Abstract. Inspired by a question of Colin de Verdière and Truc we study the dynamics of a classical charged particle moving in a bounded planar domain Ω under the influence of a magnetic field B which blows up at the boundary of the domain. We prove that under appropriate blow-up conditions the particle will never reach the boundary. As a corollary we obtain completeness of the magnetic flow. Our blow-up condition is that B should not be integrable along normal rays to the boundary, while its tangential derivative should be integrable along those same rays. Summary

We propose a method to continually monitor the energy of a quantum system. We show that by having some previous knowledge of the system's dynamics, but not all of it, one can use the measured energy to determine many other quantities, such as the work performed on the system, the heat exchanged between the system and a thermal reservoir, the time dependence of the Hamiltonian of the system as well as the total entropy produced by its dynamics. We have also analyzed how this method is dependent on the quality factor of the measurements employed.

We consider a class of magnetic fields defined over the interior of a manifold M which go to infinity at its boundary and whose direction near the boundary of M is controlled by a closed 1-form σ ∞ ∈ Γ(T * ∂M ). We are able to show that charged particles in the interior of M under the influence of such fields can only escape the manifold through the zero locus of σ ∞ . In particular in the case where the 1-form is nowhere vanishing we conclude that the particles become confined to its interior for all time.

We investigated the dynamics of Bose-Einstein condensates (BECs) under an external periodic potential. We consider two such systems, the first being made of exciton-polaritons in a nanoribbon of transition metal dichalcogenides (TMDCs), such as MoSe2, embedded in a microcavity with a spatial curvature, which serves as the source of the external periodic potential. The second, made of bare excitons in a nanoribbon of twisted TMDC bilayer, which naturally creates a periodic Moiré potential that can be controlled by the twist angle. We proved that such systems behave as semiclassical Time Crystals (TCs). This was demonstrated by the fact that the calculated BEC spatial density profile shows a non-trivial long-range two-point correlator that oscillates in time. These BECs density profiles were calculated by solving the quantum Lindblad master equations for the density matrix within the mean-field approximation. We then go beyond the usual mean-field approach, by adding a stochastic term to the master equation, which corresponds to quantum corrections, and we show that the TC phase is still present.