  In the paper we consider a functional-difference operator $H=U+U^{-1}+V$,
where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with
$q=e^{\pi i\tau}$ and $\tau>0$. The operator $H$ has applications in the
conformal field theory and in the representation theory of quantum groups.
Using modular quantum dilogarithm - a $q$-deformation of the Euler's
dilogarithm - we define the scattering solution and the Jost solutions, derive
an explicit formula for the resolvent of the self-adjoint operator $H$ in the
Hilbert space $L^{2}(\mathbb{R})$, and prove the eigenfunction expansion
theorem. The latter 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 the operator $H$.
