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.
The research supported by this grant has produced a variety of results in the intersecting areas of mathematical physics, analysis, and partial differential equations. It has been the basis for the PhD theses of two graduate students, and to the research work of one postdoctoral researcher mentored by the PI and his collaborator, N. Pavlovic. The PI has also supervised several undergraduate and graduate individual reading courses on the subject matter. Many of these results have been adopted and applied by other researchers, leading to further progress in the field. The PI has presented research talks on those outcomes at national and international conferences, at seminars, and at a summer school. The first main research topic addresses the dynamics of dilute interating Bose gases. This is an important class of quantum mechanical systems for which Bose-Einstein condensation has been experimentally verified in the mid 1990's, leading to the 2001 Nobel Prize in physics. In collaboration with N. Pavlovic, the PI has initiated a research program to investigate the well-posedness of the initial value problem for the so-called Gross-Pitaevskii (GP) hierarchy (extending important previous works of Spohn and Erdoes-Schlein-Yau). This is an infinite system of coupled partial differential equations which emerges in the derivation of the nonlinear Schrodinger (NLS) equation as a mean-field limit. One of the PI's main objectives is to generalize methods from nonlinear dispersive PDE's for the analysis of quantum field theories (QFT's). Among the research results obtained with support of this grant are (all joint with N. Pavlovic): (1) The derivation of the quintic NLS in dimensions 1,2. (2) Proof of energy conservation and blowup of solutions for focusing GP hierarchies(joint with N. Tzirakis). (3) Derivation of higher order energy functionals, and application to the proof ofglobal well-posedness of solutions. (4) Derivation of the cubic defocusing GP hierarchy in dimension 3, in the spaces introduced by Klainerman and Machedon. This work was subsequently extended by the PI together with his PhD student, K. Taliaferro. (5) New proof of unconditional uniqueness using the quantum de Finetti theorem, and the proof of scattering for GP hierarchies (joint with C. Hainzl and R. Seiringer). The second main topic addresses non-relativistic QED, the physical theory of non-relativistic quantum mechanical matter (electrons, atoms, molecules) interacting with the energy quanta (photons) of light. It is based on results from the PI's previous work (in part joint with V. Bach, J. Froehlich, A. Pizzo, and I.M. Sigal), which resolved the infrared mass renormalization problem, and yielded the construction of infraparticle scattering states.The main research results obtained with support of this grant are: (1) Proof of the limiting absorption principle (joint with J. Faupin, J. Froehlich, and I.M. Sigal). (2) Determination of the effective dynamics of an electron in an external potential(joint with V. Bach, J. Faupin, J. Froehlich, and I.M. Sigal). (3) Determination of the hydrogen ground state energy with very high precision(joint with J.-M. Barbaroux, V. Vougalter and S.A. Vugalter); this paper earned theAnnales Poincare Prize 2010. Other works supported by this grant include: (1) Four papers on the analysis of the semi-relativistic Schrodinger-Poisson system, which describes a hot plasma, with focus on its dynamics and stability (joint with W.A. Abou Salem and V. Vougalter). (2) Analysis of an ideal Euler fluid, and derivation of a single-exponential Beale-Kato-Majda estimate that controls the possible singularity formation of solutions (joint with N. Pavlovic). (3) Derivation of the Boltzmann equation as a kinetic scaling limit for the dynamics of a gas of electrons in a random medium with mean field interactions (joint with I. Rodnianski).
