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 $UV=q^{2}VU$, $q=e^{\pi i\tau}$, $\tau>0$. 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 $L^{2}(\mathbb R)$, 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 $H$.