EXACT FRIEDEL OSCILLATIONS IN THE G=1 2 LUTTINGER LIQUID/

A single impurity in the one-dimensional Luttinger model creates a loc al modification of the charge density analogous to the Friedel oscilla tions. In this paper, we present an exact solution of the case g=1/2 ( the equivalent of the Toulouse point) at any temperature T and impurit y coupling, expressing the charge density in terms of a hypergeometric function. We find in particular that at T=0 the oscillatory part of t he density goes as Int at small distance and x(-1/2) at large distance .