Proposal: DMS-9801474 Principal Investigator: Yoram Last Abstract: The main aim of this project is the study of fundamental relations between dynamical properties of quantum mechanical systems, spectral properties of the corresponding operators, spatial properties of their eigenfunctions, and various other spectral related and transport related properties. The key model problem to be considered is that of a d-dimensional (time independent) Schroedinger operator and its associated free time evolution. The operators of interest here are mainly those with long-range potentials that do not converge to infinity, such that rich spectral phenomena and dynamical behaviors may arise. The study is expected to be applicable to a more general context of problems involving unitary time evolution. Moreover, the project is expected to provide effective tools for spectral analysis of unitary and self adjoint (in particular, Schroedinger) operators, as well as for the study of dynamical properties of the corresponding quantum systems. The project aims at some fundamental questions of quantum mechanics that are expected to be important to modern physics. The core mathematical questions are also of interest in more general contexts, such as problems arising in classical chaos theory. The project would seek relations between various fundamental quantities that are associated with the systems under consideration. This is expected to provide improved and coherent understanding of such systems as a whole, as well as to provide effective analytical tools for the study of the properties themselves. The need to develop such general understanding arises in connection with various rich mathematical phenomena which until recently where mainly considered as theoretical curiosities. Recent studies show, however, that such phenomena occur in many natural models for physical systems that are now within experimental reach.
