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

塔尔博特现象的调和、数论和偏微分方程分析

• 批准号: 0410012
• 负责人: Konstantin Oskolkov
• 金额: $ 13.3万
• 依托单位: University South Carolina Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0410012&HistoricalAwards=false
• 关键词: Harmonic; Number; theoretic; PDE; Analysis

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.

本研究运用调和分析、解析数理论和偏微分方程组等方法对Talbot效应进行了研究。该项目将详细研究具有周期性初始数据的薛定谔方程和亥姆霍兹方程的解的局部和全局性质。这项工作的重点是解决方案的自相似性质，如量子复苏和极低密度的区域。势函数的光滑性在这些性质中所起的作用将被理解。为此，我们将研究具有振荡Wigner矩阵的无穷常微分方程组的解。本课题研究了衍射光栅中Talbot现象的数学理论。这种光学现象存在于具有可变离散比例因子的原始空间周期图像的自我再现(复活)。现在可以理解的是，Talbot的比例因子是二次高斯指数和，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，由于它在创造量子计算机方面的潜力，该现象最近重新引起了物理学家的注意。这项工作的目标是发展塔尔博特现象的数学理论，重点是技术应用。该项目汇集了几个数学分支--调和分析、解析数论和偏微分方程--用于解决量子力学和光学问题。