Unitary representations of $U_{q}(\mathfrak{sl}(2,\RR))$, the modular double, and the multiparticle q-deformed Toda chains

TL;DR

This paper explores the representation theory of quantum groups related to the q-deformed Toda chain, revealing dualities and explicit formulas for wave functions using advanced mathematical functions and integral transforms.

Contribution

It introduces explicit formulas for Whittaker vectors and wave functions of the q-deformed Toda chain, highlighting the role of modular duality in noncompact quantum groups.

Findings

Explicit Whittaker vector formulas using double sine functions

Wave functions expressed as Mellin-Barnes type integrals

Derivation of dual Baxter equations for the periodic chain

Abstract

The paper deals with the analytic theory of the quantum q-deformed Toda chain; the technique used combines the methods of representation theory and the Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny is the role of the modular duality concept (first discovered by L.Faddeev) in the representation theory of noncompact semisimple quantum groups. Explicit formulae for the Whittaker vectors are presented in terms of the double sine functions and the wave functions of the N-particle q-deformed open Toda chain are given as a multiple integral of the Mellin-Barnes type. For the periodic chain the two dual Baxter equations are derived.