  We present inelastic neutron scattering results of phonons in
(Pb$_{0.5}$Sn$_{0.5}$)$_{1-x}$In$_x$Te powders, with $x=0$ and 0.3. The $x=0$
sample is a topological crystalline insulator, and the $x=0.3$ sample is a
superconductor with a bulk superconducting transition temperature $T_c$ of 4.7
K. In both samples, we observe unexpected van Hove singularities in the phonon
density of states at energies of 1--2.5 meV, suggestive of local modes. On
cooling the superconducting sample through $T_c$, there is an enhancement of
these features for energies below twice the superconducting-gap energy. We
further note that the superconductivity in
(Pb$_{0.5}$Sn$_{0.5}$)$_{1-x}$In$_x$Te occurs in samples with normal-state
resistivities of order 10 m$\Omega$~cm, indicative of bad-metal behavior.
Calculations based on density functional theory suggest that the
superconductivity is easily explainable in terms of electron-phonon coupling;
however, they completely miss the low-frequency modes and do not explain the
large resistivity. While the bulk superconducting state of
(Pb$_{0.5}$Sn$_{0.5}$)$_{0.7}$In$_{0.3}$Te appears to be driven by phonons, a
proper understanding will require ideas beyond simple BCS theory.
