Mathematical Sciences: Mathematical Studies in Quantum Mechanics

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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.