MR2538279 Masatake MIYAKE and Kunio ICHINOBE Masatake MIYAKE and Kunio ICHINOBE Irregularity for Singular System of Ordinary Differential Equations in Complex Domain Funkcialaj Ekvacioj. Serio Internacia 52 2009 53--82 http://fe.math.kobe-u.ac.jp/FE/FullPapers/52-1/52_53.pdf http://www.ams.org/mathscinet-getitem?mr=MR2538279 An irregularity of an ordinary differential equation with an irregular singular point is defined by the maximal rate of exponential growth of all solutions of the associated homogeneous equation. We shall study first order singular systems of Poincaré rank $p (\geq 0)$ and characterize its irregularity from two different viewpoints; the first one is from the order of zeros of coefficient matrices, and the second one is from the comparison theorem of indices of the system on formal Gevrey spaces. Singular system, Irregularity, Exponential growth order, Formal Gevrey space, Index formula. Primary 34M25, Secondary 34A30. 52-53 2009 Irregularity for Singular System of Ordinary Differential Equations in Complex Domain Masatake MIYAKE and Kunio ICHINOBE Masatake MIYAKE and Kunio ICHINOBE
1 Adjamagbo, K. Déterminant sur les anneaux filtré C. R. Acad. Sci. Paris S\ér. I Math. 293 1981 447-449 MR0646862
2 Adjamagbo, K., Fondements de la théorie des déterminants sur un domaine de Ore, Thése Doctrat d'État és Sciences Mathématiques (1991), Université Paris 6
3 Horn, J. Zur Theorie der Systeme linearer Differentialgleichungen mit einer unabhängingen Veränderlichen, II Math. Ann. 40 1892 527-550 MR1510732
4 Hufford, G. On the characteristic matrix of a matrix of differential operators J. Differential Equations 1 1965 27-38 MR0176228
5 Jurkat, W. B.; Lutz, D. A. On the order of solutions of analytic linear differential equations Proc. London Math. Soc. (3) 22 1971 465-482 MR0287062
6 Kitagawa, K. L'irregularité en un point singulier d'un systèmes d'équations différentielles linéaires d'ordre 1 J. Math. Kyoto Univ. 23 1983 427-440 MR0721377
7 Komatsu, H. On the index of ordinary differential operators J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 1971 379-398 MR0303568
8 Lutz, D. A. Some characterization of system of linear differential equations having regular singular solutions Trans. Amer. Math. Soc. 126 1967 427-441 MR0206370
9 Malgrange, B. Sur les points singuliers des équations differentielles linéaires Enseignement Math. 20 1970 146-176
10 Miyake, M. On Cauchy-Kowalevski's theorem for general systems Publ. Res. Inst. Math. Sci. 15 1979 315-337 MR0555658
11 Miyake, M. On the determinant of matrices of ordinary differential operators and an index theorem Funkcial. Ekvac. 26 1983 155-171 MR0736898 0736898 12 Miyake, M. On the irregularity for general system of differential equations in the complex domain Funkcial. Ekvac. 26 1983 211-230 MR0736902 0736902 13 Miyake, M.; Yoshino, M. Toeplitz operators and an index theorem for differential operators on Gevrey spaces Funkcial. Ekvac. 38 1995 329-342 MR1356332 1356332
14 Miyake, M.; Yoshino, M. Fredholm property for differential operators on formal Gevrey space and Toeplitz operator method C. R. Acad. Bulgare Sci. 47 1994 21-26 MR1319686
15 Mizohata, S. Some remarks on the Cauchy problem J. Math. Kyoto Univ. 1 1961 109-127 MR0170112
16 Moser, J. The order of a singularity in Fuch's theory Math. Z. 72 1960 379-398 MR0117375
17 Okubo, K.; Kohno, M. Asymptotic expansions Kyoiku Shuppann 1976 (in Japanese)
18 Ramis, J. P. Dévissage Gevrey Astérisque 59/60 1978 173-204 MR0542737
19 Ramis, J. P. Théoremes d'indices Gevrey pour les équations différentielles ordinaires Mem. Amer. Math. Soc., 48 1984 MR0733946 20 Sibuya, Y. Ordinary differential equations in complex domain Kinokuniya, 1976 (in Japanese); Linear Differential Equations in the Complex domain: Problem of Analytic Continuation, Translations of Mathematical Monographs, 82, American MMathematical Society, Providence, RI 1990 MR1084379
21 Volevič, L. R. On general system of differential equations Dokl. Acad. Nauk SSSR 132 1960 20-23; Soviet Math. Dokl., 1 (1960), 458-461 MR0131045