A general method to evaluate rigorously concise norm bounds on the difference between the original exponential operators and their corresponding exponential product formulas is proposed, in order to evaluate the convergence speed of exponential product formulas for a new kind of exponential operator, exp(x²[A, B]). One of the remarkable results on this issue is given by the following formula: e^[A,B] is equal to the n → ∞ limit of the product ${(\exp (iA/n)\exp (-iB/n)\exp (-iA/n)\exp (iB/n))}^{n^2}$ for the Hermitian operators A and B. The convergence speed of this formula is proved rigorously to be O(1/n) even for unbounded operators A and B under the condition that the third-order free Lie elements of A and B should be bounded in norm.

中文翻译：

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指数乘积公式收敛速度的量子分析——简明范数界上的微分减法和交换积分法

提出了一种评估原始指数算子与其对应指数乘积公式之间差异的严格简明范数界限的通用方法，以评估一种新型指数算子的指数乘积公式的收敛速度，exp( x ² [ A , B ])。下面的公式给出了关于这个问题的显着结果之一：e ^{[ A , B ]}等于乘积的n → ∞ 极限${(\mathrm{经验值}(\mathrm{一世}\mathrm{一种}/n)\mathrm{经验值}(-\mathrm{一世}乙/n)\mathrm{经验值}(-\mathrm{一世}\mathrm{一种}/n)\mathrm{经验值}(\mathrm{一世}乙/n))}^{n^2}$对于厄米算符A和B。该式的收敛速度被严格地证明是O（1 / Ñ）即使对于无界算子甲和乙这的三阶自由烈元件的条件下，甲和乙应在范数有界的。