  Within schematic models based on the Tamm-Dancoff Approximation and the
Random-Phase Approximation with separable interactions, we investigate the
physical conditions which determine the emergence of the Pygmy Dipole Resonance
in the E1 response of atomic nuclei. We find that if some particle-hole
excitation manifests a different, weaker residual interaction, an additional
mode will appear, with an energy centroid closer to the distance between two
major shells and therefore well below the Giant Dipole Resonance. This state,
together with Giant Dipole Resonance, exhausts all the transition strength in
the Tamm-Dancoff Approximation and all the Energy Weighted Sum Rule in the
Random-Phase Approximation. Thus, within our scheme, this mode, which could be
associated with the Pygmy Dipole Resonance, is of collective nature. By
relating the coupling constants appearing in the separable interaction to the
symmetry energy value at and below saturation density we explore the role of
the density dependence of the symmetry energy on the low energy dipole
response.
