Unitary Representations of the Quantum Lorentz Group and Quantum Relativistic Toda Chain

We give a group theory interpretation of the three types of q-Bessel functions. We consider a family of quantum Lorentz groups and a family of quantum Lobachevsky spaces. For three values of the parameter of the quantum Lobachevsky space, the Casimir operators correspond to the two-body relativistic open Toda-chain Hamiltonians whose eigenfunctions are the modified q-Bessel functions of the three types. We construct the principal series of unitary irreducible representations of the quantum Lorentz groups. Special matrix elements in the irreducible spaces given by the q-Macdonald functions are the wave functions of the two-body relativistic open Toda chain. We obtain integral representations for these functions.