Mathematical Sciences: Mathematical Studies in Quantum Mechanics

9403401哈格多恩题目：量子力学中的数学研究派计划研究数学量子力学中的两类问题。第一类问题涉及研究周期薛定谔算子，这些算子被空间中的位势扰动。这些微扰在纯晶体的电子能谱的带隙中产生离散本征值。PI计划开发数值算法来计算这些杂质能级，并研究它们的性质。他计划将关于这类系统的已知结果从一维扩展到n维。需要研究的特性包括本征值捕获、级联和阈值行为。第二类问题涉及分子系统的含时薛定谔方程的研究。需要研究的一些问题包括：1)建立一个模型来描述避免电子能级交叉引起的非绝热效应。2)分子与电磁辐射的相互作用。3)高阶Born-Oppenheimer修正项的计算。4)碰撞参数模型在分子散射中的有效性和高能行为研究。PI计划研究量子力学中的几个数学问题。这个项目的一部分涉及对一个模型的研究，该模型描述了含有杂质的晶体中电子的能级。这些水平决定了某些晶体的颜色，如红宝石和蓝宝石。该项目的其余部分涉及在分子水平上研究化学反应动力学。

9403401 Hagedorn Title: Mathematical Studies in Quantum Mechanics The PI plans to study two types of problems in mathematical quantum mechanics. The first type of problem involves studying periodic Schroedinger operators that are perturbed by potentials that are localized in space. These perturbations give rise to discrete eigenvalues in the gaps of the pure crystal's electron energy spectrum. The PI plans to develop numerical algorithms for the calculation of these impurity levels and to study their properties. He plans to extend known results about such systems from one dimension to n dimensions. Among the properties to be studied are eigenvalue trapping, cascading, and threshold behavior. The second type of problem involves study of the time-dependent Schroedinger equation for molecular systems. Some of the problems to be studied are: 1) Development of a model to describe non-adiabatic effects caused by avoided crossings of electron energy levels. 2) Interactions of molecules with electromagnetic radiation. 3) Computation of high order Born-Oppenheimer correction terms. 4) Study of the validity and high energy behavior of the impact parameter model in molecular scattering. The PI plans to study several mathematical problems in quantum mechanics. Part of this project involves the study of a model that describes the energy levels of electrons in crystals that have impurities. These levels determine the colors of some crystals, such as rubies and sapphires. The rest of the project involves the study of chemical reaction dynamics at the molecular level.