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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 asuperconductor with a bulk superconducting transition temperature $T_c$ of 4.7K. In both samples, we observe unexpected van Hove singularities in the phonondensity of states at energies of 1--2.5 meV, suggestive of local modes. Oncooling the superconducting sample through $T_c$, there is an enhancement ofthese features for energies below twice the superconducting-gap energy. Wefurther note that the superconductivity in(Pb$_{0.5}$Sn$_{0.5}$)$_{1-x}$In$_x$Te occurs in samples with normal-stateresistivities of order 10 m$\Omega$~cm, indicative of bad-metal behavior.Calculations based on density functional theory suggest that thesuperconductivity is easily explainable in terms of electron-phonon coupling;however, they completely miss the low-frequency modes and do not explain thelarge 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, aproper understanding will require ideas beyond simple BCS theory.

physical review. b./physical review. b

Unusual phonon density of states and response to the superconducting  transition in the In-doped topological crystalline insulator  Pb$_{0.5}$Sn$_{0.5}$Te