  This paper focuses on the spectral properties of a bounded self-adjoint
operator in $L_2(\mathds R^d)$ being the sum of a convolution operator with an
integrable convolution kernel and an operator of multiplication by a continuous
potential converging to zero at infinity. We study both the essential and the
discrete spectra of this operator. It is shown that the essential spectrum of
the sum is the union of the essential spectrum of the convolution operator and
the image of the potential. We then provide a number of sufficient conditions
for the existence of discrete spectrum and obtain lower and upper bounds for
the number of discrete eigenvalues. Special attention is paid to the case of
operators possessing countably many points of the discrete spectrum. We also
compare the spectral properties of the operators considered in this work with
those of classical Schr\"odinger operators.
