An exotic calculus of Berezin-Toeplitz operators

This paper generalizes composition formulae of Berezin-Toeplitz operators for quantizations of smooth functions on a compact K\"ahler manifold to certain exotic symbol classes. This is accomplished via careful analysis of the kernel of these operators using Melin and Sj\"ostrand's method of complex stationary phase. This provides a functional calculus result, a trace formula, and a parametrix construction for a larger symbol class. These results are used in proving a probabilistic Weyl-law for randomly perturbed Toeplitz operators.Comment: 37 page

A probabilistic Weyl-law for perturbed Berezin-Toeplitz operators

This paper proves a probabilistic Weyl-law for the spectrum of randomly perturbed Berezin-Toeplitz operators, generalizing a result proven by Martin Vogel in 2020. This is done following Vogel's strategy using an exotic symbol calculus developed by the author in a recent paper.Comment: 21 pages, 3 figure

Particle Trajectories for Quantum Maps

We study the trajectories of a semiclassical quantum particle under repeated indirect measurement by Kraus operators, in the setting of the quantized torus. In between measurements, the system evolves via either Hamiltonian propagators or metaplectic operators. We show in both cases the convergence in total variation of the quantum trajectory to its corresponding classical trajectory, as defined by propagation of a semiclassical defect measure. This convergence holds up to the Ehrenfest time of the classical system, which is larger when the system is less chaotic. In addition, we present numerical simulations of these effects.Comment: 35 pages, 7 figure

Magic angle (in)stability and mobility edges in disordered Chern insulators

Why do experiments only exhibit one magic angle if the chiral limit of the Bistritzer-MacDonald Hamiltonian suggest a plethora of them? - In this article, we investigate the remarkable stability of the first magic angle in contrast to higher (smaller) magic angles. More precisely, we examine the influence of disorder on magic angles and the Bistritzer-MacDonald Hamiltonian. We establish the existence of a mobility edge near the energy of the flat band for small disorder. We also show that the mobility edges persist even when all global Chern numbers become zero, leveraging the $C_{2z}T$ symmetry of the system to demonstrate non-trivial sublattice transport. This effect is robust even beyond the chiral limit and in the vicinity of perfect magic angles, as is expected from experiments