The Floquet-Magnus expansion of unbounded operators

Abstract: The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.

• The paper presents a rigorous, iterative framework to construct effective Hamiltonians from unbounded operators, achieving error control up to O(T^(L+2)).
• It derives explicit error estimates for high-frequency approximations, ensuring controlled deviations over extended times without relying on full series convergence.
• Under mild spectral and self-adjointness conditions, the framework reconciles traditional FM expansion results with practical applications in quantum control and Floquet engineering.

The Floquet–Magnus Expansion of Unbounded Operators: An Analytical Framework

Introduction and Motivation

The Floquet–Magnus (FM) expansion plays a central role in the analysis of periodically-driven quantum systems. For bounded Hamiltonians, it provides a systematic procedure to approximate the time evolution by an effective time-independent Hamiltonian, underpinning much of contemporary quantum control and Floquet engineering. However, the physically relevant regime in many-body and quantum optical applications involves unbounded Hamiltonians, where the standard FM expansion lacks rigorous justification and is plagued by subtle issues related to self-adjointness, domain properties, and convergence. This work develops a comprehensive, nonperturbative, and mathematically controlled approach to extend the FM expansion to a broad class of unbounded, time-periodic Hamiltonians.

Framework and Main Results

The authors formulate their analysis in a general setting on a separable Hilbert space $\mathcal{H}$, with a reference self-adjoint operator $H_0$ setting the energy scale and a family $(H(t))_{t \in \mathbb{R}}$ of unbounded, time-periodic, possibly non-self-adjoint operators. The time evolution is described by strongly continuous unitary propagators $(U^{(T)}(t))_{t \in \mathbb{R}}$ generated by rescaled $T$-periodic Hamiltonians $H^{(T)}(t) = H(t/T)$. The authors introduce a rigorous regime of regularity and relative boundedness conditions with respect to $H_0$ (generalizing Kato’s hyperbolic case) ensuring the existence, uniqueness, and differentiability structure required for the subsequent analysis.

The central technical innovation is the use of an iterated integration-by-parts expansion for propagators, originally formulated for bounded operators, to derive recursive error-controlled approximations in the unbounded setting. This approach yields:

1. Recursive Construction of Effective Hamiltonians: For arbitrary truncation order $L$, the adapted FM ansatz

$$H_{\mathrm{eff},L}^{(T)} = \sum_{l=0}^L T^l H_{\mathrm{eff}}^{[l]}$$

is constructed with explicit formulas for each $H_{\mathrm{eff}}^{[l]}$, guaranteeing that the effective stroboscopic dynamics approximates the exact propagator up to $H_0$0.

1. Error Bounds in the High-Frequency Limit: The rigorous error bounds established do not require convergence of the full FM series, a restriction that is already highly nontrivial even in the bounded case. Instead, for any fixed order, the error in the evolution after $H_0$1 periods is bounded by $H_0$2 as $H_0$3. During times $H_0$4, the deviation from the true dynamics remains controlled.
2. Self-Adjointness and Domain Considerations: Under mild assumptions on spectral localization and conjugation (time-reversal symmetry) properties, the effective Hamiltonians constructed are shown to be symmetric and to admit self-adjoint extensions at every order. This ensures the unitarity of the approximate evolution.
3. Consistency with the Standard Floquet–Magnus Expansion: In the regime where the formal FM expansion for unbounded Hamiltonians makes sense, the authors demonstrate that their recursive construction reproduces the FM terms order by order. Thus, for each truncation, their effective Hamiltonian coincides with the corresponding FM expansion, while being well-defined and possessing rigorous error bounds without requiring convergence of the full infinite series.

Strong Numerical and Analytical Claims

• Rigorous High-Frequency Approximation: For any truncation order, the effective dynamics generated by $H_0$5 approximates the stroboscopic evolution to accuracy $H_0$6 for stroboscopic times and $H_0$7 for times $H_0$8, uniformly as $H_0$9. This holds regardless of convergence properties of the FM series and for a wide class of unbounded Hamiltonians.
• Minimal Domain and Spectral Restrictions: The sufficient conditions required for the error estimates (relative boundedness, commutator control, spectral locality, and conjugation symmetry) are explicit and can be checked for various physical models, as illustrated by the quantum Rabi model and driven oscillator examples.
• Constructive Error Control: The framework provides explicit computable error bounds at each order, which can be specialized to concrete models to yield sharper estimates than existing general-purpose bounds.

Applications and Illustrative Examples

The analysis is concretely implemented on physically relevant models:

• Quantum Rabi Model (Interaction Picture): The leading effective Hamiltonian recovers the rotating-wave approximation, and higher-order terms systematically yield corrections such as the Bloch-Siegert shift. The expansion matches known physics while providing rigorous justification and explicit error bounds.
• Periodically Driven Quantum Harmonic Oscillator: The approach allows for explicit computation of high-frequency expansions in the presence of unbounded (quadratic) driving terms, including all domain and self-adjointness subtleties.

Implications and Outlook

This work fundamentally advances the rigorous mathematical tools for analyzing periodically driven quantum systems with unbounded Hamiltonians. The framework not only establishes the validity and applicability of FM-type expansions beyond the bounded case but also provides a template for constructive and quantitative error control in scenarios involving unbounded generators—ubiquitous in quantum field theory, many-body physics, and quantum optics.

Theoretical implications include a new paradigm for effective Hamiltonian analysis in infinite dimensions, and a path to understanding effective descriptions and heating effects in the high-frequency regime far beyond the limitations of previous methods based on norm convergence. The constructive nature of the framework facilitates future extensions, including adaptation to multiple timescales, dissipative dynamics, and many-body localization regimes.

Practical repercussions involve enabling rigorous design and interpretation of quantum control protocols and Floquet engineering schemes in strongly driven systems with unbounded operators, which are the norm in cutting-edge experimental platforms.

Future directions will likely include optimization of error bounds for specific systems, exploration of convergence properties in the unbounded setting, and potential generalizations to nonperiodic or quasi-periodic drive scenarios, as well as applications to dispersive and non-Markovian open quantum systems.

Conclusion

The paper furnishes a complete, error-controlled, and fully constructive theory of the Floquet–Magnus expansion for a substantially general class of unbounded, time-periodic Hamiltonians. It provides a robust mathematical foundation for effective Hamiltonian methods in the study of driven quantum systems and establishes both theoretical clarity and practical tools for ongoing research at the interface of mathematical physics, quantum optics, and many-body dynamics (2605.23734).