非交换最速下降法的一致渐近研究

项目名称： 非交换最速下降法的一致渐近研究

项目编号： No.10871212

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 赵育求

作者单位： 中山大学

项目金额： 24万元

中文摘要： 非交换的最速下降法是目前应用分析研究的一个重要课题，在组合论、随机矩阵论及数学物理问题中有日渐广泛的应用。本项研究拟考虑这方面的以下问题：1确定非交换最速下降法一致渐近的范围和边界。2.应用此方法研究卷积型积分方程解的整体一致渐近 3.与Jacobi算子相结合，讨论二阶差分方程解析解的整体一致渐近。为此，我们拟从两类正交多项式的渐近研究入手，借鉴Berry和Howls关于积分Stokes现象的研究，得到一致渐近的范围和边界刻划。关于卷积型方程, 拟从用最速下降法考虑椭圆型方程边值问题解的渐近性质入手，进而考虑renewal equation和某类Toeplitz算子的整体一致渐近。关于差分方程, 拟与Jacobi算子结合，讨论二阶差分方程对应的测度,进而用最速下降法，讨论方程解析解的整体一致渐近。本课题的提出和解决，当有助于深化对非交换的最速降法本身的研究及拓展该方法的进一步应用。

中文关键词： 非交换最速下降法；一致渐近；卷积型积分方程；正交多项式；二阶差分方程的解析解

英文摘要： Non-commutative steepest descent method is a significant topic in applicable analysis, and finds its wide-spreading applications in combinatics, random matrix theory, and other problems in mathematical physics. The objective of the present project is as follows: 1.Determine the first level uniform domain and boundary of the steepest descent method. 2. Use the approach developed in 1 to investigate the globally uniform asymptotics of the convolution integral equations, and 3. Investigate the uniform asymptotics of the analytic solutions of second order difference equations related to the Jacobi operators. To serve these purposes, we take two sets of orthogonal polynomials as our starting point, putting into consideration the work of Berry and Howls on the Stokes phenomena of integrals, so as to characterize the uniform domains and boundaries. As for convolution equations, we plan to consider the asymptotic properties of the solution to the boundary value problems of elliptic PDEs, and further to consider renew equations and the uniform asympcs of a class of Toplitz operators. For difference equations, combining with known facts on Jacobi operators, we will discuss the measures associated with the equations,and then use the non-commutative steepest descent method to study the asymptotics of the analytic solutions to the equations. Solving the problems raised above would benefit the deeper understanding of the method itself, as well as the wider extending of its further applications.

英文关键词： Non-commutative steepest descent method; uniform asymptotics; convolution integral equation; orthogonal polynomials; analytic solution to 2nd order di