Qualitative Properties of Eigenfunctions for some Selfadjoint and Non-selfadjoint partial differential equations

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

This project considers eigenfunctions, which are objects that give information about, for example, the vibrations of a ringing drum head. In other contexts, these eigenfunctions can describe, for example, how fluids mix on extremely small length scales, with potential applications to improved efficiency in medical drug delivery and dialysis. These eigenfunctions are fundamental building blocks to understand problems in mathematics, physics, chemistry, and even biomedical engineering. It is precisely these important fundamental connections between the proposed research and STEM subjects that makes this research have broader significance within the larger context of the STEM areas. The PI has been developing simple, instructional model problems related to most of his research for students and younger researchers. This project concerns research in the deep relationships between solutions to partial differential equations, differential geometry, dynamical systems, and mathematical physics. These different areas of mathematics are often tied together by problems in spectral theory and microlocal analysis; that is, problems concerned with eigenvalues, eigenfunctions, phase space localization, and the generalizations thereof. The research in this proposal is divided between selfadjoint and non-selfadjoint eigenfunction problems. The study of the behaviour of solutions to partial differential equations has a rich connection to the underlying geometry and classical phase space dynamics. For example, it is well known that eigenfunctions tend to concentrate along geodesics. If there are isolated periodic geodesics, one might expect a subsequence of eigenfunctions to concentrate along such geodesics. Understanding the rate of concentration is an extremely important question in quantum chaos. On the other hand, if the geodesic flow is sufficiently chaotic, one might expect the eigenfunctions to be equidistributed in phase space. It is important to investigate how robust these phenomena are, through phase space estimates, restriction estimates, and perturbations. For example, in the chaotic case, the PI and his collaborators are working to understand restrictions of eigenfunctions to hypersurfaces. With mild geometric assumptions on a hypersurface, they conjecture the mass of such restrictions is bounded above and below, independent of the eigenvalue. The methods of investigation will introduce new microlocal energy techniques, demonstrating how to generalize a problem, previously only understood on arithmetic surfaces using number theory, to very general geometric situations. Some properties of eigenfunctions tend to be stable under small complex perturbations, which means one can also understand some non-selfadjoint problems. For non-selfadjoint problems with larger imaginary component, perturbation techniques are no longer strictly valid. The PI is working to develop a general theory of geometric control adapted to degenerate advection-diffusion type equations of any order. This has straightforward applications to the Fokker-Planck equation and similar equations from statistical mechanics. This theory also has applications to certain models in theoretic micro-fluidics, with potential applications to efficient geometrically localized drug delivery and dialysis.