  In this paper, using the quantum multistream model, we develop a method to
study the electronic band structure of plasmonic excitations in streaming
electron gas with arbitrary degree of degeneracy. The multifluid quantum
hydrodynamic model is used to obtain $N$-coupled pseudoforce differential
equation system from which the energy band structure of plasmonic excitations
is calculated. It is shown that inevitable appearance of energy bands separated
by gaps can be due to discrete velocity filaments and their electrostatic mode
coupling in the electron gas. Current model also provides an alternative
description of collisionless damping and phase mixing, i.e., collective
scattering phenomenon within the energy band gaps due to mode coupling between
wave-like and particle-like oscillations. The quantum multistream model is
further generalized to include virtual streams which is used to calculate the
electronic band structure of one-dimensional plasmonic crystals. It is remarked
that, unlike the empty lattice approximation in free electron model, energy
band gaps exist in plasmon excitations due to the collective electrostatic
interactions between electrons. It is also shown that the plasmonic band gap
size at first Brillouin zone boundary maximizes at the reciprocal lattice
vector, $G$, close to metallic densities. Furthermore, the electron-lattice
binding and electron-phonon coupling strength effects on the electronic band
structure are discussed. It is remarked that inevitable formation of energy
band structure is a general characteristics of various electromagnetically and
gravitationally coupled quantum multistream systems.
