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__Effective approximation for the linear time-dependent Schrödinger equation__

P. Singh, P. Bader, A. Iserles and K. Kropielnicka

**Abstract**

The computation of the linear Schrödinger equation presents major challenges because of the presence of a small parameter. Assuming periodic boundary conditions, the standard approach consists of semi-discretisation with a spectral method, followed by an exponential splitting. We follow an alternative strategy: our analysis commences from the investigation of the free Lie algebra generated by the operations of differentiation and multiplication with the interaction potential. It turns out that this algebra possesses structure that renders it amenable to a very effective form of {asymptotic splitting:\/} exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. The number of terms of the splitting increases linearly with time accuracy. This leads to methods that attain high spatial and temporal accuracy and whose cost scales like $O(N\log N)$, where $N$ is the number of degrees of freedom.