Solving Schrödinger equation in semiclassical regime with highly oscillatory time-dependent potentials

Journal article

Schrödinger equations with time-dependent potentials are of central importance in quantum physics and theoretical chemistry, where they aid in the simulation and design of systems and processes at atomic and molecular scales. Numerical approximation of these equations is particularly difficult in the semiclassical regime because of the highly oscillatory nature of solution. Highly oscillatory potentials such as lasers compound these difficulties even further. Altogether, these effects render a large number of standard numerical methods less effective in this setting. In this paper we will develop a class of exponential splitting schemes that allow us to use large time steps in our schemes even in the presence of highly oscillatory potentials and solutions. These are derived by combining the advantages of integral-preserving simplified-commutator Magnus expansions with those of symmetric Zassenhaus splittings. The efficacy of these methods is demonstrated through 1D, 2D and 3D numerical examples.

Authors: Iserles, A; Kropielnicka, K; Singh, PS

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Journal of Computational Physics
• Volume: 376
• Pages: 564-584
• Publication date: 2018-10-02
• Acceptance date: 2018-09-27
• DOI: 10.1016/j.jcp.2018.09.047
• ISSN: 0021-9991
• Keywords: time dependent potentials; highly oscillatory potentials; Schrödinger equation; semiclassical regime; symmetric Zassenhaus splittings; magnus expansion