Spectral gaps for many-particle systems, semiclassical analysis of graphene, and quantum dynamics

Abstract

This thesis consists of three parts with five chapters. All results presented in this thesis are within the field of mathematical quantum mechanics and investigate theoretical problems as well as computational problems for (non)-linear Schrödinger equations: The first two results, presented in the first chapter, are concerned with spectral gaps of classical many-particle systems, the mean-field O(n) model and the chain of oscillators. We study the optimal constant in the Poincar´e inequality of the associated Gibbs measure. For the mean-field O(n)-model we use renormalization ideas and semiclassical methods to relate the problem to the study of an auxiliary Schrödinger operator. For the chain of oscillators we map the Fokker-Planck operator to an associated non self-adjoint eigenvalue problem for an auxiliary discrete Schrödinger operator. In the second part, we study a disordered version of the almost Mathieu operator and the analogous tight-binding operator on the honeycomb lattice. Using semiclassical methods, we analyze the self-similarity in the Hofstadter butterfly and the metal/-insulator transition in both models. In the second chapter of part two, we analyze, using again semiclassical techniques, the squeezing of bands in a model of twisted bilayer graphene and related properties such as the so-called magic angles. In the final part, we study numerical aspects of quantum dynamics. Our first result is concerned with rates of quantum evolution under energy-constraints and continuity estimates on entropies and capacities. The second result of this chapter is on numerical methods and address the open problem of determining which classes of time-dependent linear Schrödinger equations and non-linear Schrödinger equations (NLS) on unbounded domains can be computed by an algorithm.