The ground state configurations of the one-dimensional Falicov-Kimball model are studied exactly with numerical calculations revealing unexpected effects for small interaction strength. In neutral systems we observe molecular formation, phase separation and changes in the conducting properties; while in non-neutral systems the phase diagram exhibits Farey tree order (Aubry sequence) and a devil's staircase structure. Conjectures are presented for the boundary of the segregated domain and the general structure of the ground states.

The Falicov-Kimball model of spinless quantum electrons hopping on a 1-dimensional lattice and of immobile classical ions occupying some lattice sites, with only intrasite coupling between those particles, have been studied at zero temperature by means of well-controlled numerical procedures. For selected values of the unique coupling parameter U the restricted phase diagrams (based on all the periodic configurations of localized particles (ions) with period not greater than 16 lattice constants, typically) have been constructed in the grand-canonical ensemble. Then these diagrams have been translated into the canonical ensemble. Compared to the diagrams obtained in other studies our ones contain more details, in particular they give better insight into the way the mixtures of periodic phases are formed. Our study has revealed several families of new characteristic phases like the generalized most homogeneous and the generalized crenel phases, a first example of a structural phase transition and a tendency to build up an additional symmetry -the holeparticle symmetry with respect to the ions (electrons) only, as U decreases.

We present some exact results for the ground states of the Falicov-Kimball model outside the symmetry point. Three ion configurations have been considered: the chessboard configuration and the completely empty and the fully occupied configurations.Using a method based onTchebycheff-Markov inequalities, we determine domains in the plane of chemical potentials p, and p, "where these configurations are ground states.

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