U.S.-Brazil Cooperative Research on the Theory of Disordered Systems (Applied Mathematics)

该奖项为加州大学欧文分校的Abel Klein教授与巴西圣保罗大学的J.Fernando Perez教授参与的合作研究项目提供支持。这项研究将涉及我们对随机薛定谔算符知识的扩展，这是一个与物质的量子力学描述相关的主题，特别是与无序系统中电子的运动有关的主题。这个项目建立在Klein教授的研究基础上，该研究目前由NSF数学科学部提供支持。首先，关于量子力学粒子在非理想晶格上运动的Anderson模型将扩展到二维和三维。这样的扩展将有助于描述量子力学系统的导电性和光学性质。随后，这种运动的Wegner模型--安德森模型的替代模型，也可以描述扩展的状态--将被用来研究无序系统中这种状态的性质。Klein教授和Perez教授之间的合作已经持续了大约六年，导致了数学物理领域的许多联合出版物，现在提出的研究应该会使所有试图了解非常小系统中的传导现象的物理学家和工程师受益。

This award provides support for the participation of Professor Abel Klein of the University of California, Irvine in a program of cooperative research with Professor J. Fernando Perez of the University of Sao Paulo, Brazil. The research will deal with extensions to our knowledge of random Schrodinger operators, a topic of relevance to the quantum-mechanical description of matter, and in particular to the motion of electrons in disordered systems. This project builds on the research of Prof. Klein currently supported by the Division of Mathematical Sciences of NSF. At first the Anderson model for the motion of quantum- mechanical particles on imperfect lattices will be extended to two and three dimensions. Such an extension would facilitate the description of both conduction and optical properties of quantum-mechanical systems. Subsequently, the Wegner model for such motion--an alternative to Anderson's, which can also describe extended states--will be used to investigate the properties of such states in disordered systems. The collaboration between Professors Klein and Perez, which is already of some six years' duration, has led to numerous joint publications in the field of mathematical physics, and the research now proposed should benefit all physicists and engineers who are seeking to understand the phenomena of conduction in very small systems.