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

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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.