Harmonic, Number-theoretic, and PDE Analysis of Talbot's Phenomenon

点击翻译按钮获取中文摘要

This research applies methods of harmonic analysis, analytic number theory, and partial differential equations to the study of Talbot's effect. The project will furnish a detailed study of local and global properties of solutions to Schrodinger and Helmholtz equations with periodic initial data. The work focuses on self-similarity properties of solutions, such as quantum revivals and domains of extremely low density. The role played in these properties by the smoothness of the potential function will be understood. For this purpose, solutions of infinite systems of ordinary differential equations with oscillatory Wigner matrices will be studied.This project investigates the mathematical theory of Talbot's phenomenon in diffraction gratings. This optical phenomenon consists in self-reproduction (revival), with variable discrete scaling factors, of an original space-periodic image. It is now understood that Talbot's scaling factors are quadratic Gauss exponential sums, classical objects of analytic number theory. Talbot's phenomenon is also a typical feature of the solutions of the Schrodinger equation of quantum mechanics, and the phenomenon recently returned to the attention of physicists because of its potential in the creation of quantum computers. The objective of this work is to develop the mathematical theory of Talbot's phenomenon, with a focus on technological applications. The project brings together several branches of mathematics - harmonic analysis, analytic number theory, and partial differential equations - in the solution of problems of quantum mechanics and optics.