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

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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.