Ordinarily, quantum phenomena are exhibited on very small micro-scales, while on large macro-scales nature is well described by classical, Newtonian mechanics. One of the principal subjects of investigation in this project is a Bose-Einstein condensate (BEC), for which macroscopic quantum phenomena become apparent. A BEC is a new state of matter that was first predicted theoretically by Bose and Einstein in 1924, and was produced experimentally in 1995 by Cornell and Wieman. Particles of a gas, cooled very close to absolute zero, occupy the lowest quantum state; then the gas forms a BEC. This state of matter has very unusual properties: for example, light in a BEC can be stopped entirely or slowed down significantly to the velocity of 17 meters per second. This project investigates dynamical properties of important quantum systems from a rigorous mathematical viewpoint, by using a wide variety of methods from mathematical analysis, applied mathematics, and mathematical physics. One of the phenomena to be studied theoretically is the emergence of quantum friction, when a particle passes through a BEC. The purpose of this work is to advance the understanding of physical phenomena based on first principles, whereby giving theoretical physics a rigorous mathematical foundation.
The mathematical study of interacting Bose gases is an active research topic at the interface between dispersive nonlinear PDEs and Mathematical Physics. The mean-field description of a BEC is given by the nonlinear Schrödinger or nonlinear Hartree equations. In this project, the dynamics of a quantum mechanical tracer particle in interaction with a BEC will be investigated. Of particular interest in this model is the emergence of quantum friction. Furthermore, the dynamics of thermal fluctuations around a BEC will be examined based on the Hartree-Fock-Bogoliubov equation that is obtained from the quasifree reduction of the original system. Other continuing projects focus on the dynamics of electrons in a weak random potential (describing materials such as semiconductors), and dynamics of a system of infinitely many fermions in the vicinity of a thermal equilibrium. As a new research direction, the well-posedness problem for the Boltzmann equation will be studied in its Wigner-transformed representation. This makes the model accessible to methods developed for the analysis of nonlinear Schrödinger equations and Gross-Pitaevskii hierarchies, such as Strichartz estimates for density matrices. These projects involve collaborations with various leading senior researchers, with a postdoctoral researcher, and with graduate students.
