The Joy and Pain of Skew Symmetry

Authors: Arieh Iserles

Publish Date: 2016/05/02

Volume: 16, Issue: 6, Pages: 1607-1630

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Abstract

In this paper we review recent progress on two related issues Firstly the discretisation of partial differential equations of quantum mechanics in a semiclassical regime Due to the presence of a small parameter such equations exhibit high oscillations and multiscale behaviour rendering them difficult to discretise We describe a methodology using symmetric Zassenhaus splittings in a free Lie algebra which allows for their exceedingly fast and accurate numerics The imperative of preserving the unitarity of the underlying flow takes us to the second theme of this paper approximation of derivatives by skewsymmetric matrices Here we identify a gap in the elementary theory of finitedifference approximations in the presence of Dirichlet boundary conditions it is impossible to approximate the derivative to order higher than two on a uniform grid This motivates the investigation of skew symmetry on nonuniform grids an endeavour which although still in its infancy is already replete with interesting results We conclude by discussing a number of generalisations and open problemsVarious parts of this paper are based on joint research with my colleagues Philipp Bader La Trobe Ernst Hairer Geneva Karolina Kropielnicka Gdańsk and Pranav Singh Cambridge I wish to acknowledge not just their mathematical contribution but also the great pleasure of collaborating with them I also wish to thank a number of colleagues for very fruitful discussions in particular Helge Dietert Cambridge and Caroline Lasser Munich

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