Using elementary means, we derive the three most popular splittings of $e^{(A+B)}$ and their error bounds in the case when $A$ and $B$ are (possibly unbounded) operators in a Hilbert space, generating strongly continuous semigroups, $e^{tA}$, $e^{tB}$ and $e^{t(A+B)}$. The error of these splittings is bounded in terms of the norm of the commutators $[A, B]$, $[A, [A, B]]$ and $[B, [A, B]]$.

翻译：我们利用基本方法推导了 $e^{(A+B)}$ 的三种最常用的分裂形式及其误差界，其中 $A$ 和 $B$ 是希尔伯特空间中的（可能无界）算子，生成强连续半群 $e^{tA}$、$e^{tB}$ 和 $e^{t(A+B)}$。这些分裂的误差由交换子 $[A, B]$、$[A, [A, B]]$ 和 $[B, [A, B]]$ 的范数界定。