  Phonon softening is a ubiquitous phenomenon in condensed matter systems which
is often associated with charge density wave (CDW) instabilities and
anharmonicity. The interplay between phonon softening, CDW and
superconductivity is a topic of intense debate. In this work, the effects of
anomalous soft phonon instabilities on superconductivity are studied based on a
recently developed theoretical framework that accounts for phonon damping and
softening within the Migdal-Eliashberg theory. Model calculations show that the
phonon softening in the form of a sharp dip in the phonon dispersion relation,
either acoustic or optical (including the case of Kohn-type anomalies typically
associated with CDW), can cause a manifold increase of the electron-phonon
coupling constant $\lambda$. This, under certain conditions, which are
consistent with the concept of optimal frequency introduced by Bergmann and
Rainer, can produce a large increase of the superconducting transition
temperature $T_c$. In summary, our results suggest the possibility of reaching
high-temperature superconductivity by exploiting soft phonon anomalies
restricted in momentum space.
