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

含时薛定谔方程的解的结构和某些类发展的非线性偏微分方程解的结构研究该项目提出了一种新的方法来建立周期外场中单个粒子的含时薛定谔方程中离散Floquet谱的缺失，这些谱不一定很小。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及其合作者开发。它们现在可以应用到现实的量子系统中，比如外场中的含时氢原子。另一组将由广义Borel和分析的问题是对非线性偏微分方程奇点的存在唯一性特别是形成的严格研究。这些方法的其他扩展也将被调查。含时量子现象和非线性偏微分方程组的性质在物理、化学以及其他科学和技术中起着关键作用。根据这笔赠款将开发的数学方法将为这些方程的解的分析和近似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.