PI: Alexander Kiselev Proposal: DMS-0129470 Institution: University of Chicago
The research will focus mainly on two directions. The first objective is to further study and apply to concrete quantum systems of interest the criteria relating behavior of solutions of Schroedinger equation, spectrum and dynamics. In particular, the applications may shed new light on the dynamics and spectrum of certain classes of random and quasiperiodic operators, such as Fibonacci Hamiltonian, as well as provide a better general understanding of the quantum transport phenomena. The second objective is to continue the study of advanced WKB methods for one-dimensional Schroedinger operators, using the tools of harmonic analysis. WKB methods are a classical part of quantum mechanics, and their main goal is to find a good approximation for the wave function of the system under certain natural assumptions. This project will pursue the development of new WKB-type techniques, which can be applied to obtain information about dynamics of a class of Schroedinger operators. Extensions to higher dimensions will also be sought.
The spectral and dynamical theory of Schroedinger operators is the cornerstone of Quantum Mechanics. This theory describes the laws which govern behavior of the quantum particles, such as electrons, atoms and molecules. Much of the fundamental scientific knowledge about many important physical processes (such as, for example, chemical reactions or conduction properties of various materials) comes from the theory of Schroedinger operators. This project focuses on the development of new methods in spectral and dynamical theory of Schroedinger operators which may allow a new approach to some long-standing problems in Quantum Mechanics. These problems concern, in particular, the conductance properties of materials with impurities and of quasicrystals, and have direct applications to modern engineering devices, wave guides and transistors to name two. The project will also involve development of a modern mathematical methods curriculum for undergraduate students majoring in biology and chemistry. Given rapidly evolving interface between mathematics and these sciences, a new set of ideas and material needs to be incorporated into such service courses. In addition, graduate courses will be developed with the main objective to help beginning graduate students make a transition to independent research.
