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Number theoretic and algebraic research of divergent formal solution

发散形式解的数论和代数研究

基本信息

批准号：
07640250
负责人：
YOSHINO Masafumi
金额：
$ 0.51万
依托单位：
CHUO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07640250/
关键词：
Toeplitz formal solutions hypoellipticity WKB method Newton polygone vector fields normal form formal solutions divergent Toeplitz operators WKB analysis Riemann-Hilbert Small denomi nators global hypoellipticity

项目摘要

In this research we introduced a new method based on Toeplitz operator theory, WKB analysis and generalized implicit function theorem in dealing with divergent formal power series solutions. Our method can be applied to global and local hypoelipticity of operators, index formula of a system of (partial and ordinary) differential operators on various function spaces and (local)normal forms of commuting systems of singular vector fields and diffeomorphisms. These results revealed transcendental phenomena in the theory of differential operators and the role or rapidly convergent iteration method with infinite loss of derivatives. This research will continue as a collaborations with Italian and English researchers. We will give short items of our results.1. Index formula of systems of irregular singular ordinary differential operators in formal Gevrey spaces via Toeplitz symbols.2. Necessary and sufficient conditions for irregular singular type partial differential operators with two independent variables.3. To determine critial Gevrey indices for solvability and regularity of degenerate parabolic operators via Newton polygones.4. Necessary and sufficient conditions for the reduction to their normal forms of commuting systems of resonant singular vector fields.5. WKB analysis to global solvability and regularity of mixed type operators on the torus.

本文提出了一种基于Toeplitz算子理论、WKB分析和广义隐函数定理的处理发散形式幂级数解的新方法。该方法可应用于算子的全局和局部亚椭圆性，各种函数空间上（偏微分算子和常微分算子）系统的指标公式，奇异向量场和微分同态交换系统的（局部）正规形式。这些结果揭示了微分算子理论中的超越现象以及具有无穷导数损失的快速收敛迭代法的作用。这项研究将继续与意大利和英国的研究人员合作。我们将简短地介绍一下我们的结果。通过Toeplitz符号的形式Gevrey空间中不规则奇异常微分算子系统的指标公式。二自变量不规则奇异型偏微分算子的充分必要条件。利用牛顿多边形确定退化抛物算子的可解性和正则性的临界Gevrey指标。共振奇异向量场交换系统归约为其正规的充分必要条件。环面上混合型算子的全局可解性和正则性的WKB分析。