Date of Award
Spring 2024
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
First Advisor
Schlag, Wilhelm
Abstract
This thesis consists of four parts, each studying a different class of Schrödinger equations. Though the setting and relevant questions vary between chapters, a theme that runs through all of them is the study of a Hamiltonian with a potential that is not small enough to be treated perturbatively. Indeed, in all of the Schrödinger equations below, the potential is, in some sense, large enough that it may affect the asymptotic behavior of the system.In Chapters 2 and 3, we investigate the scattering theory of Schrödinger operators with potentials that have anisotropic decay. More specifically, we assume that the potential in question decays in a short-range way, but only along a collection of rays in R^d. This generalizes the classical setting of short-range scattering in which the potential decays along all rays to infinity. For these operators, we give a dynamical characterization of the scattering states and a corresponding description of their complement. Heuristically, we show that any state decomposes into an asymptotically free piece and a piece that may interact with the potential for long times. Chapter 3 considers the general setting of anisotropic short-range potentials, whereas Chapter 2 focuses on potentials that are short-range outside of a subspace of R^d. In this latter case and others, we show a more refined description of the scattering states and their complement, which we call the surface subspace. The results of Chapters 2 and 3 are agnostic to the structure of the potential, so in Chapter 4 we specialize to potentials that are periodic in some coordinate directions and compactly supported in the others. For such potentials, we show that a dense set of states in the surface subspace exhibit directional ballistic transport. This means that they evolve ballistically in the directions in which the potential is periodic while being confined in the directions in which it decays. In Chapter 5, we study a Hamiltonian with a repulsive Coulomb potential on R^3. We show that radial solutions of the corresponding Schrödinger equation obey an L1 → L∞ estimate with the natural 3/2 decay rate. To accomplish this, we compute the kernel of the evolution via a distorted Fourier transform and then perform a detailed analysis of its asymptotics. This thesis contains joint work with Tal Malinovitch, David Damanik, Giorgio Young, Ebru Toprak, Jiahua Zou, and Bruno Vergara.
Recommended Citation
Black, Adam, "On Schrödinger Equations With Non-Perturbative Potentials" (2024). Yale Graduate School of Arts and Sciences Dissertations. 1453.
https://elischolar.library.yale.edu/gsas_dissertations/1453
