  The electronic structure of Pb$_{1-x}$Sn$_{x}$Te is studied by using the
relativistic Korringa-Kohn-Rostoker Green function method in the framework of
density functional theory. For all concentrations $x$, Pb$_{1-x}$Sn$_{x}$Te is
a direct semiconductor with a narrow band gap. In contrast to pure lead
telluride, tin telluride shows an inverted band characteristic close to the
Fermi energy. It will be shown that this particular property can be tuned,
first, by alloying PbTe and SnTe and, second, by applying hydrostatic pressure
or uniaxial strain. Furthermore, the magnitude of strain needed to switch
between the regular and inverted band gap can be tuned by the alloy
composition. Thus, there is range of potential usage of Pb$_{1-x}$Sn$_{x}$Te
for spintronic applications.
