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

We consider the functional-difference operator H = U + U^-1 + V, where U and V are the Weyl self-adjoint operators satisfying the relation UV = q²VU, q = e^πiτ, τ > 0. The operator H has applications in the conformal field theory and representation theory of quantum groups. Using the modular quantum dilogarithm (a q-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 H on the Hilbert space L² (ℝ), and prove the eigenfunction expansion theorem. This theorem is a q-deformation of the well-known Kontorovich-Lebedev transform in the theory of special functions. We also present a formulation of the scattering theory for H.