Problems in Spectral Theory and Analysis

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The primary goal of this project is to develop analytic tools to study Schrodinger operators, the main object of quantum mechanics. The theory describes how a physical system, say a bunch of particles subject to certain forces, will change over time. This is important to predict the behavior of the quantum particles, such as electrons, atoms and molecules. The principal investigator will develop some fundamental methods to understand the spectra and quantum dynamical behavior of Schrodinger operators. In particular, the project focuses on the conductance properties and transport of quasi-periodic media. The results not only provide a good explanation for some phenomena in physics and chemistry, but may also have fruitful applications to modern engineering devices such as semiconductors. Undergraduate and graduate students will have opportunities to participate in some of the research.This project aims at studying several aspects in mathematics by methods of functional, harmonic and geometric analysis. Quasi-periodic Schrodinger operators describe the conductivity of electrons in a two-dimensional crystal layer subject to an external magnetic field of flux acting perpendicular to the lattice plane. The principal investigator will investigate the spectral theory of quasi-periodic operators, including spectral transitions, structure of eigenfunctions in the pure point spectrum regime, and quantum dynamics of spectral measures in the singular continuous spectrum regime. The principal investigator will also study the spectral theory of Laplacians on noncompact complete Riemannian manifolds. The focus is on the existence of eigenvalues or singular continuous spectra embedded into essential spectra of Laplacians on asymptotically flat and on asymptotically hyperbolic manifolds, as characterized by the radial curvature. The goal is to better understand relations between geometric quantities and properties of eigensolutions. Lastly the principal investigator plan to study the independence of the time-frequency translates of various classes of functions, which is stated as HRT conjecture. The priority is to prove the HRT conjecture for special configurations and HRT conjecture for exponentially decaying functions.