On the spectral theory of a functional-difference operator in conformal field theory

Abstract

We consider the functional-difference operator , where and are the Weyl self- adjoint operators satisfying the relation , , . The operator has applications in the conformal field theory and representation theory of quantum groups. Using the modular quantum dilogarithm (a -deformation of the Euler dilogarithm), we define the scattering solution and Jost solutions, derive an explicit formula for the resolvent of the self- adjoint operator on the Hilbert space , and prove the eigenfunction expansion theorem. This theorem is a -deformation of the well-known Kontorovich–Lebedev transform in the theory of special functions. We also present a formulation of the scattering theory for .