Mathematical Analysis of Dispersion and Transport in Quantum Dynamics

University Of Texas At Austin, Austin TX

Investigators

Abstract

This research addresses problems emerging from the physics of large systems of interacting quantum mechanical particles (of boson or fermion type), analyzed with rigorous mathematical methods. The PI will study the effects of impelling a heavy quantum tracer particle into a gas of bosons, and investigate the dynamical behavior of quantum fluctuations when the phenomenon of Bose-Einstein condensation occurs. In addition, the PI will study the motion of electrons (which are fermions) in disordered media, describing systems such as semiconductors. In another line of research, the PI and his collaborators have recently introduced an approach to study equations describing classical gases and fluids with methods in quantum mechanics, using a mathematical device known as the Wigner transform; this method will be further developed in this project. The projects includes training opportunities for graduate students and postdoctoral researchers. The PI will study the fluctuation dynamics around a Bose-Einstein condensate on the level of the Hartree-Fock-Bogoliubov approximation and beyond, using a combination of methods from nonlinear partial differential equations and renormalization in quantum field theory. Moreover, the PI will study the derivation of effective equations for systems of fermions propagating in a random medium, in the context of the weakly disordered manybody Anderson model with mean field interaction. The analysis of classical Boltzmann equations via Wigner transform and methods of dispersive nonlinear partial differential equations will be further developed. An important outcome of the analysis of these problems is the development of new mathematical techniques merging methods from nonlinear partial differential equations, harmonic analysis, and quantum field theory. 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.