This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrödinger equations; a detailed stability and local error analysis is provided for the relevant special case where the Hamilton operator comprises the Laplacian and a regular space-time-dependent potential. In the context of nonautonomous linear ordinary differential equations, CFQM exponential integrators are composed of exponentials involving linear combinations of certain values of the associated time-dependent matrix; this approach extends to nonautonomous linear evolution equations given by unbounded operators. An inherent advantage of CFQM exponential integrators over other time integration methods such as Runge–Kutta methods or Magnus integrators is that structural properties of the underlying operator family are well preserved; this characteristic is confirmed by a theoretical analysis ensuring unconditional stability in the underlying Hilbert space and the full order of convergence under low regularity requirements on the initial state. Due to the fact that convenient tools for products of matrix exponentials such as the Baker–Campbell–Hausdorff formula involve infinite series and thus cannot be applied in connection with unbounded operators, a certain complexity in the investigation of higher-order CFQM exponential integrators for Schrödinger equations is related to an appropriate treatment of compositions of evolution operators; an effective concept for the derivation of a local error expansion relies on suitable linearisations of the evolution equations for the exact and numerical solutions, representations by the variation-of-constants formula and Taylor series expansions of parts of the integrands, where the arising iterated commutators determine the regularity requirements on the problem data.

中文翻译：

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非自治线性Schrödinger方程的高阶无换向拟Magnus指数积分器的收敛性分析

这项工作致力于推导适用于非自治线性Schrödinger方程的高阶无换向子准Magnus（CFQM）指数积分器的收敛结果。针对相关特殊情况提供了详细的稳定性和局部误差分析，其中汉密尔顿算子包括拉普拉斯算子和规则的时空相关势。在非自治线性常微分方程中，CFQM指数积分器由涉及相关时间依赖矩阵的某些值的线性组合的指数组成。这种方法扩展到由无界算子给出的非自治线性演化方程。CFQM指数积分器相对于其他时间积分方法（例如Runge–Kutta方法或Magnus积分器）的固有优势是，底层操作员族的结构属性得到了很好的保留；这一特征通过理论分析得到了证实，该理论分析确保了基础希尔伯特空间中的无条件稳定性以及在初始状态下低规则性要求下的全部收敛阶。由于用于矩阵指数乘积的便捷工具（例如Baker–Campbell–Hausdorff公式）涉及无限级数，因此无法与无界算子结合使用，因此对于Schrödinger的高阶CFQM指数积分器的研究存在一定的复杂性方程与进化算子的组成的适当处理有关；