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The optical reflectivity of Pb$_{0.865}$Sn$_{0.135}$Se andPb$_{0.75}$Sn$_{0.25}$Se solid solutions was measured in the THz spectralregion energetically corresponding to bulk optical phonon excitations and inthe temperature range from 40 K to 280 K. The analysis ofPb$_{0.75}$Sn$_{0.25}$Se data performed within the dynamic dielectric functionformalism revealed a new effect due to the electron-phonon coupling resultingin resonant changes of LO phonon frequency for energy gap equal to zero or toLO phonon energy. This effect is absent for Pb$_{0.865}$Sn$_{0.135}$Se thatexhibits an open energy gap with trivial band ordering at all temperatures.These results show that reflectivity in the THz range constitute a versatileexperimental method for precise determination of band inversion in narrow-gaptopological materials. For Pb$_{0.75}$Sn$_{0.25}$Se the transition from trivialinsulator to topological crystalline insulator phase takes place at temperatureT$_0$ = (172 $\pm$ 2) K.

Electron-phonon coupling and a resonant-like optical observation of a band inversion in topological crystalline insulator Pb$_{1-x}$Sn$_x$Se