  Novel methods to integrate the time-dependent Schr\"odinger equation within
the framework of multiscale approximation is presented. The methods are based
on symplectic splitting algorithms to separate the kinetic and potential parts
of the corresponding propagator. The semigroup associated with the
free-particle Schr\"odinger operator is represented in a multiwavelet basis.
The propagator is effectively discretized with a contour deformation technique,
which overcomes the challenges presented by previous discretization methods.
The discretized operator is then employed in simple numerical simulations to
test the validity of the implementation and to benchmark its precision.
