The class of commutator-free Magnus integrators is known to provide a favourable alternative to standard interpolatory Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. A high-order commutator-free Magnus integrator is given by a composition of several exponentials that comprise certain linear combinations of the values of the defining operator at specified nodes. Due to the fact that previously proposed commutator-free Magnus integrators of order five or higher involve negative coefficients in the linear combinations, severe instabilities are observed for spatially semi-discretised partial differential equations of parabolic type or for master equations describing dissipative quantum systems, respectively. In order to remedy this issue, two different approaches for the design of efficient Magnus integrators of orders four, five, and six are pursued: (i) the study of commutator-free Magnus integrators involving complex coefficients with positive real part, and (ii) the study of unconventional Magnus integrators that comprise in addition a single exponential involving a commutator. Numerical experiments for test equations of Schrödinger and parabolic type confirm that the identified novel Magnus integrators are superior to Magnus integrators previously proposed in the literature.[-]

We discuss the 2D Schrödinger equation for periodic potentials with the symmetry of a hexagonal tiling of the plane. We first review joint work with CL Fefferman on the existence of Dirac points, conical singularities in the band structure, and the resulting effective 2D Dirac dynamics of wave-packets. We then focus on periodic potentials which are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure, corresponding to the single electron model of graphene and its artificial analogues. We prove that for sufficiently deep potentials (strong binding) the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, are uniformly close to those of the celebrated two-band tight-binding model, introduced by PR Wallace (1947) in his pioneering study of graphite. We then discuss corollaries, in the strong binding regime, on (a) spectral gaps for honeycomb potentials with PT symmetry-breaking perturbations, and (b) topologically protected edge states for honeycomb structures with "rational edges. This is joint work with CL Fefferman and JP Lee-Thorp. Extensions to Maxwell equations (with Y Zhu and JP Lee-Thorp) will also be discussed.[-]

The development of quantum information processing and quantum computation goes hand in hand with the ability of addressing and manipulating quantum systems. Quantum Control Theory has provided a successful framework, both theoretical and experimental, to design and develop the control of such systems. In particular, for finite dimensional quantum systems or finite dimensional approximations to them. The theory for infinite dimensional systems is much less developed.
In this talk I propose a scheme of infinite dimensional quantum control on quantum graphs based on interacting with the system by changing the self-adjoint boundary conditions. I will show the existence of solutions of the time-dependent Schrödinger equation, the stability of the solutions and the (approximate) controllability of the state of a quantum system by modifying the boundary conditions on generic quantum graphs.[-]