Proposal: DMS-9970323 Principal Investigator: Laszlo Erdos
Abstract: This research project has two parts. The objective of the first part is to study the ground state properties of large atoms in a strong nonhomogeneous magnetic field. On physical grounds it is expected that the magnetic version of the classical Thomas-Fermi theory correctly describes the atom as the nuclear charge tends to infinity. The following specific problems need to be solved to prove this rigorously: (1) finding the proper Lieb-Thirring type inequality for magnetic fields; (2) describing the zero energy eigenstates of the three dimensional Dirac operator; (3) proving a semiclassical asymptotics result on the sum of the negative eigenvalues of the Pauli operator by constructing appropriate magnetic coherent states. The second part of the project aims at finding kinetic scaling limits of the long time Schrodinger evolution. Random impurities, phonons and electron-electron interactions modify the free evolution of an electron cloud via collisions. The goal is to characterize the large scale effects of these collision mechanisms by proving that the Schrodinger evolution converges to a macroscopic transport evolution described by the Boltzmann equation. The collision kernel needs to be determined separately in each case: it is linear in case of impurity and phonon effects, and it is nonlinear for many-body interactions.
First principles of physics (e.g., the famous Pauli or Schrodinger equations) tell us exactly what mathematical equation governs the behavior of any quantum system. Due to the size and complexity of the real systems, these equations are usually far too complicated to solve with even the most advanced of present-day computers. Approximate problems and models need to be set up and studied that retain the relevant features of the true problem but are simple enough to be computationally feasible. Physics and engineering generally use such simpler models (e.g., the Thomas-Fermi theory in atomic physics or the Boltzmann equation in electron transport theory). Besides the computational and numerical aspects of such models, it is equally important to investigate the extent to which these simplified models are consistent with the real ones; i.e., to determine the regimes of physical parameters in which it is justified to use them. Rigorous mathematical study not only gives a solid theoretical justification of the often ad hoc simplifying procedures, but very often helps to find the correct model as well. This is precisely the objective of the present proposal in the case of two important physical problems: (1) behavior of atoms in strong magnetic fields; (2) electron transport in an impure medium. Both problems have direct applications in semiconductor design. In addition, the first problem is of interest from an astrophysical perspective, for strong magnetic fields occur naturally on the surfaces of neutron stars and are responsible for such unusual physical phenomena as highly nonspherical atoms. This award is jointly funded by the Analysis Program in the Division of Mathematical Sciences and the Mathematical Physics Program in the Division of Physics.
