  We present the calculation of the coherent spectral functions and density of
states (DOS) for excitonic systems in the frame of the three dimensional
extended Falicov-Kimball model. By using gauge-invariant U(1) transformation to
the usual fermions, we represent the electron operator as a fermion attached to
the U(1) phase-flux tube. The emergent bosonic gauge field, related to the
phase variables is crucial for the Bose-Einstein condensation (BEC) of
excitons. Employing the path-integral formalism, we manipulate the bosonic and
fermionic degrees of freedom to obtain the effective actions related to
fermionic and bosonic sectors. Considering the normal and anomalous excitonic
Green functions, we calculate the spectral functions, which have the forms of
convolutions in the reciprocal space between bosonic and fermionic
counterparts. For the fermionic incoherent part of the DOS we have found the
strong evidence of the hybridization-gap in DOS spectra. Furthermore,
considering Bogoliubov coherence mechanism, we calculate the coherent DOS
spectra. For the coherent normal fermionic DOS, there is no hybridization-gap
found in the system due to strong coherence effets and phase stiffness. The
similar behavior is observed also for the condensate part of the anomalous
excitonic DOS spectra. We show that for small values of the Coulomb
interaction, fermionic DOS exhibits a Bardeen-Cooper-Schriffer (BCS) -like
double-peak structure. In the BEC region of the BCS-BEC crossover, the
double-peak structure disappears totally for both: coherent and incoherent DOS
spectra. We discuss also, temperature dependence of DOS functions.
