Karolina Kropielnicka

Effective Approximation for the Semiclassical Schrödinger Equation

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 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

Implicit difference methods for evolution functional differential equations

{\mathcal {O}}\!\left( M\log M\right)

Compact schemes for laser–matter interaction in Schrödinger equation based on effective splittings of Magnus expansion

OMlogM, where

Differential difference inequalities related to hyperbolic functional differential systems and applications

M

Difference methods for parabolic functional differential problems of the Neumann type

M is the number of degrees of freedom in the discretisation.

Implicit difference functional inequalities and applications

We build efficient and unitary (hence stable) methods for the solution of the linear time-dependent Schrödinger equation with explicitly time-dependent potentials in a semiclassical regime. The Magnus–Zassenhaus schemes presented here are based on a combination of the Zassenhaus decomposition (Bader et al. 2014 Found. Comput. Math. 14, 689–720. (doi:10.1007/s10208-013-9182-8)) with the Magnus expansion of the time-dependent Hamiltonian. We conclude with numerical experiments.