Structure of Solutions of the Time Dependent Schroedinger Equation and of Certain Classes of Evolution Nonlinear PDEs

本课题提出了在不一定小的周期外场中建立单粒子时变薛定谔方程离散Floquet谱不存在的新方法。这样的结果暗示了包括电离在内的物理现象。这些基于广义Borel求和的方法目前正在由PI和他的合作者开发。它们现在可以应用于现实的量子系统，如外场中随时间变化的氢原子。用广义Borel求和分析的另一组问题是对非线性偏微分方程的存在性、唯一性，特别是奇点的形成的严格研究。我们还将研究这些方法的其他扩展。非线性偏微分方程的时变量子现象和性质在物理、化学以及其他科学和技术中起着关键作用。在这笔赠款下发展的数学方法将为分析和近似这些方程的解提供一些创新的方法。

Structure of Solutions of the time-dependent Schroedinger equation and of certain classes of evolution nonlinear PDEs Abstract of Proposed ResearchOvidiu Costin The project proposes new methods for establishing absence of discrete Floquet spectrum in the time-dependent Schroedinger equation of one particle in periodic external fields, which are not necessarily small. Such results imply physical phenomena including ionization. These methods, based on generalized Borel summation, are currently being developed by the PI and his collaborators. They can now be applied to realistic quantum systems such as the time-dependent Hydrogen atom in external fields. Another set of questions that will be analyzed by generalized Borel summation is the rigorous study of existence, uniqueness and especially formation of singularities in nonlinear partial differential equations. Other extensions of these methods will also be investigated. Time dependent quantum phenomena and properties of nonlinear partial differential equations play a key role in physics, chemistry as well as in other sciences and in technology. The mathematical methods to be developed under this grant will provide some innovative approaches to the analysis and approximation of solutions of these equations.