  In this paper we present a novel multiscale splitting approach to solve
multiscale Schroedinger equation, which have large different time-scales. The
energy potential is based on highly oscillating functions, which are magnitudes
faster than the transport term. We obtain a multiscale problem and a highly
stiff problem, while standard solvers need to small time-steps. We propose
multiscale solvers, which are based on operator splitting methods and we
decouple the diffusion and reaction part of the Schroedinger equation. Such a
decomposition allows to apply a large time step for the implicit
time-discretization of the diffusion part and small time steps for the explicit
and highly oscillating reaction part. With extrapolation steps, we could reduce
the computational time in the highly-oscillating time-scale, while we relax
into the slow time-scale. We present the numerical analysis of the extrapolated
operator splitting method. First numerical experiments verified the benefit of
the extrapolated splitting approaches.
