Multiscale Computational Methods for Semiclassical Schrodinger Equations

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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.