We study convergence rates of the Trotter-Kato splitting $e^{A+L} = \lim_{n \to \infty} (e^{L/n} e^{A/n})^n$ in the strong operator topology. In the first part, we use complex interpolation theory to treat generators $L$ and $A$ of contraction semigroups on Banach spaces, with $L$ relatively $A$-bounded. In the second part, we study unitary dynamics on Hilbert spaces and develop a new technique based on the concept of energy constraints. Our results provide a complete picture of the convergence rates for the Trotter splitting for all common types of Schr\"odinger and Dirac operators, including singular, confining and magnetic vector potentials, as well as molecular many-body Hamiltonians in dimension $d=3$. Using the Brezis-Mironescu inequality, we derive convergence rates for the Schr\"odinger operator with $V(x)=\pm |x|^{-a}$ potential. In each case, our conditions are fully explicit.

翻译：我们研究了Trotter-Kato分裂$e^{A+L} = \lim_{n \to \infty} (e^{L/n} e^{A/n})^n$在强算子拓扑下的收敛速率。在第一部分中，我们利用复插值理论处理Banach空间上压缩半群的生成元$L$和$A$，其中$L$相对于$A$是有界的。在第二部分中，我们研究了Hilbert空间上的酉动力学，并基于能量约束的概念发展了一种新技术。我们的结果为所有常见类型的Schrödinger算子和Dirac算子的Trotter分裂收敛速率提供了完整的描述，包括奇异势、约束势和磁矢势，以及三维（$d=3$）分子多体Hamilton量。利用Brezis-Mironescu不等式，我们推导了具有势能$V(x)=\pm |x|^{-a}$的Schrödinger算子的收敛速率。在每种情况下，我们的条件都是完全显式的。