Quantum analysis on the convergence speed of exponential product formulas — differential-subtraction and exchange-integration method on concise norm bounds

Abstract

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.