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arXiv:2505.09559 · Modern mathematical physics

The Magnus Expansion for Non-Hermitian Hamiltonians

Yair Mulian

Published Jul 24, 2026• 0 citations
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Abstract

The Magnus expansion offers a powerful geometric method to express a time-ordered exponential operator solution, typical of time-dependent linear differential equations, as an ordinary operatorial exponential. This representation possesses advantageous theoretical and qualitative features, such as preserving the underlying Lie algebraic structure of the system while remaining an exact solution to the original differential equation. For any finite-dimensional Hermitian Hamiltonian governing a quantum system, the standard Magnus expansion guarantees a manifestly unitary representation at every finite order of truncation. However, this foundational property is no longer preserved if the underlying Hamiltonian is non-Hermitian. In this work, we derive a generalized, unitarized expansion framework that systematically restores and maintains the property of geometric unitarity for all bounded finite-dimensional non-Hermitian Hamiltonians.

DOI
10.3390/mmphys2030006

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The Magnus expansion offers a powerful geometric method to express a time-ordered exponential operator solution, typical of time-dependent linear differential equations, as an ordinary operatorial exponential. This representation possesses advantageous theoretical and qualitative features, such as preserving the underlying Lie algebraic structure of the system while remaining an exact solution to the original differential equation. For any finite-dimensional Hermitian Hamiltonian governing a quantum system, the standard Magnus expansion guarantees a manifestly unitary representation at every finite order of truncation. However, this foundational property is no longer preserved if the underlying Hamiltonian is non-Hermitian. In this work, we derive a generalized, unitarized expansion framework that systematically restores and maintains the property of geometric unitarity for all bounded finite-dimensional non-Hermitian Hamiltonians.
Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time ab initio quantum dynamics, and open system quantum dynamics.
We propose a unified approach for different exponential perturbation techniques used in the treatment of time-dependent quantum mechanical problems, namely the Magnus expansion, the Floquet--Magnus expansion for periodic systems, the quantum averaging technique and the Lie--Deprit perturbative algorithms. Even the standard perturbation theory fits in this framework. The approach is based on carrying out an appropriate change of coordinates (or picture) in each case, and can be formulated for any time-dependent linear system of ordinary differential equations. All the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation.
We propose a unified approach for different exponential perturbation techniques used in the treatment of time-dependent quantum mechanical problems, namely the Magnus expansion, the Floquet–Magnus expansion for periodic systems, the quantum averaging technique, and the Lie–Deprit perturbative algorithms. Even the standard perturbation theory fits in this framework. The approach is based on carrying out an appropriate change of coordinates (or picture) in each case, and it can be formulated for any time-dependent linear system of ordinary differential equations. All of the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation.
The Magnus expansion provides an exponential representation of one-parameter operator families, expressed as a series expansion in their generators. This is particularly relevant in quantum mechanics for describing a unitary evolution determined by a time-dependent Hamiltonian generator of the dynamics. The solution is constructed as a series expansion in terms of increasingly complex nested commutators that rapidly become challenging to compute directly. This work establishes a universal upper bound scaling as O((A(t)/ξ)nn−3/2)O\big((A(t)/ξ)^n n^{-3/2}\big)O((A(t)/ξ)nn−3/2) on the error incurred when the Magnus expansion is truncated at an arbitrary order. The factor n−3/2n^{-3/2}n−3/2 sharpens the exponential suppression governed by the integrated norm of the generator A(t)A(t)A(t) and the convergence radius ξξξ, and we prove that this dependence on the order nnn is optimal for the corresponding majorant. The main technical ingredient of the proof is the binary tree representation introduced by Iserles and Norsett from which we derive a recursion formula to delimit the magnitude of any term in the expansion. We complement our analytic bound with explicit computation of the first tree coefficients. With these findings we aim to contribute to the understanding of the accuracy and limitations of the Magnus expansion technique, and to provide a sharper bound for approximating quantum dynamics when information on the generator is not available.
For almost 75 years, the general solution for the Schrödinger equation was assumed to be generated by an exponential or a time-ordered exponential known as the Dyson series. We study the unitarity of a solution in the case of a singular Hamiltonian and provide a new methodology that is not based on the assumption that the underlying space is L2(R). Then, an alternative operator for generating the time evolution that is manifestly unitary is suggested, regardless of the choice of Hamiltonian. The new construction involves an additional positive operator that normalizes the wave function locally and allows us to preserve unitarity, even when dealing with infinite dimensional or non-normed spaces. Our considerations show that Schrödinger and Liouville equations are, in fact, two sides of the same coin and together they provide a unified description for unbounded quantum systems.