权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

WKB analysis for high-order ordinary differencial equations with a large parameter

大参数高阶常微分方程的WKB分析

基本信息

批准号：
12640195
负责人：
AOKI Takashi
金额：
$ 1.6万
依托单位：
KINKI UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640195/
关键词：
differential equations of higher-order differntial equations of infinite order exact WKB analysis local theory turning points Stokes curves connection problems WKB solutions 無段階微分方程式高階微分方程式最急降下法ストークス現象積分表示レベル交叉 WKB解析

项目摘要

The purpose of this research was to establish the exact WKB analysis for ordinary differential equations of higher-order with a large parameter. Regarding local theory, we have achieved this purpose. The results we have obtained are as follows:(1) We established a new method of describing Stokes geometry for differential equations of higher-order with a large parameter. This method is called the exact steepest descent method. By using this method, we can find complete Stokes geometry for linear ordinary differential equations of arbitrary order with quadratic coefficients. This is a natural generalization of the steepest descent method.(2) We consider linear ordinary differential equation of infinite order with a large parameter satisfying the following condition: Regarding the large parameter as a differential operator with respect to the variable of the Borel plane, they are microdifferential operators of order 0 and they do not contain that variable. The category of such operators c … More ontains linear ordinary differential operators of arbitrary order with the large parameter. For such an operator, we have defined the notions of WKB solutions, turning points and Stokes curves. We have also introduced the notion of simplicity of turning points. These notions are natural extension of that for finite-order case,(3) Such an equation in the class mentioned above may admit infinitely many phases. After fixing one of it, we have constructed the WKB solution of the difierential equation. This construction is applicable to linear ordinary differential equations of arbitrary order.(4) We have established local decomposition theorem near turning points. If we consider a simple turning point of a differential equation of infinite order, we can decompose the relevant differential operator into the product of two operators; one is invertible and another is of second order whose turning point is exactly the same as the simple turning point. Moreover, the phase of the second order operator is the coincident with the original phase. This implies that an infinite-order differential equation is reduced to a second-order equation near a simple turning point. Thus we have obtained local connection formulas of infinite-order equations near simple turning points. Less

本研究的目的是建立高阶大参数常微分方程的精确WKB分析。关于地方理论，我们已经达到了这个目的。我们得到的结果如下：（1）建立了高阶大参数微分方程的Stokes几何的一种新的描述方法。这种方法被称为精确最速下降法。利用这种方法，我们可以找到任意阶二次系数线性常微分方程的完全Stokes几何。这是最速下降法的自然推广。(2)考虑带大参数的无穷阶线性常微分方程满足以下条件：当大参数作为关于Borel平面变量的微分算子时，它们是0阶微微分算子，且不含该变量.这类经营者的类别c ...更多信息得到了任意阶的大参数线性常微分算子.对于这样的算子，我们定义了WKB解、转向点和Stokes曲线的概念。我们还介绍了转折点的简单性概念。这些概念是有限阶情形的自然推广。（3）这类方程可以有无穷多个相。在确定了其中一个解后，我们构造了该微分方程的WKB解.这种构造方法适用于任意阶线性常微分方程。(4)建立了转向点附近的局部分解定理。如果我们考虑无穷阶微分方程的一个简单转向点，我们可以将相关的微分算子分解为两个算子的乘积;一个是可逆的，另一个是二阶的，其转向点与简单转向点完全相同。此外，二阶算符的相位与原相位一致。这意味着一个无限阶微分方程在一个简单的转折点附近化为一个二阶方程。从而得到了无穷阶方程在简单转向点附近的局部联络公式。少