New analytic techniques in differential and difference systems

许多分析人员正在开发基于广义Borel求和和反渐近展开的新方法，并正在形成一个新的研究领域，即“可分析性理论”。这些方法使人们能够比通常处理的更一般地处理奇点处的方程的解，并为以前的形式计算提供了严格的基础。本提案旨在开发和应用这些新技术。pi将继续在偏微分方程、差分方程和偏微分方程的背景下进行研究。应用包括在不规则奇点处的非线性ode的分类，确定它们的可积性以及它们对更简单系统的可约性，绝热不变性的研究，以及差分方程的painlevel分析。这些技术将用于在任意强度的周期性扰动下继续严格地研究时变薛定谔问题和共振理论。该项目是一项正在展开的事业的一部分，旨在弥合从正式计算到严格证明的差距，并推进似乎正在出现的单一行为的一般理论。数不胜数的应用存在于数学和物理现象的研究中。

New methods based on generalized Borel summation and trans-asymptoticexpansions are being developed by a number of analysts and are emergingas a new field of study, "analyzability theory". The methods enable oneto treat solutions of equations at singularities much more general thenusually dealt with and provide a rigorous foundation to previouslyformal calculations. The present proposal aims at developing andapplying these new techniques. The PIs will continue their research inthe context of ODEs, difference equations and PDEs. Applicationsaddressed include the classification of nonlinear ODEs at irregularsingularities, determining their integrability properties as well astheir reducibility to simpler systems, study of adiabatic invariance,and Painleve' analysis for difference equations. These techniques willbe used to continue rigorous research of the time dependent Schroedingerproblem under periodic perturbations of arbitrary strength and also ofresonance theory.The project is a piece in an unfolding enterprise to bridge the gap fromformal calculations to rigorous proofs, and in the advance of whatappears to be emerging as a general theory of singular behavior.Countless applications exist in mathematics as well as in the study ofphysical phenomena.