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

在这个项目中，泛函分析将用于理解系统的形状如何与能级相联系，无论是作为线性方程的本征值出现，如描述量子力学粒子的薛定谔方程，还是从非线性方程中出现，如描述半导体中载流子输运的方程。 小参数经常存在，无论是普朗克常数还是物理尺寸，为了阐明区域几何的作用，需要仔细的渐近性。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属性。这种严格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规模的电子设备，包括量子线和量子波导。 这些设备的属性，如电导率和能级，对它们的形状和配置很敏感。 该项目将使这些影响定量化，并有助于指导设备的设计。 这项工作也对类似方程描述的其他各种各样的现象产生了影响。 这些包括实际应用的渗流的流体和稳定性的散装物质，以及纯粹的数学应用几何。

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.