RKKY range function of a one-dimensional noninteracting electron gas

The apparently straightforward evaluation of the Ruderman-Kittel-Kasuya-Yosida RKKY range function, or more generally, of the linear density modulation n x induced at zero temperature in a noninteracting onedimensional electron gas by a localized static impurity modeled with a function potential has proven surprisingly troublesome. Although the original calculation gave an incorrect answer,1 more recent investigations appear to suggest that only a certain procedure, known to lead to a physically sensible answer, should be employed.2 The original and most popular procedure is based on the standard theory of linear response.3 If the impurity potential is assumed to be of the form U x = 2u /2m x where u is a suitable wave vector , then one has