Dynamics of complex quantum systems with randomness and nonlinearities

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This project addresses research problems at the boundary of analysis, applied mathematics, and mathematical physics. In a first part of the project, we study properties of a single electron in a random medium (weakly disordered Anderson model) in the framework of nonrelativistic Quantum Electrodynamics (QED). In a second part, we address the dynamics of a gas of electrons in a weak random potential (describing materials such as semiconductors), where the interactions between the electrons are modeled in dynamical Hartree-Fock theory. In a third part of the project, we investigate the Cauchy problem for the Gross-Pitaevskii hierarchy, which is a many body mean-field theory describing a gas of interacting Bose particles. We study dynamical properties of solutions of general form, and compare them to dynamical properties as predicted by the nonlinear Schrodinger equation obtained for solutions of factorized type.Non-relativistic quantum electrodynamics describes electrons, atoms, and molecules moving at ordinary speeds and interacting with the energy quanta of light (photons). It is the fundamental theory for the description of processes in molecular physics and quantum chemistry. In this research project, we use the framework of non-relativistic quantum electrodynamics to study the motion of an electron in a random medium (e.g., materials including impurities) when exposed to light or lattice vibrations. Moreover, we study the effect of the interaction between electrons on the dynamics of an electron gas in a random medium. The analysis of these questions is crucial for the understanding of electric properties of semiconductors. Another main part of this project investigates the dynamics of systems of many Bose particles, which play a role in the phenomenon of superfluidity. In this work, we investigate the question of how well the non-relativistic quantum electrodynamic description of a Bose gas matches predictions of other models.