Abstract Harrell 9622730 This project is concerned with the mathematical analysis of several problems in spectral theory of differential operators arising in quantum physics and reaction-diffusion. It is hoped that the results will be of real use to physicists, while remaining in the realm of rigorous and original functional analysis or differential-equations theory. Many of the particular issues have to do with resonances and metastability, or with connections among geometry, operator algebra, and spectra. The first specific topic in the proposal concerns the eigenvalues of Schroedinger operators where the potential energy involves curvature. This sounds abstract, but it actually originates in the analysis of the shapes of phase regions of bistable metallic alloys. The second topic looks at certain purely algebraic identities recently discovered by Harrell and Stubbe, which turn out to apply, sharply, to the partition function (= trace of the heat kernel), among other spectral expressions. The third topic has to do with semiclassical analysis, an area in which Harrell has worked for several years, but new features include magnetic fields and time-dependent interactions. The final topic has to do with local properties of eigenfunctions, such as their nodes, and some inverse problems involving them. Mathematical machinery to be called upon ncludes perturbation theory, asymptotics, and variational analysis. This project is concerned with the mathematical analysis of several problems in physics, especially the physics of atoms, molecules, semiconductors, and metallic alloys. Many of the particular problems have to do with resonances and with metastable (slowly changing) phenomena, and others have to do with the role of geometry in these areas of physics. The goal of the research is to produce something new and useful to working physicists and at the same time to elucidate the mathematical structure which the physical theory relies on. The first specific topic in the proposa l is to make estimates of the "normal modes" of an equation like those in quantum mechanics, but where the potential energy involves curvature. This sounds like an abstract mathematical problem, but it actually originates in the analysis of metallic regions with two different stable phases, when one tries to find the shapes of the regions with one or other phase. The second topic looks at certain purely algebraic identities recently discovered by Harrell and Stubbe, which turn out to apply to one of the central functions of statistical mechanics. The third topic has to do with the differences between the theory of quantum mechanics and the theory of classical physics, which applies to our world on a larger scale. Harrell has worked in this area for several years; new innovations include incorporating magnetism. The final topic has to do with some poorly understood local properties of quantum mechanical wavefunctions, and how measuring them might establish what sorts of forces are present. The methods used come from both pure and applied mathematics, and physics.
