An exact power series representation of the Baker-Campbell-Hausdorff formula
An exact power series representation of the Baker-Campbell-Hausdorff formula
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DOI：
10.1088/1751-8121/abcbae
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发表时间：
2021-01-08
影响因子：
2.1
通讯作者：
Long, M. W.
中科院分区：
文献类型：
--
作者：
Moodie, Jordan C.;Long, M. W.
构建了贝克 - 坎贝尔 - 豪斯多夫公式在两个变量中仅一个变量下的幂级数精确表达式。通过双曲函数找到了该级数的封闭形式系数，其中包含了对第二个变量的所有依赖关系。有人认为，如果只有微扰变量很小，那么这个精确级数就可以被截断，并有望对完整展开给出良好的近似。这改进了现有的公式，现有的公式要求两个变量都很小。提供了几种不同的表达式，并重点关注其中一个矩阵是对角矩阵的情况，在这种情况下得到了一个特别易于使用的公式。
An exact representation of the Baker-Campbell-Hausdorff formula as a power series in just one of the two variables is constructed. Closed form coefficients of this series are found in terms of hyperbolic functions, which contain all of the dependence on the second variable. It is argued that this exact series may then be truncated and be expected to give a good approximation to the full expansion if only the perturbative variable is small. This improves upon existing formulae, which require both to be small. Several different representations are provided and emphasis is given to the situation where one of the matrices is diagonal, where a particularly easy to use formula is obtained.