Exact WKB analysis and microlocal analysis

精确的 WKB 分析和微局部分析

• 批准号: 11440042
• 负责人: KAWAI Takahiro
• 金额: $ 4.67万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440042/
• 关键词: exact WKB analysis; microlocal analysis; Stokes geometry; exact steepest descent path; exact steepest descent method; virtual turning point; infra-red divergence; natural boundaries; ストークス曲線; ストークス幾何; ボレル和; 非断熱近似; Borel変換; 鞍点; Landau-Zener /

1°Concerning the Stokes geometry for higher order linear ordinary differential equations with a large parameter,(1) we first made a concrete and detailed study of Laplace-type equations with the help of the ordinary steepest descent method ([AKT5]), and then by musing on the WKB-theoretic meaning of the obtained results reflectively from the viewpoint of the Borel resummation,(2) we proposed in [AKT3] the exact steepest descent method that makes use of the newly invented notion "exact steepest descent paths" so that we may describe the Stokes geometry for general operators.(3) Some concrete but delicate issues in the Stokes geometry are examined by the exact steepest descent method in [AkoT] and [KoT].In view of the spiritual target of this project, the introduction of the exact steepest descent method into the exact WKB analysis is quite important, as it clearly exemplifies the complementary character of the exact WKB analysis and microlocal analysis, it shows that the global aspect o … More f the quantized Legendre transformation can be described in terms of the exact steepest descent paths.2° Non-adiabatic transition probabilities for Landau-Zener type problems are calculated on the basis of microlocal analysis of operators with multiple characteristics ([AKT1]). Important in its own right is the concrete algorithm for detecting virtual turning points for the operators in question.3° Microlocal structure of the S-matrix is studied ia [KS] when infra-red divergence is relevant.4° Natural boundaries of solution of non-liner ordinary differential equations are studied in [K] from the viewpoint of WKB analysis. Microlocal study of natural boundaries of Dirichlet series was also made in [KStr].5° Local theory of the exact WKB analysis for the infinite series of differential operators with a large parameter was developed in [AKKT] with the help of a quantized contact transformation, one of the basic notions in microlocal analysis.6° [T1] constructed the exact WKB analysis for systems of differential equations, and we are currently (2002) trying to apply it to the study of higher order Painlev」 equations (Noumi equation etc.). Less

1°关于高阶大参数线性常微分方程的Stokes几何，（1）我们首先用最速下降法（[AKT 5]）对Laplace型方程作了具体而详细的研究，然后从Borel重积的观点反思所得结果的WKB理论意义，（2）我们在[AKT 3]中提出了精确最速下降法，它利用了新发明的“精确最速下降路”的概念，从而可以描述一般算子的Stokes几何。(3)[AkoT]和[KoT]用精确最速下降法研究了Stokes几何中的一些具体而微妙的问题。鉴于本课题的精神目标，将精确最速下降法引入精确WKB分析是十分重要的，因为它清楚地阐明了精确WKB分析和微局部分析的互补性，表明了精确WKB分析的全局性，表明了精确WKB分析和微局部分析的互补性。[AkoT]和[KoT]中的精确最速下降法是一种非常有效的方法 ...更多信息 f的量子化勒让德变换可以用精确的最陡下降路径来描述.基于多特征算子的微局部分析，计算了Landau-Zener型问题的2 °非绝热跃迁概率（[AKT 1]）.重要的是在其本身的权利是具体的算法来检测虚拟的转折点的运营商在问题中。3 °微局部结构的S-矩阵的研究在[KS]时，红外发散是相关的。4 °自然边界的非线性常微分方程的解在[K]中研究的WKB分析的观点。[KStr]也对Dirichlet级数的自然边界进行了微局部研究。[AKKT]借助微局部分析的基本概念之一量子化接触变换，发展了5 °大参数微分算子无穷级数的精确WKB分析的局部理论。[T1]构造了微分方程组的精确WKB分析，我们目前（2002）正试图将其应用于高阶Painlev方程（Noumi方程等）的研究。少

项目成果

Y.Takei: "On a double turning point problem for systems of linear ordinary differential equations"

Y.Takei：“关于线性常微分方程组的双转点问题”

[AKKT]T. Aoki, T. Kawai, T. Koike and Y. Takei: "On the exact WKB analysis of operators admitting infinitely many phases"RIMS preprint. 1350. (2002)

[AKKT]T。

[T2]Y. Takei: "On an exact WKB approach to Ablowitz-Segur's connection problem for the second Painleve equation"ANZIAM J. Australian Math. Soc.. 44. .111-119 (2002)

[T2]Y。

T.Aoki, T.Kawai, Y.Takei: "On the exact steepest descent method : a new method for the description of Stokes curves"J. Math. Phys.. 42. 3691-3713 (2001)

T.Aoki，T.Kawai，Y.Takei：“论精确最速下降法：一种描述斯托克斯曲线的新方法”J.

T.Aoki, T.Kawai, Y.Takei: "On a complete description of the Stokes geometry for higher order ordinary differential equations with a large parameter via integral representations"Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear

T.Aoki、T.Kawai、Y.Takei：“通过积分表示对具有大参数的高阶常微分方程的斯托克斯几何的完整描述”走向微分方程（线性或非线性）的精确 WKB 分析