  The RKKY interaction is an important theoretical model for indirect exchange
interaction in magnetic multilayer. The expression for RKKY range function in
three dimension and lower has been derived in the 1950s. However, the
expression for one dimension is still studied in recent years, due to its
strong singularity. By using an adiabatic limit of retarded Green's function
form that directly related to RKKY interaction in one dimension, we decompose
the singularity and recover the range function. Furthermore, we show in
adiabatic limit, RKKY interaction also induces one-dimensional spin pumping and
distance-dependent magnetic damping.
