Only a handful of quantum systems can be solved analytically: in most realistic models, approximations are being used to derive effective models which are easier to solve. Although there are plenty of methods for deriving effective models, many of them are not understood well enough. Examples for this are the rotating-wave approximation, higher-order corrections from the Floquet--Magnus expansion, Trotterization, and finite-dimensional cutoffs of infinite dimensional models. In the recent past, we have studied the validity of several approximation methods by computing error bounds and rigorously proving convergence of the errors in the relevant parameter regime. A key aspect of this program is the following: whenever infinite-dimensional systems are studied, the relevant energy scale for the error is determined by the specific initial state one considers—that is, the same approximation can be more or less effective depending on the initial condition.

Selected publications:

• D. Burgarth, P. Facchi, R. Hillier, M. Ligabò, "Taming the Rotating Wave Approximation". Quantum 8 (2024), 1262.

• A. Dey, D. Lonigro, K. Yuasa, D. Burgarth, "Error bounds for the Floquet–Magnus expansion and their application to the semiclassical quantum Rabi model". Phys. Rev. A 112 (2025), 053723.

• F. Fischer, D. Burgarth, D. Lonigro, "Quantum particle in the wrong box (or: the perils of finite-dimensional approximations)". Quantum 10 (2026), 1985.

• L. Richter, D. Burgarth, D. Lonigro, "Quantifying the rotating-wave approximation of the Dicke model". J. Phys. A: Math. Theor. 59 (2026), 075203.

The Schrödinger equation generated by a given time-independent Hamiltonian admits a unique solution for any choice of initial condition if and only if the Hamiltonian is self-adjoint. While verifying the latter for finite-dimensional matrices is an easy task, the question of self-adjiontness becomes much more involved in infinite-dimensional systems, such as the ones involving boson degrees of freedom: a given formal Hamiltonian might have multiple self-adjoint extensions, each generating a distinct evolution, or no self-adjoint extension at all. We address this problem in relevant cases. These include bosonic systems with higher-order terms, which are routinely used for continuous-variable quantum computing, as well as light–matter interactions involving ultraviolet divergences to be renormalized. However, even the simplest textbook examples, like the quantum harmonic oscillator, still offer surprises.

Selected publications:

• D. Lonigro, "Generalized spin-boson models with non-normalizable form factors". J. Math. Phys. 63 (2022), 072105.

• D. Lonigro, A. Hahn, D. Burgarth, "On the Liouville–von Neumann Equation for Unbounded Hamiltonians". Open Sys. Inf. Dyn. 31 (2024), 2450018.

• F. Fischer, D. Burgarth, D. Lonigro, "Self-adjoint realizations of higher-order squeezing operators". J. Phys. A: Math. Theor. 59 (2026), 255203.

• D. Burgarth, P. Facchi, "Orbital Angular Momentum Can Take Non-Integer Values in a Closed Universe". arXiv:2506.03254 [quant-ph].

Quantum control investigates the manipulation of quantum systems via externally tunable time-dependent terms. Fundamental questions include whether arbitrary state-to-state transitions can be achieved, how quickly and robustly this can be done, and what constraints arise from the unbounded nature of the system or the structure of the controls. We have contributed to this field in multiple ways. We proved quantum speed limits that are connected to the symmetries of the system, proved sufficient criteria for convergence of numerical optimization methods for control pulses, studied dynamical decoupling for large classes of infinite dimensional systems, and extended known spectral criteria for controllability to cases where the drift and control Hamiltonians do not share a common invariant domain, for example in the presence of time-dependent boundary conditions.

Selected publications:

• D. Burgarth, P. Facchi, and R. Hillier, "Control of Quantum Noise: On the Role of Dilations". Ann. Henri Poincaré 24 (2023), 325–347.

• A. Balmaseda, D. Lonigro, and J.M. Pérez-Pardo, "On Global Approximate Controllability of a Quantum Particle in a Box by Moving Walls". SIAM J. Control Optim. 62 (2024), 826–852.

• A. Hahn, D. Burgarth, D. Lonigro. "Efficiency of Dynamical Decoupling for (Almost) Any Spin-Boson Model". SciPost Phys. 19 (2025), 035.

• M. Wiedmann, D. Burgarth, "Quantum Speed Limits from Symmetries in Quantum Control". J. Phys. A: Math. Theor. 59 (2026), 065301.

Quantum systems often retain memory of their past interactions with the environment, giving rise to genuinely non-Markovian dynamics that cannot be captured by simple memoryless models. Our research develops rigorous mathematical tools to characterize these memory effects, identify hidden forms of non-Markovian behaviour, and understand their impact on quantum technologies. At the same time, we investigate the fundamental structure of quantum stochastic processes and multi-time measurements, addressing questions such as whether quantum dynamics can be described by underlying trajectories and how temporal correlations differ from their classical counterparts. This line of research connects open quantum systems, probability theory, and the mathematical foundations of quantum mechanics.

Selected publications:

• D. Burgarth, P. Facchi, M. Ligabò, D. Lonigro, "Hidden non-Markovianity in open quantum systems". Phys. Rev. A 103 (2021), 012203.

• D. Burgarth, P. Facchi, M. Fraas, R. Hillier, "Non-Markovian noise that cannot be dynamically decoupled by periodic spin echo pulses". SciPost Phys. 11, 027 (2021).

• D. Lonigro, D. Chruściński, "Quantum regression in dephasing phenomena". J. Phys. A: Math. Theor. 55 (2022), 225308.

• D. Lonigro, F. Sakuldee, Ł. Cywiński, D. Chruściński, P. Szańkowski, "Double or nothing: a Kolmogorov extension theorem for (bi)probabilities in quantum mechanics". Quantum 8 (2024), 1447.