  The density, spin and isospin correlation functions in nuclear matter with a
neutron-proton ($np$) condensate are calculated to study the possible
signatures of the BEC-BCS crossover in the low-density region. It is shown that
the criterion of the crossover (Phys. Rev. Lett. {\bf 95}, 090402 (2005)),
consisting in the change of the sign of the density correlation function at low
momentum transfer, fails to describe correctly the density-driven BEC-BCS
transition at finite isospin asymmetry or finite temperature. As an unambiguous
signature of the BEC-BCS transition, there can be used the presence (BCS
regime) or absence (BEC regime) of the singularity in the momentum distribution
of the quasiparticle density of states.
