  A phonon-roton dispersion relation is proposed for the elementary excitations
of a quantum biexcitonic liquid in semiconductors. The proposed dispersion
relation is used as a starting point for a calculation of the photoluminescence
spectrum of the liquid and an analysis of its behavior under variation of the
temperature and density of the biexcitonic liquid. The parameters of the
dispersion curve of elementary excitations of the quantum biexcitonic liquid
are evaluated by fitting the calculated photoluminescence spectrum to the
experimental spectrum of the biexcitonic liquid of semiconducting $\beta
$-ZnP$_{2}$ crystals. Experimental studies of how the photoluminescence
spectrum of a biexcitonic liquid in $\beta $-ZnP$_{2}$ depends on the
temperature and the intensity of the laser excitation confirm the initial
theoretical model. The dependence of the temperature of the crystals on the
excitation intensity is measured, and for some of the samples an anomalous
dependence is found: the temperature of the crystal decreases as the excitation
intensity increases. This effect is probably a consequence of the giant thermal
conductivity of the superfluid biexcitonic liquid in $\beta $-ZnP$_{2}$
crystals.
