Unitary representations of the modular and two-particle \(q\)-deformed Toda chains (Q2758423)

The author studies the analytic theory of the quantum two-particle \(q\)-deformed Toda chains and the corresponding theory of joint eigenfunctions of the quantum Toda Hamiltonians that are called the generalized Whittaker functions. This is the simplest nontrivial example clarifying the role of the modular duality concept in the representation theory of noncompact semisimple quantum groups.NEWLINENEWLINEIn Section 1 the elementary representation of the algebra \(U_q(\text{sl}(2,\mathbb R))\) is considered. Then the theory of Whittaker vectors and Whittaker functions for the corresponding modular double and with the 2-particle \(q\)-deformed open Toda chain is given. The explicit formulae for the Whittaker vectors in terms of the double sine functions of Barnes are obtained. Moreover, the integral representations for solutions to the one-parameter family of two-particle relativistic Toda chains in the framework of representation theory are derived; all these solutions enjoy a dual symmetry. NEWLINENEWLINEThe appendix is devoted to the analytical properties of the double sine functions.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00056].