This work deals with the spectral properties of the functional-dierence equations, that
arise in a number of applications in the diraction of waves and quantum scattering. Their link
with some of the spectral properties of perturbations of the Mehler operator is addressed. The latter
naturally arise in studies of functional-dierence equations of the second order with a meromorphic
potential which depend on a characteristic parameter. In particular, this kind of equations is frequently
encountered with in the asymptotic treatment of eigenfunctions of the Robin Laplacians in wedge-
or cone-shaped domains. The unperturbed selfadjoint Mehler operator is studied by means of the
modied MehlerFock transform. Its resolvent and spectral measure are described. These results
are obtained by use of some additional analysis applied to the known Mehler formulas. For a class of
compact perturbations of this operator, sucient conditions of existence and niteness of the discrete
spectrum are then discussed. Applications to the functional-dierence equations are also addressed.
An example of a problem leading to the study of the spectral properties for a functional-dierence
equation is considered. The corresponding eigenfunctions and characteristic values are found explicitly
in this case.