  We propose a numerical scheme to solve the time dependent linear
Schr\"odinger equation. The discretization is carried out by combining a
Runge-Kutta time-stepping scheme with a finite element discretization in space.
Since the Schr\"odinger equation is posed on the whole space $\R^d$ we combine
the interior finite element discretization with a convolution quadrature based
boundary element discretization. In this paper we analyze the resulting fully
discrete scheme in terms of stability and convergence rate. Numerical
experiments confirm the theoretical findings.
