Multiscale Computational Methods for Semiclassical Schrodinger Equations

作者建议发展有效的数值方法来解决量子动力学中的几个问题。具体而言，我们选择了以下三个主题：量子散射的半经典方法、表面跳跃方法和基于Bloch分解的周期晶格量子动力学计算方法。这些都是具有挑战性的计算问题，涉及高频波和量子-经典耦合。数学和计算方法在提高我们对这些问题的理解和模拟能力方面发挥着重要的作用。这些问题出现在固体物理、半导体器件建模、玻色-艾斯坦凝聚(BEC)、太阳能和功能材料如石墨烷等领域，因此发展起来的计算方法将有广泛的应用，其中一些具有重大的国家利益。部分研究成果将为未来应用数学和科学计算的研究生课程提供优秀的补充，从而有助于改进应用数学的研究生课程，以便更好地培养未来的现代应用数学研究生。

The proposer proposes to develop efficient numerical methodsfor several problems in quantum dynamics. Specifically, the followingthree topics are selected: Semiclassical methods for quantumscattering, surface hopping, and Bloch-decomposition based computationalmethods for quantum dynamics in periodic lattice. These are challengingcomputational issues that involve high frequency waves andquantum-classical coupling. Mathematical and computationalmethods can play important roles to enhance our understanding as wellas our ability to simulate these problems.These problems arise in solid state physics, semiconductor devicemodeling, Bose-Eistein Condensations (BEC), solar energy, andfunctional materials such as graphane, thus the developed computationalmethods will have a wide range of applications, some of which ofsignificant national interests. Some of the research results willprovide excellent additions for future graduate courses in appliedmathematics and scientific computing, thus will help toimprove the graduate curriculum in applied mathematics in order tobetter train future graduate students in modern applied mathematics.