  We present pretty detailed spectral analysis of Jacobi matrices with
periodically modulated entries in the case when $0$ lies on the soft edge of
the spectrum of the corresponding periodic Jacobi matrix. In particular, we
show that the studied operators are always self-adjoint irrespective of the
modulated sequence. Moreover, if the growth of the modulated sequence is
superlinear, then the spectrum of the considered operators is always discrete.
Finally, we study regular perturbations of this class in the linear and the
sublinear cases. We impose conditions assuring that the spectrum is absolute
continuous on some regions of the real line. A constructive formula for the
density in terms of Tur\'an determinants is also provided.
