磁Schrödinger方程的分裂方法
A splitting approach for the magnetic Schrödinger equation
AI总结：
本文提出一种分裂方法解决磁Schrödinger方程，通过将运算分为三个部分并证明Lie分裂的一阶收敛性，特别处理了动量项以保持守恒性质，数值实验验证了该方法的高效性和可靠性。
摘要：
在外部电磁场存在的Schrödinger方程是计算量子力学中的重要问题，也提供了一个微分方程流可分解为三个部分的典型案例。本文针对三个算子（其中两个为无界算子）提出分裂方法，并在此框架下证明Lie分裂的一阶收敛性。结果应用于磁Schrödinger方程，将其分解为势能、动能和输运部分，后者需特殊处理以保持方案的守恒性质。讨论了多种选项。一维、二维和三维空间中的数值实验表明，特征方法结合非等距快速傅里叶变换（NFFT）提供了一种快速且可靠的实现离散质量守恒的技术。
英文摘要：
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.
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