Spectra, Geometry, and Asymptotics of Some Differential Equations of Mathematical Physics

数学物理中一些微分方程的谱、几何和渐近

• 批准号: 0204059
• 负责人: Evans Harrell
• 金额: $ 7.5万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204059&HistoricalAwards=false
• 关键词: Spectra; Geometry; Asymptotics; Some; Differential

In this project functional analysis will be used to understand how the shapes of systems are connected with energy levels, whether arising as eigenvalues of linear equations, such as the Schroedinger equation which describes quantum mechanical particles, or arising from nonlinear equations like those describing carrier transport in semiconductors. Small parameters are often present, whether Planck's constant or a physical dimension, and careful asymptotics are required in order to elucidate the role of the geometry of a domain. It is intended to seek useful estimates of energies and other physical quantities in terms of curvature, and to determine the shapes and geometric properties that optimize those quantities.This rigorous mathematical research will contribute to nanotechnology and quantum mechanics. Laboratories are beginning to produce very small-scale electrical devices, including quantum wires and quantum waveguides. The properties of these devices, such as conductivity and energy levels, are sensitive to their shape and configuration. This project will make these effects quantitative and help guide the design of devices. The work also has implications for other quite diverse phenomena described by similar equations. These include practical applications to the seepage of fluids and the stability of bulk matter, as well as purely mathematical applications to geometry.

在这个项目中，功能分析将被用来理解系统的形状是如何与能级联系在一起的，无论是作为线性方程的特征值产生的，比如描述量子力学粒子的薛定谔方程，还是从描述半导体载流子输运的非线性方程产生的。无论是普朗克常数还是物理维度，经常存在小参数，并且为了阐明域的几何形状的作用，需要仔细的渐近。它的目的是寻求能量和其他物理量在曲率方面的有用估计，并确定优化这些量的形状和几何性质。这项严谨的数学研究将有助于纳米技术和量子力学。实验室开始生产非常小的电子设备，包括量子线和量子波导。这些器件的特性，如电导率和能级，对它们的形状和结构很敏感。本项目将使这些影响量化，并有助于指导器件的设计。这项工作对其他由类似方程描述的相当不同的现象也有启示。这包括流体的渗流和大块物质的稳定性的实际应用，以及几何学的纯数学应用。