectral variable, extending known dualities in quantum integrable systems.

## Contribution

It introduces a new dual difference equation system for hyperoctahedral Whittaker functions associated with the quantum Toda chain with boundary perturbations, extending classical bispectral duality.

## Key findings

- Hyperoctahedral Whittaker functions satisfy dual difference equations.
- The functions are entire in the spectral variable.
- Extension of bispectral duality to boundary-perturbed quantum Toda chains.

## Abstract

We show that hyperoctahedral Whittaker functions---diagonalizing an open quantum Toda chain with one-sided boundary potentials of Morse type---satisfy a dual system of difference equations in the spectral variable. This extends a well-known bispectral duality between the nonperturbed open quantum Toda chain and a strong-coupling limit of the rational Macdonald-Ruijsenaars difference operators. It is manifest from the difference equations in question that the hyperoctahedral Whittaker function is entire as a function of the spectral variable.