指数算子正分解的结构
The Structure of Positive Decompositions of Exponential Operators
摘要：
许多物理演化方程的解可以表示为两个或多个算子作用于初始数据的指数。通过将指数分解为乘积形式，可以系统地导出精确的解。对于时间可逆的方程，如汉密尔顿方程或薛定谔方程，分解系数是否为正并不重要。事实上，大多数求解经典动力学问题的辛算法都含有一些负系数。对于时间不可逆系统，如福克-普朗克方程或量子统计传播子，只有正系数分解，尊重扩散核的时间不可逆性，可以产生实用的算法。这些正的时间步长，向前分解，是一类非常有效的因式分解算法。这项工作介绍了一个框架，了解这些算法的结构。通过适当的分解系数的表示，我们表明，特定的误差项和阶条件可以解决{ \it解析}。使用这个框架，我们可以超越Sheng-Suzuki定理，并推导出误差系数$e_{VTV}$的下界。通过微扰推广的框架，我们可以进一步证明，它是不可能有一个六阶前向算法只包括交换子$[VTV]\equiv[V,[T,V]]$。这些高阶前向算法的模式是，在从（2n）$^{\rm th}$到（2n+2）$^{\rm th}$阶的过程中，必须在分解过程中包括新的交换子$[VT^{2n-1}V]$。
英文摘要：
The solution of many physical evolution equations can be expressed as an exponential of two or more operators acting on initial data. Accurate solutions can be systematically derived by decomposing the exponential in a product form. For time-reversible equations, such as the Hamilton or the Schrödinger equation, it is immaterial whether or not the decomposition coefficients are positive. In fact, most symplectic algorithms for solving classical dynamics contain some negative coefficients. For time-irreversible systems, such as the Fokker-Planck equation or the quantum statistical propagator, only positive-coefficient decompositions, which respect the time-irreversibility of the diffusion kernel, can yield practical algorithms. These positive time steps only, forward decompositions, are a highly effective class of factorization algorithms. This work introduce a framework for understanding the structure of these algorithms. By a suitable representation of the factorization coefficients, we show that specific error terms and order conditions can be solved {\it analytically}. Using this framework, we can go beyond the Sheng-Suzuki theorem and derive a lower bound for the error coefficient $e_{VTV}$. By generalizing the framework perturbatively, we can further prove that it is not possible to have a sixth order forward algorithm by including only the commutator $[VTV]\equiv[V,[T,V]]$. The pattern of these higher order forward algorithms is that in going from the (2n)$^{\rm th}$ to the (2n+2)$^{\rm th}$ order, one must include a new commutator $[VT^{2n-1}V]$ in the decomposition process.
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