Dynamical Properties of Quantum Systems with Infinitely Many Degrees of Freedom

University Of California-Davis, Davis CA

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The goal of this research program is to obtain mathematical results concerning the dynamical properties of quantum systems with infinitely many degrees of freedom. More specifically, our interest will be focused on two main areas. 1) Spectroscopy of systems of non-relativistic, quantum mechanical point-like nuclei and electrons interacting with the quantized radiation field. Because of the growing applications of quantum optics, physicists have become increasingly interested in experimental situations where the theoretical explanation must go beyond regular QED (quantum electrodynamics) perturbation theory. This calls for a more refined analysis of the mathematical theory of non-relativistic QED that describes the interaction of charged quantum matter and the quantized radiation at low energy scales related to those phenomena. Our main goal within this project is to improve the mathematical control on the dynamics of metastable states where a non-perturbative analysis is necessary. To this end, we plan to push forward the analysis of spectral, scattering, and expansion methods developed in recent years also with the contribution of the PI. 2) Irreversible processes and transport equations in open quantum systems. Transport theory is intimately related to the study of many-body systems, for which a detailed control of the dynamics is beyond both analytic and numerical methods. Therefore we are forced to deal with effective equations for coarse-grained physical quantities, but one one would like to derive them from the underlying quantum dynamics. This is a mathematically challenging program that links some basic physical concepts to rigorous results expressed in mathematical proofs which hold, at least, in simple models with a non-trivial micro-dynamics. We will extend some first results that we have recently attained regarding quantum diffusion. The concepts and techniques involved in this research project connect different branches of analysis and mathematical physics. Our general strategy is to address mathematical physics problems from different perspectives, because we are confident that solid physical intuition can make links across different mathematical contexts. Conversely we want to gain more insight in the mathematical structures and techniques starting from the solution of physics problems. These problems will provide doctoral research projects for students in mathematics and in physics. The projects represent a valuable training for both students who will pursue academic research in mathematical physics as well as students who will apply tools from analysis and probability to model complex systems in physical applications.