Unitary representations of $U_{q}(\mathfrak{sl}(2,\RR))$, the modular double, and the multiparticle q-deformed Toda chains
S. Kharchev, D. Lebedev, M. Semenov-Tian-Shansky
TL;DR
The paper develops an analytic framework for the quantum $q$-deformed Toda chain using the modular double of $U_q( rak{sl}(2,b R))$, unifying representation theory with the Quantum Inverse Scattering Method. It constructs principal series representations on dual noncommutative tori, introduces Whittaker vectors and functions via dual nilpotent generators, and expresses these objects through the double sine $S_2$ with Mellin–Barnes-type integral representations. Extending to $N$ particles, it uses a twisted trigonometric $R$-matrix and a lattice Lax representation to obtain inductive integral representations for open-chain wave functions, then derives a dual Baxter framework for the periodic chain. The results reveal a deep duality between the $q$-Toda system and its modular dual, with explicit spectral data and connections to relativistic Toda models, providing a robust analytic handle on the spectrum and eigenfunctions of multiparticle $q$-Toda chains.
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.