BC Toda chain II: symmetries. Dual picture

TL;DR

This paper advances the understanding of quantum BC Toda chain by proving Baxter operators' commutativity, demonstrating wave function symmetries, and establishing their relation to hyperoctahedral Whittaker functions.

Contribution

It proves key properties of wave functions and Baxter operators, and links the model to hyperoctahedral Whittaker functions, extending previous integral representations.

Findings

Baxter operators are shown to commute.

Wave functions are symmetric under signed permutations.

Wave functions satisfy dual difference equations and match hyperoctahedral Whittaker functions.

Abstract

In the previous paper we derived Gauss-Givental integral representation for the wave functions of quantum BC Toda chain and also introduced Baxter operators for this model. In the present paper we prove commutativity of Baxter operators, as well as show that the constructed wave functions are symmetric with respect to signed permutations of spectral parameters and diagonalize Baxter operators. Furthermore, we derive Mellin-Barnes integral representation for the wave functions. With its help we show that wave functions satisfy dual system of difference equations with respect to spectral parameters and coincide with hyperoctahedral Whittaker functions. Finally, we give heuristic proofs of orthogonality and completeness of the wave functions.