| This dissertation is an examination of various aspects of the Schrödinger evolution and associated Hamiltonians. The text is divided into two parts.; In Part 1, the focus is on random operators which arise in models of the motion of a particle or quasi-particle in condensed matter physics. The randomness may be inherent, as in a disordered alloy, or may represent an effective, but unknown, interaction with other particles. In Chapter 2 a mathematical analysis of Anderson localization is presented. The phenomenon of band edge localization provides a motivation for Chapter 3 in which a novel method for constructing Hamiltonians with spectral gaps is presented.; In Part 2, the subject is Hamiltonians which vary slowly with time, and the focus is on understanding the “adiabatic limit” in which the time scale over which the Hamiltonian changes becomes arbitrarily large. In chapter 4, a new perspective on the adiabatic limit is proposed, in which the strong operator topology is shown to provide a natural notion of convergence for an adiabatic theorem without a gap condition. The final chapter, Chapter 5, is devoted to the study of adiabatic transport in 2D. Results from Chapter 2 are used here used along with methods from adiabatic theory to provide a rigorous derivation for a formula of relevance to the theory of quantized Hall conductance. |
