  We present a family of algorithms for the numerical approximation of the
Schr\"odinger equation with potential concentrated at a finite set of points.
Our methods belong to the so-called fast and oblivious convolution quadrature
algorithms. These algorithms are special implementations of Lubich's
Convolution Quadrature which allow, for certain applications in particular
parabolic problems, to significantly reduce the computational cost and memory
requirements. Recently it has been noticed that their use can be extended to
some hyperbolic problems. Here we propose a new family of such efficient
algorithms tailored to the features of the Green's function for Schr\"odinger
equations. In this way, we are able to keep the computational cost and the
storage requirements significantly below existing approaches. These features
allow us to perform reliable numerical simulations for longer times even in
cases where the solution becomes highly oscillatory or seems to develop finite
time blow-up. We illustrate our new algorithm with several numerical
experiments.
