  In this paper we propose fast solution methods for the Cauchy problem for the
multidimensional Schr\"odinger equation. Our approach is based on the
approximation of the data by the basis functions introduced in the theory of
approximate approximations. We obtain high-order approximations also in higher
dimensions up to a small saturation error, which is negligible in computations,
and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous
equation. The proposed method is very efficient in high dimensions if the
densities allow separated representations. We illustrate the efficiency of the
procedure on different examples, up to approximation order 6 and space
dimension 200.
