Linear Differential Equations in the Complex Domain
Linear Differential Equations in the Complex Domain
复制标题
复域中的线性微分方程
DOI：
10.1007/978-3-030-54663-2
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Y. Haraoka
中科院分区：
文献类型：
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作者：
Y. Haraoka
复域上的微分方程是数学和物理学的基本对象。在物理学中，各种领域，如力学，电磁学和量子力学使用微分方程作为基本工具。特别地，贝塞尔微分方程、勒让德微分方程、高斯超几何微分方程、库默合流超几何微分方程和厄米特微分方程作为这些领域中的基本方程出现。这些微分方程用于描述物理现象，因此是真实的变量方程;然而，我们可以将它们视为复变量方程，这一观点将加深对这些微分方程的理解。在数学中，复变量线性常微分方程不仅出现在分析中，而且出现在数论、代数几何、微分几何、表示论等领域，并在各个领域发挥着重要作用。这种多样性使得复域中的线性微分方程变得重要和有吸引力。另一方面，除了与这些不同领域的关系之外，复域线性微分方程的研究本身也得到了发展。特别是完整系统理论和微分方程变形理论的发展，对数学和物理学的许多领域产生了重要影响。此外，尼古拉斯·M.卡茨在1996年出版的一个决定性的发展理论的线性Fuchsian常微分方程。Fuchsian常微分方程是复射影直线P1上（或一般的黎曼曲面上）的线性微分方程，其奇点只有正则奇点。局部理论研究正则奇点邻域内解的行为，在世纪就已经完成。另一方面，研究在正则奇点处的解的解析延拓的整体理论是非常困难的，在整体理论中我们没有统一的分析方法。然而，存在一类特殊的Fuchsian常微分方程，称为刚性的，我们有明确的描述的整体行为的解决方案。因此，在本发明中，
Differential equations in the complex domain are fundamental objects in mathematics and physics. In physics, various fields such as mechanics, electromagnetism, and quantum mechanics use differential equations as fundamental tools. In particular, Bessel’s differential equation, Legendre’s differential equation, Gauss’s hypergeometric differential equation, Kummer’s confluent hypergeometric differential equation, and Hermite differential equation appear as fundamental equations in these fields. These differential equations are used in describing physical phenomena and hence are equations in real variables; however, we may regard them as equations in complex variables, and this viewpoint will bring deep understandings of these differential equations. In mathematics, linear ordinary differential equations in complex variables appear not only in analysis but also in number theory, algebraic geometry, differential geometry, representation theory, and so on and play substantial roles in each field. This diversity makes linear differential equations in the complex domain important and attractive. On the other hand, apart from the relations to these various fields, the study of linear differential equations in the complex domain has been developed by itself. In particular, the theory of holonomic systems and the deformation theory of differential equations have made big progresses, and the results influenced many fields in mathematics and physics. Moreover, the book “Rigid Local Systems” by Nicolas M. Katz published in 1996 brought a decisive development to the theory of linear Fuchsian ordinary differential equations. We shall explain the development in detail.A Fuchsian ordinary differential equation is a linear differential equation on the complex projective line P1 (or on a Riemann surface, in general) having only regular singular points as singular points. The local theory studies behaviors of solutions in a neighborhood of a regular singular point and has been completed already in the nineteenth century. On the other hand, the global theory, which studies analytic continuations of a solution specified at a regular singular point, is very difficult, and we have no uniform method of analysis in the global theory. However, there exists a special class of Fuchsian ordinary differential equations called rigid, for which we have explicit descriptions of global behaviors of solutions. Thus,