Describing, approximating, and controlling quantum systems with unbounded Hamiltonians

Summary. Any quantum-mechanical model poses three tasks: it has to be described, by an observable that generates its dynamics; its dynamics has to be approximated, since it can rarely be computed exactly; and it may have to be controlled, whether to exploit some of its degrees of freedom or to suppress them. In finite dimensions all three reduce to finite-dimensional algebraic problems. When the Hilbert space is infinite-dimensional and the generator of the dynamics—the Hamiltonian—is unbounded, as is the case for every bosonic system, each of the three tasks acquires difficulties of its own: a formal expression may determine several dynamics or none, an approximation may converge only state by state or not at all, and exact control is generally out of reach. These difficulties are of mathematical interest, but they are also practical ones, with measurable consequences for the models now being built as quantum hardware. This thesis treats all three.

Chapter 1 develops this picture, and Chapter 2 collects the operator-theoretic background. Chapter 3 classifies the self-adjoint realisations of higher-order squeezing operators and of multiphoton light–matter Hamiltonians, and studies their behaviour under small regularising perturbations. Chapter 4 gives a checkable characterisation of the domain of the Liouvillian corresponding to an unbounded Hamiltonian, and with it a precise form of the Liouville–von Neumann equation. Chapter 5 derives, from a single integration-by-parts identity, explicit error bounds for the rotating-wave approximation or the Floquet–Magnus expansion in four settings: for bounded generators at every order, for the Dicke model with an error governed by the total angular momentum, for unbounded generators at every order with the effective Hamiltonians shown to admit self-adjoint realisations, and for spin–boson models with a structured field. Chapter 6 shows that finite-dimensional truncations of a semibounded Hamiltonian converge to the dynamics of a Friedrichs extension, which need not be the intended one—a particle in a box, truncated in a basis that carries no boundary information, is simulated with Dirichlet walls whatever boundary condition was intended—and that truncations of higher-order squeezing operators do not converge at all, oscillating between two distinct realisations unless a Kerr term restores essential self-adjointness. Chapter 7 proves that dynamical decoupling of a qubit from a thermal boson field works at the optimal rate O(1/N) for spin–boson models under suitable regularity assumptions, with all dependence on the field entering through its partition function. Chapter 8 establishes approximate controllability for systems in which the control acts through a singular perturbation, such as a point interaction of tunable strength, with an explicit bound on the control effort.