Spectral Theory and Dynamics of Ergodic Schrodinger Operators

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The goal of this research project is to develop dynamical techniques for the spectral analysis of the ergodic Schroedinger operators, which arise in modeling the motion of quantum particles in certain disordered media. A large part of the theory of dynamical systems deals with long-term behaviors of typical trajectories in certain mathematical or physical systems, such as hyperbolic systems or Hamiltonian systems. A key task of spectral analysis of the ergodic Schroedinger operators is to study the asymptotic behaviors of the solutions of the associated eigenvalue equations. Bridges between two different areas can then be built since ``asymptotic behaviors of solutions'' may be interpreted as ``long-term behaviors of certain systems''. The goal of this project is to develop dynamical techniques that are driven by building such bridges and that may benefit both areas.Different disordered media lead to different type of ergodic base systems. The famous and intensively studied Anderson model corresponds to i.i.d. random variables which can be generated by full shift. Two types of base systems with which this project is concerned are quasi-periodic systems, typical almost periodic systems, and hyperbolic systems, classic type of strongly mixing systems. Various levels of randomness may be detected by a dynamical object, the Lyapunov exponent, which is the main object of study of this project. One focus of this project is the study of positivity and large deviation estimates of the Lyapunov exponent. These properties are super sensitive to the randomness of the base dynamics, are generally difficult to obtain, and are thus among central topics in dynamics systems. From the side of spectral theory, they are strong indications of the Anderson Localization phenomenon and imply immediately certain regularity of both the Lyapunov exponent and the integrated density of states. Deep investigation of the two properties for both quasi-periodic and hyperbolic base dynamics may shed light on how to obtain positive Lyapunov exponent of the standard map. This is one of the most notorious unsolved problems in dynamical systems where the difficulty lies exactly in the complicated coexistence of both elliptic and hyperbolic behaviors. Another fundamental relation between dynamical systems and spectral theory is the Cantor Spectrum phenomenon. The most famous physical example regarding this phenomenon is the Hofstadter's butterfly. In dynamical systems, Cantor spectrum phenomenon may be viewed as some kind of ubiquity of uniformly hyperbolic systems. Another focus of this project is then to investigate Cantor spectrum.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.