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

Bialynicki‐Birula, Iwo; Mielnik, Bogdan; Plebański, Jerzy F.

doi:10.1016/0003-4916(69)90351-0

Cited by 121 publications

(67 citation statements)

References 4 publications

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“…Prato and Lamberti [198] give explicitly the fifth order using an algorithmic point of view. One can also find in the literature quite involved explicit expressions for an arbitrary order [16,167,206,219,220]. In the next subsection we describe a recursive procedure to generate the terms in the expansion.…”

Section: Formulae For the First Terms In Magnus Expansionmentioning

confidence: 99%

The Magnus expansion and some of its applications

Blanes¹,

Casas²,

Oteo³

et al. 2009

Physics Reports

1,091

1,300

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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution.The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.

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Section: Formulae For the First Terms In Magnus Expansionmentioning

confidence: 99%

The Magnus expansion and some of its applications

Blanes¹,

Casas²,

Oteo³

et al. 2009

Physics Reports

1,091

1,300

View full text Add to dashboard Cite

show abstract

“…(2) evaluated with the Hamiltonian of Eq. (12). Proceeding in this manner, and after some algebraic manipulations, we arrive at:…”

Section: Formalismmentioning

confidence: 99%

“…This is briefly what we shall need to know about the Magnus expansion for its present application; further details about the formalism and recursive procedures for building up the successive terms can be found in the specific literature [11,12,13]. Here, we use the first and second order Magnus approximation to seek solutions to the problem of 2ν oscillations in a medium with an arbitrary density profile, which is symmetric with respect to the middle point of the neutrino trajectory.…”

Section: Introductionmentioning

confidence: 99%

Perturbative exponential expansion and matter neutrino oscillations

2008

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We derive an analytical description of neutrino oscillations in matter based on the Magnus exponential representation of the time evolution operator. Our approach is valid in a wide range of the neutrino energies and properly accounts for the modifications that the respective probability transitions suffer when neutrinos originated in different sources traverse the Earth. The present approximation considerably improves over other perturbative treatments existing in the current literature. Furthermore, the analytical expressions derived inside the Magnus framework are remarkably simple, which facilitates their practical use. When applied to the calculation of the day-night asymmetry in the solar neutrino flux our result reproduces the numerical calculation with an accuracy better than 1% for the first order approximation. When the approximation is extended to the second order, the accuracy of the method is further improved by almost one order of magnitude, and it is still better than 5% even for neutrino energies as large as 100 MeV. In the GeV regime characteristic of atmospheric and accelerator neutrinos this accuracy is complemented by a good reproduction of the position of the maxima in the flavor transition probabilities.

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“…We can write it as Ω(t) = Ω 1 (t) + Ω 2 (t) + Ω 3 (t) + Ω 4 (t) + · · · , where the term Ω n (t) is a sum of n-fold integrals of n − 1 nested commutators. Explicit expressions for the Ω n (t) are given by Bialynicki-Birula, Mielnik and Plabański [2], Chacon and Fomenko [4], and Iserles and Nørsett [9]. The Magnus series can be used to derive the Baker-Campbell-Hausdorff (BCH) formula for the product of two matrix exponentials.…”

Section: Introductionmentioning

confidence: 99%

Convergence of the Magnus Series

2007

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The Magnus series is an infinite series which arises in the study of linear ordinary differential equations. If the series converges, then the matrix exponential of the sum equals the fundamental solution of the differential equation. The question considered in this paper is: When does the series converge? The main result establishes a sufficient condition for convergence, which improves on several earlier results.

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

Cited by 121 publications

References 4 publications

The Magnus expansion and some of its applications

The Magnus expansion and some of its applications

Perturbative exponential expansion and matter neutrino oscillations

Convergence of the Magnus Series

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