  The inverse seesaw mechanism provides an attractive approach to generate
small neutrino mass, which origins from a tiny $U(1)_L$ breaking. In this
paper, we work in the supersymmetric version of this mechanism, where the
singlet-like sneutrino could be an asymmetric dark matter (ADM) candidate in
the maximally $U(1)_{L}$ symmetric limit. However, even a tiny $\delta m$, the
mass splitting between sneutrino and anti-sneutrino as a result of the tiny
$U(1)_{L}$ breaking effect, could lead to fast oscillation between sneutrino
and anti-sneutrino and thus spoils the ADM scenario. We study the evolution of
this oscillation and find that a weak scale sneutrino, which tolerates a
relatively larger $\delta m\sim 10^{-5}$ eV, is strongly favored. We also
investigate possible natural ways to realize that small $\delta m$ in the
model.
