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2 results on '"Analytic prolongation"'

1. On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues.

Author

Biagi, Stefano, Bonfiglioli, Andrea, and Matone, Marco

Subjects
*BANACH algebras, *MATRICES (Mathematics), *LOGARITHMS
Abstract

We investigate some topics related to the celebrated Baker-Campbell-Hausdorff Theorem: a non-convergence result and prolongation problems. Given a Banach algebra A with identity I, and given X , Y ∈ A , we study the relationship of different issues: the convergence of the BCH series ∑ n Z n (X , Y) , the existence of a logarithm of e X e Y , and the convergence of the Mercator-type series ∑ n (− 1) n + 1 (e X e Y − I) n / n which provides a selected logarithm of e X e Y . We fix general results, among which we provide a non-convergence result for the BCH series, and (by suitable matrix counterexamples) we show that various pathologies can occur. These are related to some recent results, of interest in physics, on closed formulas for the BCH series: while the sum of the BCH series presents several non-convergence issues, these closed formulas can provide a prolongation for the BCH series when it is not convergent. On the other hand, we show by suitable counterexamples that an analytic prolongation of the BCH series can be singular even if the BCH series itself is convergent. [ABSTRACT FROM AUTHOR]

Published
2020
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2. On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues

Author

Stefano Biagi, Andrea Bonfiglioli, Marco Matone, and Stefano Biagi, Andrea Bonfiglioli, Marco Matone

Subjects
High Energy Physics - Theory, Primary: 15A16, Pure mathematics, media_common.quotation_subject, logarithms, convergence of the BCH series, FOS: Physical sciences, 010103 numerical & computational mathematics, Hwa-Long Gau, Secondary: 34A25, 01 natural sciences, Convergnece of the BCH series, High Energy Physics - Phenomenology (hep-ph), FOS: Mathematics, 40A30, Analytic prolongation, 0101 mathematics, prolongation of the BCH series, Banach *-algebra, Mathematical Physics, Condensed Matter - Statistical Mechanics, Computer Science::Information Theory, media_common, Mathematics, Algebra and Number Theory, Statistical Mechanics (cond-mat.stat-mech), Primary: 15A16, 15B99, 40A30. Secondary: 34A25, 15B99, analytic prolongation, Baker-Campbell-Hausdorff Theorem, matrix algebras, Prolongation, Mathematics - Rings and Algebras, Mathematical Physics (math-ph), High Energy Physics - Phenomenology, High Energy Physics - Theory (hep-th), Rings and Algebras (math.RA), Baker–Campbell–Hausdorff formula, Identity (philosophy), Convergence (relationship)
Abstract

We investigate some topics related to the celebrated Baker-Campbell-Hausdorff Theorem: a non-convergence result and prolongation issues. Given a Banach algebra $\mathcal{A}$ with identity $I$, and given $X,Y\in \mathcal{A}$, we study the relationship of different issues: the convergence of the BCH series $\sum_n Z_n(X,Y)$, the existence of a logarithm of $e^Xe^Y$, and the convergence of the Mercator-type series $\sum_n {(-1)^{n+1}}(e^Xe^Y-I)^n/n$ which provides a selected logarithm of $e^Xe^Y$. We fix general results and, by suitable matrix counterexamples, we show that various pathologies can occur, among which we provide a non-convergence result for the BCH series. This problem is related to some recent results, of interest in physics, on closed formulas for the BCH series: while the sum of the BCH series presents several non-convergence issues, these closed formulas can provide a prolongation for the BCH series when it is not convergent. On the other hand, we show by suitable counterexamples that an analytic prolongation of the BCH series can be singular even if the BCH series itself is convergent., Comment: 18pp Additional results and comments added

Published
2018

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