Connection problem for Birkhoff-Okubo equations (Yoshishige Haraoka) Department of Mathematics Kumamoto University 50. $\Lambda$ $n\c

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1239: 1-10 Issue Date 2001-11 URL http://hdl.handle.

1 Title Connection problem for Birkhoff-Oku systems and hypergeometric systems) Author(s) 原岡喜重 Citation 数理解析研究所講究録 (2001) 1239: 1-10 Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

$$\Lambda$ $n\cross n$ $A$ $n\cross n$ $(ti_{n}- \Lambda)\frac{d\mathrm{Y}}{dt}=A\mathrm{Y}$ (1) Okubo equation (Okubo system system of Okubo normal form) $\mathrm{c}\mathrm{p}^{1}$ A $\infty$ Fuchs$

2 Connection problem for Birkhoff-Okubo equations (Yoshishige Haraoka) Department of Mathematics Kumamoto University 50. $\Lambda$ $n\cross n$ $A$ $n\cross n$ $(ti_{n}- \Lambda)\frac{d\mathrm{Y}}{dt}=A\mathrm{Y}$ (1) Okubo equation (Okubo system system of Okubo normal form) $\mathrm{c}\mathrm{p}^{1}$ A $\infty$ Fuchs Birkhoff Laplace Fuchs ([Bi] [I]). ([0]) [Y2] Okubo equation. Okubo equation [H1] Euler Okubo normal form $\mathrm{c}\mathrm{p}^{1}$ Fuchs. ([HY]) Fuchs $\mathrm{c}\mathrm{p}^{1}\backslash \{a_{1} \ldots a_{p+1}\}$ monodromy Dehgne. (rigid local system) rigid $a_{j}$ 1Od monodromy tuple local monodromy Simpson [Sim] $\mathrm{d}\dot{\mathrm{e}}$ligne-simpson problem $\text{ }$ rigid local system Katz [Ka] Dettweiler-Reiter[DR] Deligne-Simpson problem additive version Kostov [Kol] [K02] additive version& Schlesinger type $\frac{d\mathrm{y}}{dt}=(\sum_{j=1}^{p}\frac{a_{j}}{t-a_{j}})\mathrm{y}$ (2) Okubo equation Schlesinger type Reiter rigid Schlesinger system (2) Okubo $\cdot$ equation (1) subsystem (1) (2). Dettwe.iler-

3 2 (rigid ) Okubo equation Okubo equation& Birkhoff $\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}\mathrm{i}\mathrm{t}\dot{\mathrm{y}}$ monodromy Stokes rigid (Laplace { Okubo equation Birkhoff Stokes (\S 1 ) Ramis Galois [RM] ) monodromy Stokes \S 1 Okubo equation Birkhoff \S 2

4 $\mathrm{f}\mathrm{i}\mathrm{l}$. $\text{ }$ 3 Birkhoff $x\ovalbox{\tt\small REJECT}\otimes$ Okubo equation $x\ovalbox{\tt\small REJECT}\otimes$ $\frac{dw}{dx}=x^{r-1}(\sum_{m=0}^{\infty}\frac{a_{m}}{x^{m}})w$ (3) $x=\infty$ Birkhoff-Turrittin $A_{0}$ $\infty$ (3) $\frac{dv}{dx}=x^{r-1}(\sum_{m=0}^{r}\frac{b_{m}}{x^{m}})v$ (4) Poincare rank $r$ (4) $r=1$ $\frac{dv}{dx}=(b_{0}+\frac{b_{1}}{x})v$ (5) Laplace $\mathrm{y}(t)=\int e^{-xt}v(x)dx$ $V(x)= \int e^{xt}\mathrm{y}(t)dt$ (6) (5) $(ti-b_{0}) \frac{d\mathrm{y}}{dt}=(-b_{1}-i)\mathrm{y}$ (7) Okubo equation Poincare rank$=1$ Birkhoff Okubo equation Poincar\ e rank 1 Laplace (6) (4) Okubo equation rank reduction $u=x^{r}$ $u$ $r$ $r=1$ ([L]) Okubo equation Birkhoff [BJL] Poincare rank$=1$ Birkhoff (5) Stokes Okubo equation(7) $B_{0}=(\begin{array}{lll}\lambda_{1} \ddots \lambda_{\tau\iota}\end{array})$ $(\lambda_{i}\neq\lambda_{j}(i\neq j))$ (8) (8) $B_{0}$ $B_{0}$ \S 2 (8) (7) $t=\lambda_{1}$ $\lambda_{2}$ $\lambda_{n}$ $\infty$ $\ldots$

5 $\mathrm{y}$ 4 $t=\lambda_{j}$ 1 exponent solution $\mathrm{y}_{j}(t)=(t-\lambda_{j})^{\mu_{j}}(\tilde{f_{j}}+o(t-\lambda_{j}))$ (9) $\lambda_{j}$ $\lambda_{k}$ $\mathrm{y}_{k}(t)$ $t=\lambda_{j}$ $\langle$ $c_{jk}$ $2(t)=c_{jk}\mathrm{Y}_{j}(t)+\mathrm{r}\mathrm{e}\mathrm{g}(t=\lambda_{j})$ (10) $\mathrm{r}\mathrm{e}\mathrm{g}(t=\lambda_{j})$ $t=\lambda_{j}$ $\mathrm{y}_{j}(t)$ Okubo eqaution Laplace Birkhoff (5) : $V_{k}(x; \eta)=\int_{\gamma_{k}(\eta)}e^{xt}\mathrm{y}_{k}(t)dt$ (11) $\gamma_{k}(\eta)$ $t$ $t=\lambda_{k}$ $\infty$ $\eta$ ( 1 ) 1 $V_{k}(x;\eta)$ $x=\infty$ $(\pi/2-\eta 3\pi/2-\eta)$ $\tilde{\eta}$ Stokes $(\pi/2-\tilde{\eta} 3\pi/2-\tilde{\eta})$ $\eta\tilde{\eta}$ $V_{k}(x;\eta)$ $\}_{-}^{\wedge}\text{ }$ (9) (10) (11) $V_{k}(x ; \tilde{\eta})$ $\lambda_{j}$ 2 $V_{k}(x; \eta)-v_{k}(x.\tilde{\eta})=\frac{1-e^{2\pi 1d_{k}}}{2\pi i}.\int e^{xt}\mathrm{y}_{k}(t)dt$ (12) $=(1-e^{2\pi}):d_{k}c_{jk}V_{j}(x;\hat{\eta})$ $-(d_{k}+1)=\mu_{k}$

6 $\lambda_{k}$ 5 2 Stokes Stokes $(1-e^{2\pi\dot{\cdot}d_{k}})c_{jk}$. (7) (5) Stokes \S 2. \S Jordan-Pochhammer equation. (7) $B_{0}$ (8) $B_{1}\sim(d_{1}I_{n-1} d_{2})$ $-(d_{k}+1)=\mu_{k}$ $-B_{1}-I$ $-B_{1}-I=(_{a_{n}-\mu_{1}}^{a_{2}-\mu_{1}}a_{1}.\cdot.$ $a_{n}.-\cdot.\mu_{1}a_{1}-\mu_{1}a_{2}$. $a_{2}-.\cdot.\mu_{1}a_{1}-\mu_{1}a_{n})$.$\cdot$ $(n-1)\mu_{1}+\mu_{2}=a_{1}+a_{2}+\cdots+$ (7) Jordan- Pochhammer $\mathrm{y}(t)$ $\mathrm{y}(t)=(\begin{array}{l}(a_{1}-\mu_{1})\int_{\delta}\prod_{j=1}^{n}(\lambda_{j}-s)^{a_{j}-\mu_{1}}\cdot(t-s)_{\lambda_{1}}^{\mu_{1}}\tau^{ds}\neg-s\vdots(a_{n}-\mu_{1})\int_{\delta}\prod_{j=1}^{n}(\lambda_{j}-s)^{a_{j}-\mu_{1}}\cdot(t-s)^{\mu_{1}}\frac{\ }{(\lambda_{n}-s)}\end{array})$ $\triangle$ $t=\lambda_{j}$ exponent solution $(n-1)$ Jordan-Pochbmmer \S 1 Birkhoff (5) Stokes (5) LauriceUa s $F_{D}$ Laplace 1 [HL]

7 Okubo s system $\mathrm{i}\mathrm{i}_{4}4\beta \mathrm{g}\text{ }$ Okubo system $(ti_{4}- (\lambda_{1}i_{2} \lambda_{2}i_{2}))\frac{d\mathrm{y}}{dt}=a\mathrm{y}$ (13) $A$ $a_{1}$ $*$ $a_{2}$ $A=(\begin{array}{llllll}a_{1} * a_{2} * 1 b_{2}\end{array})\sim(\begin{array}{lll}\mu_{1}i_{2} \mu_{2} \mu_{3}\end{array})$ (14) $*$ 1 $($ $b_{9}$ $\Delta$. $\mathrm{i}\mathrm{i}_{4}$ (13) system $a_{1}+a_{2}+b_{1}+b_{2}=2\mu_{1}+\mu_{2}+\mu_{3}$. ([Yl]) (13) $\Lambda(=B_{0})$ (8) 51 6 (13) Birkhoff Stokes (13) [ (10) $cjk$ exponent solutions (10) $\mathrm{r}\mathrm{e}\mathrm{g}(t=\lambda_{j})$ 2 Stokes ( $\mathrm{b}^{1}\mathrm{j}arrow$ $\mathrm{a}\mathrm{a}$ monodromy ). monodromy monodromy rigidity rigidity monodromy (13) ([H1] [H2]) Birkhoff $[\mathrm{m}])_{\text{ }}$ (13) $($[Hl Proposition 5.10] (14) $*$ $\mathrm{y}(t)=\int_{\delta}\phi Ud\tau_{1}\wedge d\tau_{2}$ $\Phi=(1-\frac{\lambda_{2}-t}{\lambda_{2}-\lambda_{1}}\tau_{2})^{\mu 1}\tau_{2}^{-\mu_{2}}(1-\tau_{2})^{a_{1}-\mu_{1}}(1-\tau_{1}-\tau_{2})^{\mu_{1}+\mu_{2}-a_{1}-b_{1}}$ $\mathrm{x}\tau_{1^{a_{2}+b_{1}-\mu_{1}-\mu 2}}(1-\tau_{1})^{\mu_{1}+\mu-a_{2}-b_{2}}2$ $c_{1}=$ $c_{3}=$

8 7 $\triangle$ $(\tau_{1} \tau_{2})$ $\cdot$ $\frac{\lambda_{2}-t}{\lambda_{2}-\lambda_{1}}\tau_{2}=0$ 1- $\tau_{2}=01-\tau_{2}=01-\tau_{1}-\tau_{2}=0$ $\tau_{1}=01-\tau_{1}=0$ $\lambda_{1}$ $\lambda_{2}$ $t$ 2-chain 3 $\triangle_{j}$ $\Delta$ t - $=$ $\underline{-}\sigma$ \lambda \lambda $4\sim \mathrm{t}_{\mathrm{l}}>\theta$ $\mathrm{t}_{\mathrm{l}}>\mathrm{c}$ $\triangle_{j}$ 1 3 $\mathrm{y}_{j}(t)$ (i) $t=\lambda_{1}$ $\mathrm{y}_{8}(t)$ exponent $a_{1}$ $\mathrm{y}_{6}(t)$ exponent $\mathrm{y}_{13}(t)$ $\mathrm{y}_{14}(t)$ $\mathrm{y}_{15}(t)$ $\mathrm{y}_{16}(t)$ (ii) $t=\lambda_{2}$ $b_{1}$ { Y3(t) exponent exponent u $\text{ }$ $\mathrm{y}_{9}(t)$ $\mathrm{y}_{10}(t)$ $\mathrm{y}_{11}(t)$ $\mathrm{y}_{12}(t)$ $\mathrm{y}_{1}(t)$ a2 $b_{2}$ ( $\mathrm{h}.\mathrm{m}_{tarrow\lambda_{1}}\mathrm{y}_{8}(t)/(t-\lambda_{1})^{a_{1}}$ ) $\Delta_{j}$ 3 Cauchy ([A]) 1 (13) [H2] 2.3. Extended Airy equations. $p_{n}(x)$ $n$ 2 $y +p_{n}(x)y=0$ (15) $x=\infty$ $n=1$ Airy Stokes [Sibl] [Sib2] $n\leq 2$ $n\geq 3$ [Sibl] $n=1$ Airy $n=2$ parabolic cylinder Stokes [Sib2] rigid ones non-rigid ones \S 1 rank reduction (15) (5) $n=2$ $y =(x^{2}+c)y$ (16)

9 $\langle$ $-A_{\lrcorner}z$ 8 $y_{1}=y- \frac{y }{x}$ $y_{2}=y+ \frac{y }{x}$ $\mathrm{y}={}^{t}(y_{1}y_{2})$. $\mathrm{y} =xa(x)\mathrm{y}$ $A(x)= (\begin{array}{ll}1 00-1\end{array})+\frac{1}{2}(\begin{array}{ll}c-1 -c-1c-\mathrm{l} -c-1\end{array}) \frac{1}{x^{2}}$ $\overline{\mathrm{t}\backslash }$ Poincare rank 2 Poincare rank1 $z=x^{2}$ rank reduction. $V(z):=(_{z^{1/2}\mathrm{Y}(z^{1/2})}^{\mathrm{Y}(z^{1/2})})$ $V = \frac{1}{2}$ ( $A_{0}+_{z}\mathit{0}_{\underline{A}_{1}\llcorner 1}$ ) $V$ (17) $A_{0}=(\begin{array}{ll}\mathrm{l} 00-1\end{array})$ $A_{1}= \frac{1}{2}(\begin{array}{ll}-1c -c-1c-1 -c-1\end{array})$ $A= \frac{1}{2}(\begin{array}{ll}a_{1} OO A_{1}+1\end{array})$ $B= \frac{1}{2}(_{\mathit{0}}^{a_{0}}$ $A_{0}O)$ (17). $zv =(A+zB)V$ Laplace Okubo equation \S rigidity (16) Stokes $\lceil_{\mathrm{r}}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{y}\rfloor$ (16) $\mathrm{i}\mathrm{i}_{4}$ system \S 2.2 rank reduction ( ) $n=2$ $n=3$ $n=4$ $n=4$ $y =(x^{4}+ax^{3}+\ ^{2}+cx)y$ (18)

10 9 Poincare rank 3 rank reduction $z=x^{3}$ $V(z)=(\nearrow \mathrm{j}_{z^{1/3})}^{/3}z^{1/3})))$ $\mathrm{y}(x)$ (18) 2 Poincare rank1 Birkhoff $zv =(A+zB)V$ $B= \frac{1}{3}(\begin{array}{lll}a_{0} A_{1} A_{0} A_{2} A_{1} A_{0}\end{array})$ $A= \frac{1}{3}(\begin{array}{lll}a_{3} A_{2} A_{1}O A_{3}+1 A_{2}O O A_{3}+2\end{array})$ $A_{0}=(\begin{array}{ll}0 11 0\end{array})$ $A_{1}=(\begin{array}{ll}0 0a 0\end{array})$ $A_{0}=(\begin{array}{ll}0 0b 0\end{array})$ $A_{0}=(\begin{array}{ll}0 0c -2\end{array})$ Laplace A $B$ {1 1-1 $1$ $-1$ -1} Okubo equation 6 $B$ Okubo equation Fuchs $t=1$ $-1$ Fuchs Okubo equation Fuchs Okubo equation rigidity $\text{ }$ \mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{d}}\rfloor$ $n\geq 3$ $\lceil_{\mathrm{n}\mathrm{o}\mathrm{n}- (15) Fuchs Okubo equation non-rigid Acknowledgement. D. A. Lutz rank reduction References [A] K. Aomoto On the structure of integrals of power products of linear functions Sci. Papers Coll. Gen. Education Univ. of Tokyo 27 (1977) $)$ [BJL] W. Balser W. B. Jurkat and D. A. Lutz On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular $\mathrm{i}$ singularities SIAM J. Math. Anal. 12 (1981) [Bi] G. D. Birkhoff Singular points of ordinary linear differential equations Ikans. Amer. Math. Soc. 10 (1909) [DR] M. Dettweiler and S. Reiter An algorithm of Katz and its application to the inverse Galois problem Algorithm methods in Galois theory J. Symbolic Comput. 30 (2000)

$in 10 $\mathrm{b}\mathrm{o}\mathrm{m}$ $\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\overline{\mathrm{l}}\mathrm{t}$ [H1] Y.$

11 in 10 $\mathrm{b}\mathrm{o}\mathrm{m}$ $\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\overline{\mathrm{l}}\mathrm{t}$ [H1] Y. Haraoka Integral representations of solutions of differential equations ffee accessory parameters to appear in Advances in Math.. $\mathrm{i}\mathrm{i}_{4}$ $\mathrm{i}\mathrm{i}_{4}$ [H2] Y. Haraoka Connection problem for Okubo s system in preparation. preparatior [HL] Y. Haraoka and D. A. Lutz in preparation. $\mathrm{f}$ [HY] Y. Haraoka and T. Yokoyama Solutions of Riemann-Hilbert problem for rigid local systems in preparation. [I] E. L. Ince Ordinary Differential Equations Dover New York [Ka] N. M. Katz Rigid Local Systems Princeton Univ. Press Dehgne-Simpson $Pa7^{*}is$ $Pa7^{*}is_{:}$ [KO1] V. Kostov On the Deh.gne-Simpson problem C. R. Acad. Sci. $329(1999)$ Deligne-Simpson $\mathrm{a}\mathrm{g}/ $. $\mathrm{a}\mathrm{g}/\mathrm{o}$ [K02] V. Kostov On the Deh.gne-Simpson problem preprint arxiv:math. [L] D. A. Lutz On the reduction of rank of linear differential systems Pacific $Pac$ Math. $J.$ 42 (1972) [M] K. Mimachi An integral representation of the solution of afourth order Fuchsian differential equation of Okubo type $Rmk$. Ekvac. 38 (1995) [O] K. Okubo On the group of Fuchsian equations Seminar Reports of Tokyo Metropoli- $\tan$ University Tokyo [RM] J. P. Ramis and J. Martinet Theori\ e de Galois diff\ erentieue diff\ erentieue et resommation resommatit Computer alqebra al.qebra and differential equations Comput. Math. Appl. Academic Act Press $\cdot$ $[$ [Sibl] Y. Sibuya Global theory of asecond order linear h.near ordinary differential equation with apolynomial coefficient North-Holland North-HoUand $\mathrm{f}\mathrm{f}\mathrm{i}\backslash E\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{m}[]--k \mathrm{f}\}\text{ }\supset_{l}\dagger\backslash \yen \mathrm{u}^{\backslash }\backslash JP[]\triangleright\gamma_{I}\text{ }\backslash \not\in \text{ }C\text{ }$ $\text{ }\not\in \text{ }\Re \text{ _{}\ell}$ $\grave{ }\#\acute{\grave{\#}}\mathrm{a}\mathrm{e}\not\in$ [Sib2] 24 (2001) $\mathrm{g}1_{\mathrm{t}}$ [Sim] C. T. Simpson Products of Matrices in Differential Geometry Global Analysis and Topology pp Canad. Math. Soc. Cana.$\mathrm{d}$. Math. Soc. Proc. 12 Amer. Providence [Y1] T. Yokoyama On an irreducibility condition for hypergeometric systems systen Funk. Ek- $vac.$ $38$ (1995) [Y2] T. Yokoyama On extension and restriction of Okubo systems preprint.

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