Summary
A recursive algorithm is derived, allowing the expansion in λ of z = exp [x + λy] for noncommuting x and y, written as ordered product of exponentials. Such an expansion is the inverse of the usual Baker-Campbell-Hausdorff formula. The explicit form of the terms, up to third order in λ is also given. The same method provides the explicit expansion to any order for the matrix elements of z.
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Riccardi, A. A new term in the recursive expansion of the inverse Baker-Campbell-Hausdorff formula. Lett. Nuovo Cimento 40, 147–153 (1984). https://doi.org/10.1007/BF02747116
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DOI: https://doi.org/10.1007/BF02747116