Quantum and Classical Evolution.

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Proposal DMS-0400526 PI: Israel-Michael Sigal, University of Notre Dame Project title: Quantum and Classical Evolution Project abstract: In this project the PI addresses problems of long-time and asymptotic evolution in quantum and classical field theories. More specifically, he aims to (A) Determine the resonance structure of and construct the scattering theory for systems of non-relativistic particles such as atoms and molecules interacting with the quantized electromagnetic field; (B) Construct scattering theory and the theory of periodic/quasiperiodic motion of solitons (or solitaty wave solutions) for the nonlinear Schroedinger equation with an external potential; (C) Construct theory of tunneling for solitons in the nonlinear Schroedinger equation with an external potential and in the (second) quantized version of this equation. Mathematically, these problems deal with linear partial differential equations in an infinite number of variables (A) and with nonlinear partial differential equations with interplay between the nonlinearity and geometry of the external potentials (B) as well as with the problems bridging both frameworks (C). The problems above originate in our trying to describe some basic physical phenomena such as emission and absorption of radiation by quantum, non-relativistic systems (e.g., atoms and molecules) and dynamics of Bose-Einstein condensates. They also play an important role in technological applications, e.g. optical communication. It is hoped that the results the research to be undertaken will provide conceptual and unifying underpinning to a more quantative (phenomenological, numerical and experimental) physical treatment. In selecting the research projects above the PI was guided by their importance, possibility of the emergence of new mathematics and by the availability of tools and ideas to start probing them. These projects will also provide fertile ground for excellent entry problems for graduate students, postdocs and beginning faculty whom the PI will supervise and mentor.