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Annals of Physics

Explicit solution of the continuous Baker-Campbell-Hausdorff problem and a new expression for the phase operator

Iwo Białynicki‐Birula, Bogdan Mielnik, Jerzy F Plebanski

Published Jan 1, 1969• 148 citations
Publisher
DOI
10.1016/0003-4916(69)90351-0

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Explicit solution of the continuous Baker-Campbell-Hausdorff problem and a new expression for the phase operator

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The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.

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Solution of the Schr�dinger Equation for a Linear Potential using the Extended Baker-Campbell-Hausdorff Formula

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A generalization of the phase approach to scattering theory, and some numerical results

Francesco Calogero, D. G. Ravenhall

Il Nuovo Cimento•Jun 1, 1964•12 citations
The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.