A splitting approach for the magnetic Schrödinger equation

The Schrödinger equation in the presence of an external electromagnetic field is an important problem in computational quantum mechanics. It also provides a nice example of a differential equation whose flow can be split with benefit into three parts. After presenting a splitting approach for three...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2017-05, Vol.316, p.74-85
Main Authors: Caliari, M., Ostermann, A., Piazzola, C.
Format: Article
Language:English
Subjects:
Computation
Conservation
Convergence
Differential equations
Electromagnetic fields
Exponential splitting methods
Fourier techniques
Fourier transforms
Magnetic Schrödinger equation
Nonequispaced fast Fourier transform
Schroedinger equation
Splitting
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Summary:The Schrödinger equation in the presence of an external electromagnetic field is an important problem in computational quantum mechanics. It also provides a nice example of a differential equation whose flow can be split with benefit into three parts. After presenting a splitting approach for three operators with two of them being unbounded, we exemplarily prove first-order convergence of Lie splitting in this framework. The result is then applied to the magnetic Schrödinger equation, which is split into its potential, kinetic and advective parts. The latter requires special treatment in order not to lose the conservation properties of the scheme. We discuss several options. Numerical examples in one, two and three space dimensions show that the method of characteristics coupled with a nonequispaced fast Fourier transform (NFFT) provides a fast and reliable technique for achieving mass conservation at the discrete level.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2016.08.041