CAREER: Solutions, Spectrum and Dynamics of Schrodinger Oerators

这项研究将主要集中在两个方向上。第一个目标是进一步研究和应用与薛定谔方程、谱和动力学有关的解的行为的准则。特别是这些应用可能对某些类型的随机和准周期算符的动力学和谱有新的认识，并提供对量子输运现象的更好的一般理解。第二个目标是利用调和分析工具继续研究一维薛定谔算子的改进的WKB方法。WKB方法是量子力学的经典部分，其主要目的是在一定的自然假设下找到系统波函数的良好近似。这个项目将致力于开发新的WKB类型的技术，这些技术可以用来获得一类薛定谔算子的动力学信息。还将寻求更高维度的扩展。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规律，如电子、原子和分子。关于许多重要物理过程(例如，各种材料的化学反应或导电性质)的许多基础科学知识都来自薛定谔算符理论。本项目致力于发展薛定谔算符的谱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电导性质，并直接应用于现代工程设备、波导和晶体管等。该项目还将为生物和化学专业的本科生开发现代数学方法课程。鉴于数学和这些科学之间迅速发展的接口，需要将一套新的思想和材料纳入到这些服务课程中。此外，将开设研究生课程，主要目的是帮助刚毕业的学生过渡到独立研究。

ABSTRACTPI: Alexander KiselevProposal: DMS-0129470Institution: University of ChicagoThe research will focus mainly on two directions.The first objective is to further study and apply to concretequantum systems of interest the criteria relating behaviorof solutions of Schroedinger equation, spectrum and dynamics.In particular, the applications may shed new light on the dynamics and spectrum of certain classes of random and quasiperiodic operators, such as Fibonacci Hamiltonian, as well as provide a better general understanding of the quantum transport phenomena. The second objective is to continue the study of advanced WKB methods for one-dimensional Schroedinger operators, using the tools of harmonic analysis. WKB methods are a classical part of quantum mechanics, and their main goal is to find a good approximation for the wave function of the system under certain natural assumptions. This project will pursue the development of new WKB-type techniques, which can be applied to obtain information about dynamics of a class of Schroedinger operators. Extensions to higher dimensions will also be sought. The spectral and dynamical theory of Schroedinger operators is the cornerstone of Quantum Mechanics. This theory describes the laws which govern behavior of the quantum particles, such as electrons, atoms and molecules. Much of the fundamentalscientific 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 project focuses on the development of 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 conductance properties of materials with impurities and of quasicrystals, and have direct applications to modern engineering devices, wave guides and transistors to name two. The project will also involve development of a modernmathematical methods curriculum for undergraduate students majoring in biology and chemistry. Given rapidly evolving interface between mathematics and these sciences, a new set of ideas and material needs to be incorporated into such service courses. In addition, graduate courses will be developed with the main objective to help beginning graduatestudents make a transition to independent research.