The Magnus expansion and some of its applications

Abstract

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.

In-medium similarity renormalization group for closed and open-shell nuclei

“…By rearranging this relation, we obtain a differential equation for U (s) whose formal solution is given by the path-or S-ordered exponential [86,87] …”

Solution of the quasi-one-dimensional linearized Euler equations using flow invariants and the Magnus expansion

“…A simpler and more general method is proposed here to solve (3.17) based on the Magnus expansion (Magnus 1954;Blanes et al 2009). In this method, the solution of (3.17) is written as…”

“…is satisfied (see Blanes et al (2009);Casas (2007); Moan & Niesen (2006) for more details), but may diverge for larger values of the integral. In the above, || · || 2 is the 2-norm of the matrix, which for a square matrix is equal to the spectral norm (the largest singular value of the matrix), and can be calculated as…”

“…This method can be extended to compute the solution up to any frequency as shown in Blanes et al (2009). Given a certain value of Ω max , (4.12) gives the maximal value of ξ = ξ 1 , for which the solution given by (4.2) converges.…”

Dynamical decoupling in optical fibers: Preserving polarization qubits from birefringent dephasing

“…Typically, the coupling terms contributing to the decoherence errors undergo an effective renormalization. This renormalization transformation can be considered as the cancelation of the terms in the Magnus expansion of the effective Hamiltonian [26]. This analysis assumes ideal pulses having zero width, in which case it has been shown that DD sequences can be designed to make higher-order system-bath coupling terms vanish [4,22,27].…”