Bound-state transition: an analytical model

By using an accurate dielectric function for a homogeneous paramagnetic electron liquid we attempt a simple analytical treatment of the bound-state (metal-insulator) transition in a system consisting of a single proton immersed in the electron liquid, as a function of global density. The inability of the resulting effective Thomas-Fermi picture to account for the transition is remedied by the inclusion of the appropriate cusp condition that is also introduced in a simple analytical manner. The expected transition from a delocalized state (at high density) to a localized state (at low density) is shown to be the result of the combined action of a minimal number of very general principles such as overall charge neutrality, the compressibility sum rule and the so-called q(4) sum rule, at the simplest analytical level.