We extend Howland time-independent formalism to the case of completely positive and trace preserving dynamics of finite-dimensional open quantum systems governed by periodic, time-dependent Lindbladian in Weak Coupling Limit, expanding our result from previous papers. We propose the Bochner space of periodic, square integrable matrix-valued functions, as well as its tensor product representation, as the generalized space of states within the time-independent formalism. We examine some densely defined operators on this space, together with their Fourier-like expansions and address some problems related to their convergence by employing general results on Banach space-valued Fourier series, such as the generalized Carleson–Hunt theorem. We formulate Markovian dynamics in the generalized space of states by constructing appropriate time-independent Lindbladian in standard (Lindblad–Gorini–Kossakowski–Sudarshan) form, as well as one-parameter semigroup of bounded evolution maps. We show their similarity with Markovian generators and dynamical maps defined on matrix space, i.e. the generator still possesses a standard form (extended by closed perturbation) and the resulting semigroup is also completely positive, trace preserving and a contraction.

中文翻译：

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关于 CP 可分量子演化的 Howland 时间无关公式

我们将 Howland 时间无关形式主义扩展到有限维开放量子系统的完全正和跟踪保持动力学的情况，该系统由弱耦合极限中的周期性、时间相关的 Lindbladian 控制，扩展了我们之前论文的结果。我们提出了周期性平方可积矩阵值函数的 Bochner 空间，以及它的张量积表示，作为时间无关形式中的广义状态空间。我们检查了该空间上的一些密集定义的算子，以及它们的类傅里叶展开式，并通过采用 Banach 空间值傅里叶级数的一般结果（例如广义 Carleson-Hunt 定理）来解决与其收敛相关的一些问题。我们通过在标准（Lindblad-Gorini-Kossakowski-Sudarshan）形式中构建适当的与时间无关的 Lindbladian 以及有界演化图的单参数半群来在状态的广义空间中制定马尔可夫动力学。我们展示了它们与定义在矩阵空间上的马尔可夫生成器和动态映射的相似性，即生成器仍然具有标准形式（由闭扰动扩展），并且生成的半群也是完全正的，迹保持和收缩。