Friedel Oscillations for Interacting Fermions in One Dimension

We study Friedel oscillations in one-dimensional electron liquids for arbitrary electron-electron interaction and arbitrary impurity strength. Explicit results for spinless as well as spin- $\frac{1}{2}$ electrons are given. In the case of Luttinger-liquid leads, the Friedel oscillations decay as $x^{-g}$ far away from the impurity where $g$ is the interaction constant. For a weak scatterer, a slower decay is found at small-to-intermediate distances from the impurity, with a crossover to the asymptotic $x^{-g}$ decay.