@TechReport{	  it:2012-024,
  author	= {Katharina Kormann and Elisabeth Larsson},
  title		= {An {RBF}-{G}alerkin Approach to the Time-Dependent
		  {S}chr{\"o}dinger Equation},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2012},
  number	= {2012-024},
  month		= sep,
  abstract	= {In this article, we consider the discretization of the
		  time-dependent Schrödinger equation using radial basis
		  functions (RBF). We formulate the discretized problem over
		  an unbounded domain without imposing explicit boundary
		  conditions. Since we can show that time-stability of the
		  discretization is not guaranteed for an RBF-collocation
		  method, we propose to employ a Galerkin ansatz instead. For
		  Gaussians, it is shown that exponential convergence is
		  obtained up to a point where a systematic error from the
		  domain where no basis functions are centered takes over.
		  The choice of the shape parameter and of the resolved
		  region is studied numerically. Compared to the Fourier
		  method with periodic boundary conditions, the basis
		  functions can be centered in a smaller domain which gives
		  increased accuracy with the same number of points.}
}