### Abstract

The computation of the semiclassical 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. In this paper we sketch an alternative strategy. Our analysis commences with the investigation of the free Lie algebra generatedby differentiation and by multiplication with the interaction potential: it turns outthat this algebra possesses a structure which renders it amenable to a very effective form of asymptotic splitting: exponential splitting where consecutive terms are scaledby increasing powers of the small parameter. This leads to methods which attain high spatial and temporal accuracy and whose cost scales as O(M log M), where M is thenumber of degrees of freedom in the discretisation.

Original language | English |
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Pages (from-to) | 689-720 |

Number of pages | 32 |

Journal | Foundations of Computational Mathematics |

Volume | 14 |

Issue number | 4 |

Early online date | 19 Feb 2014 |

DOIs | |

Publication status | Published - 31 Aug 2014 |

### Keywords

- Exponential splittings
- Krylov subspace methods
- Semiclassical Schrödinger equation
- Spectral collocation
- Zassenhaus splitting

### ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics

## Profiles

## Cite this

*Foundations of Computational Mathematics*,

*14*(4), 689-720. https://doi.org/10.1007/s10208-013-9182-8