Convergence of Magnus and Magnus-like expansions in the Schrödinger representation

Abstract

General and greatly simplified methods are presented for calculating terms in the exponent in Magnus and Magnus-like expansions to first order in a ‘‘small’’perturbation and to infinite order in the overall Hamiltonian in the Schrödinger representation. These techniques are applied to four simple but important models and it is shown that in each case the Magnus exponent diverges for some range of the parameters of the model. This result casts serious doubt on the utility of the Magnus expansion in the Schrödinger representation. Several of the problems are also treated in the interaction representation giving results which converge throughout the useful range of parameters. There is no evidence that a similar question exists in this representation.

Sufficient conditions for the convergence of the Magnus expansion

“…converges for ω > ω 0 and diverges otherwise [10,30,22]. Several different arguments have been offered trying to explain this phenomenon [18].…”

“…It is described by the Hamiltonian 5) where σ x , σ y , σ z are Pauli matrices, and β is a coupling constant. In fact, this system constitutes a truncation in state space of a more general one, namely an atom or freely rotating molecule in a circularly polarized radiation field [30,18]. It has been previously established that when t = 2π/ω the Magnus expansion of the corresponding evolution operator U (t), solution of the Schrödinger equation…”

Graphene with time-dependent spin-orbit coupling: truncated Magnus expansion approach

“…This higher degree of accuracy is associated to the unitarity preserving property of the ME approach. Salzman [22] and Fernandez [23] also deal with the driven quantum harmonic oscillator and conclude that the ME solution will always converge for times within the first period 0 < t < T . This problem admits an exact analytical solution and the authors show that the ME solution gives a proper account of the dynamics apart for values of the driving frequency near resonances.…”

Divergence of the Floquet-Magnus expansion in a periodically driven one-body system with energy localization

“…For example, let us consider the harmonic oscillator driven by a periodic force. It can be shown that the radius of convergence of the Floquet-Magnus expansion is equal to the natural period of the oscillator [14][15][16]. This means that the expansion diverges when the system indefinitely absorbs the energy from the driving owing to the resonance.…”