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

本研究运用调和分析、解析数论和偏微分方程的方法来研究塔尔博特效应。该项目将提供一个详细的研究局部和全局性质的解薛定谔和亥姆霍兹方程与周期初始数据。这项工作的重点是解决方案的自相似性质，如量子恢复和极低密度的域。势函数的平滑性在这些性质中所起的作用将被理解。为此目的，将研究具有振荡维格纳矩阵的无穷常微分方程组的解。本课题研究了衍射光栅中塔尔博特现象的数学理论。这种光学现象包括原始空间周期图像的可变离散比例因子的自我复制（复兴）。现在我们知道，塔尔博特的标度因子是二次高斯指数和，是解析数论的经典对象。塔尔博特现象也是量子力学薛定谔方程解的一个典型特征，由于它在量子计算机的创造中具有潜力，这种现象最近又重新引起了物理学家的注意。这项工作的目的是发展塔尔博特现象的数学理论，重点是技术应用。该项目汇集了数学的几个分支-调和分析，解析数论和偏微分方程-在量子力学和光学问题的解决方案。

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.