Effective Approximation for the Semiclassical Schrödinger Equation

Abstract

The computation of the semiclassical Schrodinger 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 generated by differentiation and by multiplication with the interaction potential: it turns out that this algebra possesses a structure which 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. This leads to methods which attain high spatial and temporal accuracy and whose cost scales as $${\mathcal {O}}\!\left( M\log M\right) $$ O M log M , where $$M$$ M is the number of degrees of freedom in the discretisation.