  It has been known for a long time that the low temperature behavior shown by
the dielectric constant of quantum paraelectric $SrTiO_{3}$ can not be fitted
properly by Barrett's formula using a single zero point energy or saturation
temperature ($T_{1}$). As it was originally shown [K. A. M\"{u}ller and H.
Burkard, Phys. Rev. B {\bf 19}, 3593 (1979)] a crossover between two different
saturation temperatures ($T_{1l}$=77.8K and $T_{1h}$=80K) at $T\sim10K$ is
needed to explain the low and high temperature behavior of the dielectric
constant. However, the physical reason for the crossover between these two
particular values of the saturation temperature at $T\sim10K$ is unknown. In
this work we show that the crossover between these two values of the saturation
temperature at $T\sim10K$ can be taken as a direct consequence of (i) the
quantum distribution of frequencies $g(\Omega)\propto\Omega^{2}$ associated
with the complete set of low-lying modes and (ii) the existence of a definite
maximum phonon frequency given by the structural transition critical
temperature $T_{tr}$.
