The Baker-Campbell-Hausdorff formula and the convergence of the Magnus expansion

The authors show that for a wide class of dynamical systems (described by Hamiltonians of the form usually considered in time-dependent perturbation theory) the divergence of the Magnus expansion in the Schrodinger picture for large time intervals is due to pole singularities inherent to the Baker-Campbell-Hausdorff formula.

[1]  D. P. Burum.  Magnus expansion generator , 1981 .

[2]  Salzman.  New criterion for convergence of exponential perturbation theory in the Schrödinger representation. , 1987, Physical review. A, General physics.

[3]  James Wei,et al.  Lie Algebraic Solution of Linear Differential Equations , 1963 .

[4]  Kurt Bernardo Wolf,et al.  Lie Methods in Optics , 1986 .

[5]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[6]  R. D. Richtmyer,et al.  Expansion of the Campbell‐Baker‐Hausdorff formula by computer , 1965 .

[7]  Karl Goldberg,et al.  The formal power series for $\log e^xe^y$ , 1956 .

[8]  A. Maradudin,et al.  The Baker‐Hausdorff Formula and a Problem in Crystal Physics , 1962 .

[9]  E. B. Fel'dman.  On the convergence of the magnus expansion for spin systems in periodic magnetic fields , 1984 .

[10]  Stanly Steinberg,et al.  Lie series, Lie transformations, and their applications , 1986 .

[11]  On the convergence of exponential perturbation theories in the Schrödinger representation , 1986 .

[12]  Klarsfeld,et al.  Recursive generation of higher-order terms in the Magnus expansion. , 1989, Physical review. A, General physics.

[13]  Eduardo A. Castro,et al.  The magnus expansion for the damped harmonic oscillator , 1987 .

[14]  Comment on: The Magnus expansion for the harmonically driven harmonic oscillator , 1988 .

[15]  M. Maricq.  Convergence of the Magnus expansion for time dependent two level systems , 1987 .

[16]  R. Wilcox.  Exponential Operators and Parameter Differentiation in Quantum Physics , 1967 .

[17]  J. Oteo,et al.  Exponential infinite-product representations of the time-displacement operator , 1989 .