  When interactions between particles are strong, at low temperature, these
particles can form self-organized quantum crystals, and when the particles
interact weakly, periodic structures can be imposed by external fields, e.g. by
optical lattices. These opposite cases usually are treated separately, dealing
either with quantum crystals or with optical lattices. Here the unified theory
is developed for arbitrary particle interaction strength, treating in the frame
of the same model both the limiting cases of quantum crystals and optical
lattices, as well as the states intermediate between these two limits. Bose
particles are considered, hence at low temperature in optical lattices
Bose-Einstein condensation can happen, while it seems to be prohibited in ideal
quantum crystals that do not contain mesoscopic regions of disorder, such as
dislocations and grain boundaries.
