Scalar particles in a narrow-band periodic potential

A system of an infinite number of spinless particles in a narrow-band periodic potential is treated. The dimension of the space is arbitrary, the tight-binding approximation is used, and it is assumed that the filling fraction is nearly one electron per atom. After a preliminary discussion of the Hartree approximation, the full Schrödinger equation is considered and a rigorous spectral perturbation theory in the kinetic energy term is set up. To get rid of the lack of relative boundedness of this perturbation, a vacuum state is constructed and its energy renormalized to zero, and passage is made to an excitonic representation in which the quasiparticles appear naturally as local perturbations of the vacuum. In this representation, relative boundedness is recovered and Rayleigh-Schrödinger expansions can be used to find cluster expansions for all local observables.