Zhongwei Shen. DMS-9732894 The PI will study several problems which arise in the fields of quantum mechanics and fluid dynamics. For Schr""odinger operators with electro-magnetic potentials, work will be done on the estimates of negative eigenvalues and the non-classical asymptotics of the counting function. The objective is to establish useful estimates in the cases when the classical Cwikel-Lieb-Rosenblum bound fails. Work will also be done on the semi-classical analysis of the ground state energy. The emphasis will be on the degenerate potentials. The second line of research concerns the Pauli operator, which describes the motion of charged particles with spin. The PI will study the Lieb-Thirring type inequalities for the Pauli operator with a non-homogeneous magnetic field. Such inequalities play an important role in the study of stability of matter and semi-classical analysis in a magnetic field. Finally work will be continued on boundary value problems in domains with rough boundaries, which arise naturally in many engineering applications. In particular, the resolvent estimates and fractional powers of the Stokes operator in Lipschitz domains will be studied. The objective is to obtain a better description of the behavior of the solutions to the nonlinear Navier-Stokes equations in nonsmooth domains. This project lies at the interface of mathematical physics, harmonic analysis and partial differential equations. Its goal is to gain better understanding of the spectral properties of quantum systems, and to improve the mathematical theory, upon which the numerical approximations and applications are based, for the Navier-Stokes equations which model the fluid flow.
