  The Lindhard function represents the basic building block of many-body
physics and accounts for charge response, plasmons, screening, Friedel
oscillation, RKKY interaction etc. Here we study its non-Hermitian version in
one dimension, where quantum effects are traditionally enhanced due to spatial
confinement, and analyze its behavior in various limits of interest. Most
importantly, we find that the static limit of the non-Hermitian Lindhard
function has no divergence at twice the Fermi wavenumber and vanishes
identically for all other wavenumbers at zero temperature. Consequently, no
Friedel oscillations are induced by a non-Hermitian, imaginary impurity to
lowest order in the impurity potential at zero temperature. Our findings are
corroborated numerically on a tight-binding ring by switching on a weak real or
imaginary potential. We identify conventional Friedel oscillations or heavily
suppressed density response, respectively.
