  Topological crystalline insulators represent a novel topological phase of
matter in which the surface states are protected by discrete point
group-symmetries of the underlying lattice. Rock-salt lead-tin-selenide alloy
is one possible realization of this phase which undergoes a topological phase
transition upon changing the lead content. We used scanning tunneling
microscopy (STM) and angle resolved photoemission spectroscopy (ARPES) to probe
the surface states on (001) Pb$_{1-x}$Sn$_{x}$Se in the topologically
non-trivial (x=0.23) and topologically trivial (x=0) phases. We observed
quasiparticle interference with STM on the surface of the topological
crystalline insulator and demonstrated that the measured interference can be
understood from ARPES studies and a simple band structure model. Furthermore,
our findings support the fact that Pb$_{0.77}$Sn$_{0.23}$Se and PbSe have
different topological nature.
