Harmonic Analysis and Problems in Mathematical Physics

中威沈。PI将研究量子力学和流体动力学领域中出现的几个问题。对于具有电磁势的Schr\ odinger算子，将对负特征值的估计和计数函数的非经典渐近进行研究。目标是在经典的Cwikel-Lieb-Rosenblum边界失效的情况下建立有用的估计。我们也会做基态能量的半经典分析。重点将放在简并势上。第二项研究涉及泡利算子，它描述带自旋的带电粒子的运动。PI将研究具有非均匀磁场的泡利算子的Lieb-Thirring型不等式。这些不等式在物质稳定性研究和磁场半经典分析中起着重要的作用。最后将继续研究粗糙边界域的边值问题，这是许多工程应用中自然出现的问题。特别地，我们将研究Stokes算子在Lipschitz域中的解析估计和分数幂。目的是为了更好地描述非线性Navier-Stokes方程在非光滑域上的解的行为。本课题是数学物理、谐波分析和偏微分方程的交叉课题。它的目标是更好地理解量子系统的光谱特性，并改进基于数值近似和应用的数学理论，以模拟流体流动的Navier-Stokes方程。

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.