Ground-state analysis of the Falicov-Kimball model on complete graphs

The ground-state nature of the Falicov–Kimball model of mobile electrons and fixed nuclei on complete graphs is investigated. We give a pedagogic derivation of the eigenvalue problem and present a complete account of the ground-state energy both as a function of the number of electrons and nuclei and as a function of the total number of particles for any value of interaction U [Formula: see text] [Formula: see text]. We also study the energy gap and show the existence of a phase transition characterized by the absence of gap at the half-filled band for U < 0. The model in consideration was proposed and partially solved by Farkasovsky for finite graphs and repulsive on-site interaction U > 0. Contrary to his proposal, we conveniently scale the hopping matrix to guarantee the existence of the thermodynamic limit. We also solve this model on bipartite complete graphs and examine how sharp the Kennedy–Lieb variational estimate is as compared with the exact ground state. PACS Nos.: 71.2-b, 02.10Gd

Variational Theory of Reflection

Schwinger's variational method for the scattering phase shift produced by a central potential is adapted to reflection by a planar potential barrier (or well). The formulation is general, for an arbitrary transition between any two media, but the application here is limited to reflection at a barrier between media of equal potential energy. The simplest variational estimate for the reflection amplitude correctly tends to -1 at grazing incidence, as it must for any finite barrier. This is in contrast to the first order perturbation reflection amplitude, which diverges at grazing incidence. The same variational estimate is also correct to second order in the ratio of the interface thickness to the wavelength of the incident wave. The theory applies also to the reflection of the electromagnetic s (or transverse electric) wave at an interface between two media.