Solutions and Spectrum of Schrodinger Operators

建议：DMS-9801530首席研究员：Alexander Kiselev摘要：研究将主要集中在两个目标上。第一个是研究任意维无限区域上广义本征函数方程的解的性态、薛定谔算子(以及更一般的椭圆算子)的谱性质和量子动力学之间的一般关系。第二个目标是研究具体量子力学系统的光谱和动力学性质。我们的目标是发展一种新的谱分析技术来研究薛定谔算子本质谱的精细结构(即将本质谱分解成绝对连续的、奇异的连续的和纯点分量)。除了光谱信息，这些新的方法还可以通过广义本征函数方程的解的行为、谱测量和量子动力学之间的新关系来产生关于各种系统的量子动力学的有价值的结果。这项技术可能会在许多重要问题上有潜在的应用，并将提供进一步的洞察，特别是准晶模型，安德森模型和其他涉及随机势或遍历势的模型，以及具有缓慢衰减势的薛定谔算符。薛定谔算符的谱和动力学理论是量子力学的基石。这一理论描述了控制量子粒子行为的定律，如电子、原子和分子。许多重要物理过程的基本科学知识(例如各种材料的化学反应或导电性质)都来自薛定谔算符理论。这一建议集中于发展薛定谔算符的谱理论和动力学理论的新方法，这可能为量子力学中的一些长期存在的问题提供一种新的途径。这些问题特别涉及含杂质材料和准晶材料的导电性质，并直接应用于现代工程器件、波导和晶体管等。

Proposal: DMS-9801530 Principal Investigator: Alexander Kiselev Abstract: The research will focus mainly on two objectives. The first is to study the general relationships between the behavior of solutions of the generalized eigenfunction equation, spectral properties and quantum dynamics of Schroedinger (and more generally elliptic) operators on infinite domains in any dimension. The second objective is the investigation of spectral and dynamical properties of concrete quantum mechanical systems. The goal is the development of a new spectral analysis technique for studying fine structure in the essential spectra of Schroedinger operators (that is, the decomposition of the essential spectrum into absolutely continuous, singular continuous and pure point components). In addition to spectral information, the new methods may yield valuable results about quantum dynamics of various systems via new relations between the behavior of solutions of generalized eigenfunction equation, spectral measures, and quantum dynamics. This technique may have potential applications to and will provide further insight into many important problems, in particular such as models of quasicrystals, the Anderson model and other models involving random or ergodic potentials, and Schroedinger operators with slowly decaying potentials. The spectral and dynamical theory of Schroedinger operators is the cornerstone of Quantum Mechanics. This theory describes the laws which govern the behavior of quantum particles, such as electrons, atoms and molecules. Much of the fundamental scientific knowledge about many important physical processes (such as, for example, chemical reactions or conduction properties of various materials) comes from the theory of Schroedinger operators. This proposal focuses on the development of the new methods in spectral and dynamical theory of Schroedinger operators which may allow a new approach to some long-standing problems in Quantum Mechanics. These problems concern, in particular, the cond uctance properties of materials with impurities and of quasicrystals, and have direct applications to modern engineering devices, wave guides and transistors to name two.