ベクトルの近似直交化を用いた高階線型常微分方程式の整数型解法

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かずまさかやぬま
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1 Fuminori Sakaguchi Graduate School of Engineering, University of Fukui ; Masahito Hayashi Graduate School of Mathematics, Nagoya University; Centre for Quantum Technologies, National University of Singapore 1 $p_{m}(x)(m = 0,1, \ldots, M)$ $P(x, \frac{d}{dx})=\sum_{m=0}^{m}p_{m}(x)(_{tx}d)^{m}$ 1 ( $ODE$) $P(x, \frac{d}{dx})f=0$ Hilbert (acurracy) $M$ ( working pre [1] [2] [3] cision $P(x, \frac{d}{dx})$ ) Petrov-Galerkin ( ) $ODE$ 2 [1] [2] [3] $P(x, \frac{d}{dx})$ $ODE$ Hilbert $P(x, \frac{d}{dx})$ $ODE$ $P(x, \frac{d}{dx})$ (closed extension)

2 $\mathcal{h}^{0}$ $\mathcal{h}^{0}$ ) $\tilde{b}_{p}$ 133 $D(\tilde{A}_{P})$ $:= \{f\in C^{M}(\mathbb{R})\cap H P(x, \frac{d}{dx})f\in \mathcal{h}\}$. ) C3 $B_{P}$ $P(x, \frac{d}{dx})$ $\tilde{b}_{p}$ ( C3 ) $\overline{a}_{p}$ $\tilde{b}_{p}$ ( [1] $\overline{a}_{p}$ $\overline{a}_{p}$ ) $\triangleright$ $\Vert\cdot\Vert_{\mathcal{H}}+ \tilde{a}_{p}\cdot\vert_{\mathcal{h}}$ 5 $A_{P}$ $(P(x, \frac{d}{dx}), \mathcal{h}, \{e_{n}\}_{n=0}^{\infty}, \mathcal{h}^{\langle\rangle}, \{e_{n}^{0}\}_{n=0}^{\infty})$ $( \mathcal{h}\supset \mathcal{h} 215 (P(x, \frac{d}{dx}), H, \{e_{n}\}_{n=0}^{\infty}, \mathcal{h}^{}, \{e_{n}^{0}\}_{n=0}^{\infty})$ $\forall_{f}\in \mathcal{h},$ $\langle f,$ $f\rangle_{h^{}}\leq\langle f,$ $f\rangle\prime re)$ Hilbert Cl-3 $B_{P}$ $m$ $B_{P}$ $D( \overline{b}_{p}):=\{f\in C^{M}(\mathbb{R})\cap \mathcal{h} P(x, \frac{d}{dx})f\in \mathcal{h}$ $\}.$ $f$ 12- $ $ $ J\{f_{n};=\langle f, e_{n}\rangle_{\mathcal{h}}\}_{n=0}^{\infty}$ $m+ \ell\sum_{n=\max(0,m-\ell 0)}^{0}b_{m}^{n}f_{n}=0$ (1) $P(x, \frac{d}{dx})$ $\tilde{b}_{p}$ $\tilde{b}_{p}$ ($\Vert\cdot\Vert_{\mathcal{H}}+\Vert\tilde{B}_{P}$. $ \mathcal{h}$ $\{f_{n}:=\langle f, e_{n}\rangle_{\mathcal{h}}\}^{\infty}$ $n=0$ $\tilde{b}_{p}$ ( ) $B_{P}$ $V:= \{f^{arrow}:=\{f_{n}\}_{n=0}^{\infty} \sum_{n=0}^{\infty}b_{m}^{n}f_{n}=0(m\in Z^{+})\}$ $A_{P}\subset B_{P}$ $A_{P}$ $B_{P}$ (1) $f-$ $B_{P}f=0$ $4^{2}$- $B$ - $A_{P}f=0$ $B_{P}P$ ( ( ) [1] $)$ $A_{P}$ $B_{P}$ $C4V\cap l^{2}(z^{+})$ $\{f_{n}\}_{n=0}^{\infty}$ $\{e_{n}\}_{n=0}^{\infty}$ $\{e_{n}^{0}\}_{n=0}^{\infty}$ Cl $\forall_{n\in}z+,$ $e_{n}\in D(\tilde{B}_{P})$ 1- $\sum_{n=0}^{n}f_{n}e_{n}$ Cl-3 $\ell_{0}$ C2 $ m-n >\ell_{0}$ $b_{m}^{n}$ $:=$ $\langle B_{P}e_{n},$ C3 $e_{m}^{0}\rangle_{h })=0$ $B_{P}$ ( $e_{m}^{\langle\rangle}\in D(C_{P})$ $Narrow\infty$ $P(x, \frac{d}{dx})f=0$ $f\in C^{M}(\mathbb{R}\backslash S)\cap \mathcal{h}$ $S$ C4 ) [3] $f$ $ODE$ $D(\tilde{B}_{P})$ $\langle B_{P}f,$ $e_{m}^{(\}}\rangle_{\mathcal{h}^{}}=\langle f,$ $C_{P}e_{m}^{0}\rangle\prime\kappa$ $\mathcal{h}^{}$ $D(C_{P})$ $C_{P}$ ( C2 $m$ $n_{m}$ 2.2 $ODE$ $P(x, \frac{d}{d})f=0$ $x$ 5 $(P(x, \frac{d}{dx}), \mathcal{h}, \{e_{n}\}_{n=0}^{\infty}, \mathcal{h}$ $, \{e_{n}^{0}\}_{n=0}^{\infty})$ $CI$ -C4 (1) $\ell$2- $C^{M}(\mathbb{R})\cap \mathcal{h}$ $n>n_{m}$ $f\mapsto\{\langle f, e_{n}\rangle_{\mathcal{h}}\}_{n=0}^{\infty}$ $ODE$ $b_{m}^{n}=0$ 1 1

3 f^{arrow}\vert_{\ell^{2},k}^{2}}$ for 134 ( $j_{0}+\ell_{0}$ $j_{0}+1$ 1 ) $n\geq j_{0}+\ell_{0}+1$ $f_{n}=- \frac{1}{b_{n-\ell_{0}}^{n}}\sum_{m=n-2\ell_{0}}^{n-1}b_{n-\ell_{0}}^{m}f_{m}$ (2) $b_{m}^{n}$ 1: $\Omega(f_{g})$ 1 $\ell^{2}(z)$ $\Omega(f^{\neg})$ 2 $m$ $j_{0}$ $j_{0}$ C5 $b_{m}^{m+\ell_{0}}\neq 0$ $b_{m}^{n}$ ( $ODE$ ( 1 ) $ODE$ Fuchs ( ) $\ell^{2}(z^{+})$ ([1] Teorem 2.3) 1 (1) $\ell^{2}$- $V$ $C^{M}(\mathbb{R})\cap \mathcal{h}$ $ODE$ $C^{M}(\mathbb{R}\backslash S)\cap \mathcal{h}$ $ODE$ 1 ( $V\cap\ell^{2}(Z^{+})$ ) $M$ 1 $ODE$ Fuchs $V$ C $V\cap P^{2}(Z^{+})$ 2- $ODE$ $C^{M}(\mathbb{R}\backslash S)\cap \mathcal{h}$ [1] $\Pi_{n}$ $<e_{0},$ $e_{1},$ Fuchs $e_{n}>$ $\ldots,$ ( $n+1$ ) $j_{0}+\ell_{0}$ $1\leq K\leq N$ $K$ $N$ - $\forall f\in\ell^{2}(z^{+})$ $\Omega(f^{\neg})\geq$, $ODE$ 1 (1) $2^{2}$- $\Vert f^{arrow}\vert_{l^{2}}^{2}:=\sum_{n=0}^{\infty} f_{n} ^{2}$ $\ell^{2}(z)\cross\ell^{2}(z)$ $\sigma_{k,n}^{(\omega)}(f^{\neg})$ $:=$ $\frac{\omega(\pi_{n}f\gamma}{\vert $\underline{\sigma_{k,n}^{(\omega)}} := \min_{f^{arrow}\in V\backslash \{0\}}\sigma_{K,N}^{(\Omega)}(f7.$ $\tilde{f}\in V\backslash \{0\}(3)$ $j_{0}$ ) $B_{P}$ 1 C2, C5 $K\geq$ 1 (1) $io+\ell_{0}-1$ $f\in V\backslash \{O\}$ $\Vert f^{arrow}\vert_{\ell^{2},k}:=$

4 $\Vert\Pi_{K}f^{arrow}\Vert_{\ell^{2}}>0$ $\mathfrak{z}$ $\emptyset$ $\langle$ $\prime-$ $*\backslash$ $\overline{\backslash _{\prime}\backslash.}$ $\ell$ $\wedge^{\backslash }$ $4^{\backslash }$ $I_{1}$ 135 $W$ $\langle\vec{x},$ $y\neg\rangle_{\ell^{2},k}:=\langle\pi_{k}\tilde{x},$ $\Pi_{K}\vec{y)}\ell^{2}$ $P_{W,K}$ 2.3 $K(\geq j_{0}+\ell_{0}-1)$ $Narrow$ $\infty$ $\sup_{\vec{x}\in(\sigma_{k,n}^{(\omega)})^{-1}[\sigma_{k,n}^{(\omega)},c\sigma_{k,n}^{(\omega\rangle}]}\frac{\vert P_{V\cap\ell^{2}(),K}z+\vec{x}-\vec{x}\Vert_{l^{2},K}}{\Vert\vec{x} _{\ell^{2},k}}arrow 0.$ ( [1] (18) (19) $P_{V\cap l^{2}(),k}z+$ $P_{V,K}$ ) $Narrow\infty$ $\Pi_{K}(V\cap\ell^{2}(Z^{+}))$ $\Pi_{K}(\sigma_{K,N}^{(\Omega)})^{-1}[\sigma_{K,N}^{(\Omega)}, c\sigma_{k,n}^{(\omega)}]$ [1] $ODE$ $(\sigma_{k,n}^{(\omega)})^{-1}[\sigma_{k,n}^{(\omega)}, c\sigma_{k,n}^{(\omega)}]$ o (2) $P$ $n$ $\mathscr{x}$ 1 $T$ 2 2 $\ell^{2}(z^{+})$ $n$ 2 1 $A,$ $B$ $\star$ $N+1$ $A^{-1/2}BA^{-1/2}$. $K+1(K<$ $A$ $B$ 1 1$N)$ [1] [2] $V\cap\ell^{2}(Z^{+})$ $(\sigma_{k,n}^{(\omega)})^{-1}[\sigma_{k,n}^{(\omega)}, c\sigma_{k,n}^{(\omega)}]$ $V\cap\ell^{2}(Z^{+})$ $\acute{}\supset$ - $V\cap P^{2}(Z^{+})$ 4 $\ell^{2}(z^{+})$- $\Omega$ 1 $\Vert f^{arrow},\vec{g}\vert_{\omega,n}:=\omega(\pi_{n}f^{arrow}, \Pi_{N}\vec{g})$ $:\backslash E$ $(\Pi_{K}f\Pi_{Kg})arrow,$ $ _{-} $ 2 $V\cap\ell^{2}(Z^{+})$ $K$ ffl $\ell^{1}.$ $\ell^{2}(z^{+})$- $V\cap\ell^{2}(Z^{+})$ [2] $\Phi$ [2]

5 $($ $l\grave{\grave{1}}$ $j$ $i$ $\overline{pj}$ $I$ $f$ $\Theta i$ $l$ $ $ $V\cap\ell^{2}(Z^{+})$ [2]( 1 ) $V\cap\ell^{2}(Z^{+})$ [1] [2] 2: $\psi_{k,\ddot{n}}(x)$ $(k=2,\ddot{n}=5)$ 3 $ODE$ $p_{m}(x)$ Cl-4 $k$ $($ $(f, g)_{(k)}:= \int_{-\infty}^{\infty}f(x)\overline{g(x)}(x^{2}+1)^{k}dx$ (4) 1 $\Vert\cdot\Vert_{(k)}(\Vert f\vert_{(k)}^{2}=(f, f)_{(k)})$ $L_{(k)}^{2}(\mathbb{R});=\{f$ : $ \Vert f\vert_{(k)}<\infty\}$ $i$ $k_{1}\geq k_{2}$ $L_{(k_{2})}^{2}(\mathbb{R})$ $L_{(0)}^{2}(\mathbb{R})=L^{2}(\mathbb{R})$ $L_{(k_{1})}^{2}(\mathbb{R})\subset$ $e_{n}=\sqrt{\frac{1}{\pi}}\psi_{k_{0},\ddot{n}_{k_{0},n}},$ $\psi_{k,\ddot{n}}(x)$ $:= \frac{1}{(x+i)^{k+1}}(\frac{x-1}{x+i})^{\ddot{n}}$ 3 $\psi_{k,\ddot{n}}(x)=-\frac{i}{2}(\psi_{k-1,\ddot{n}}(x)-\psi_{k-1,\ddot{n}+1}(x))$ (5) $X \psi_{k},\ddot{n}(x)=\frac{1}{2}(\psi_{k-1,\ddot{n}}(x)+\psi_{k-1,\ddot{n}+1}(x))$ $\frac{d}{dx}\psi_{k,\ddot{n}}(x)=\ddot{n}\psi_{k+1,\ddot{n}-1}(x)-(\ddot{n}+k+1)\psi_{k+1,\ddot{n}}(x)$ ( $ODE$ $P(x,d\varpi)$ $p_{m}(x)(m=$ $0,1,$ $\ldots,$ $M)$ $p_{m}(x)(m=0,1, \ldots, M)$ $k_{0}^{0}\leq$ $x$ ) $k_{0}- \max_{m}(\deg p_{m}-m),$ $k_{0}\geq 0$ $k_{0},$ $k_{0}^{0}$ $\mathcal{h}=l_{(k_{o})}^{2}(\mathbb{r})$, $\mathcal{h}^{0}=l_{(k_{0}^{0})}^{2}(\mathbb{r})$ $e_{n}^{0}=\sqrt{\frac{1}{\pi}}\psi_{k_{0}^{0},\ddot{n}_{k_{0}^{0},n}}$ $( zu(1,1)$ [4] ; ) Cl-4 [1] $\ddot{n}_{k,n}:=l-\frac{k+1}{2}$ $+(-1)^{n+k+1} L\frac{n+1}{2}$ 2 $karrow\infty$ $x=\pm i$ ( ) C5. $k_{0}^{0}$ $\ell_{0}=2m+k_{0}$ [2] $ $ [1] $x=\pm i$ - $\{_{\tau_{\pi}^{1}}\psi_{k,\ddot{n}}(x) \ddot{n}\in Z\}$ $ $ $L_{(k)}^{2}(\mathbb{R})$ $x$ ([1] Lemma3.1) $\mathfrak{b}$ 7:. C5 $\Re$

6 $\lambda O$ $F^{1},$ $I$ $\iota$ $4 \overline{t}$ $\overline{\lrcorner f}$ $f_{\backslash $ib\grave{\grave{\}}}$ }$ $1^{I}.$ $ _{-}^{\backslash }$ $\prime\rfloor\backslash$ $\wedge^{\backslash ^{\backslash }}$ $\vec{f_{int}}^{\vec{(}d)}$ 137 $p_{m}$ ( ) $B_{P}$ $(B_{P}e_{n}, e_{m}^{\phi})_{\mathcal{h}^{}}$ $p_{m}$ [2] $\Vert f^{arrow},\vec{g}\vert_{\omega,n}=$ $\Omega(\Pi_{N}f^{arrow},$ $\Pi_{N}$ 1 Step 2 $\{F_{n}^{(1)}\}_{n=0}^{N},$ $\ldots,$ $\{F_{n}^{(d)}\}_{n=0}^{N}$ ( 1 ) Gram-Schmidt 1 3 ; ( ) $\Re$ 1 Gram- $b$ 2 Schmidt Euclid $U\grave{}$ $(Q2)$ [2] $QI$ Q2 $V\cap\ell^{2}(Z^{+})$ 1 $4 $ 2 $F$ [2] 3 Ql [2] 9 Gram-Schmidt $1f$ ( ) 1. $[\cdot]_{\mathbb{c}}$ ( $\{F_{n}^{(1)}\}_{n=0}^{N},$ $\{F_{n}^{(d)}\}_{n=0}^{N}$ $\ldots,$ ) $\vee $ $T$ $\vec{f_{int}}^{(1)}. \ldots$,.

7 $\tilde{v}_{1},\vec{v}_{2},$ 138 $h$ $\ldots$, 2 $\vec{v}_{1},\vec{v}_{2},$ $\ldots$, $\vec{e}^{(1)},$ $\ldots,\vec{e}(d)arrow$ $\sigma_{k,n}^{(\omega)}(\vec{e}^{(n)})$ (3) $\vec{e}^{(n)}$ $\vec{e}^{(n)}$ $\vec{g}^{(1)}$ $\vec{g}^{(1)}$ 4.1 $h$ $\geq$ $d$ $\vec{g}^{(1)}$ $\Pi_{N}$ $((\sigma_{k,n}^{(\omega)})^{-1}[\underline{\sigma_{k,n}^{(\omega)}},$ $\overline{1^{\gamma T}-}R^{-}d\underline{\sigma_{K,N}^{(\Omega)}}])$ $\Pi_{K}\tilde{G}^{(1)}$ 2.3 $ODE$ $\sum_{n=0}^{k}(\vec{g}^{(1)})_{n}e_{n}(x)$ [2] $V\cap\ell^{2}(Z^{+})$ 1 [2] $V\cap P^{2}(Z^{+})$ 1 1 [2] 5 $ODE$ (Wolfram Research Mathematica ) 2 $w_{n}:=\{\begin{array}{l}1 (n\leq K)e^{r(\mu_{\mathfrak{n}}-\mu\kappa)} (K<n<J)R:=e^{r(\mu_{J}-\mu\kappa)} (n\geq N)\end{array}$ $\mu_{n}:= \ddot{n}_{k_{0},n}-\frac{k_{0}+1}{2} -\frac{k_{0}+1}{2}$

8 139 $\Vert f^{arrow},\vec{g}\vert_{\omega,n}=\omega(\pi_{n}f^{arrow}, \Pi_{N}\vec{g})=\sum_{n=0}^{\infty}w_{n}f_{n}\overline{g_{n}}$ $f(x)$ $w_{n}$ $f_{n}$ 5 $\overline{\psi_{k_{0},\ddot{n}}}(x)$ $=$ $\psi_{k_{0},-\ddot{n}-k-1}(x)$ $n$ $K=2 \lfloor\frac{3(n-k_{0})}{8}\rfloor+^{\mathfrak{l}}k_{0},$ $J=$ $2 \lfloor\frac{7(n-k_{0})}{16}\rfloor+k_{0}$ $K=2 \lfloor\frac{7(n-k_{0})}{16}\rfloor+k_{0},$ $J=2 \lfloor\frac{15(n-k_{0})}{32}\rfloor+k_{0}$, $r=10^{8}$ $f(2)/f(1)$ $N+1=10000$ 1942 ( $10^{9}$ $+$ ) Weber ( Schr\"odinger ) 2 $ODE$ $(9x^{2}-6x+5)f"+(90x-30)f +126f=0$ (6) $ODE$ $L_{(k_{O})}^{2}(\mathbb{R})$ $C$ 6 $N$ $\langle f,$ $\frac{1}{2\pi}(\psi_{k_{0},0}+$ $\psi_{k_{},-k_{0}-1})\rangle_{\mathcal{h}}=1$ $C$ $N+1=7000$ $N$ $N$ ( ) $(f(x)=$ $\sum_{n}f_{n}e_{n}(x))$ $\mathfrak{t}b\frac{f_{2}}{f_{0}}$ $K$ $0$ $f(x)$ 2 $\frac{5}{2}(\frac{8}{29})^{4}= \ldots\cross 10^{-2}$ $f -x^{2}f+(2\nu+1)f=0$ (7) $\nu\in Z^{+}$ $C^{2}(\mathbb{R})\cap L^{2}(\mathbb{R})$ $\{C(\exp\frac{-x^{2}}{2})H_{\nu}(x) C\in \mathbb{c}\}$ $k_{0}\leq 3$ $\{\frac{c(3x-1)}{((3x-1)^{2}+4)^{4}} C\in \mathbb{c}\}$ $ODE$ $k_{0}=2,$ $N+1=18,24,$ $k_{0}=3$ $k_{0}\in Z^{+}$ $_{}k_{0})k=2\lfloor\frac{7(n-k_{0}) L_{(}^{2}}{16}\rfloor+$(k0) $\mathbb{r}$ $J$ $K=2 L\frac{3N}{8}\rfloor+k_{0}$ $J=2 L\frac{7N}{16}\rfloor+$ $ODE$ 4 $ODE$ $\rho:=-$ ( ) (8) $\frac{f_{1}}{f_{0}}=1_{\backslash }\frac{f_{2}}{f_{0}}=\frac{ i}{28561}$ $\frac{f_{2}}{f_{0}}$ $\circ$ 4 $N$ 100 $N=48$ 1 $N=48$ $N>48$ $=$2 $\frac{15(n-k_{o}) \nu=}{32}\rfloor+00_{\backslash }$ $\lfloor$

9 140 $\underline{f_{2}}$ 4: $f_{0}$

10 $3_{\backslash }$ $k_{0}=0,$ 141 $ODE$(7) $xarrow$ $30x$ $\frac{1}{(30)^{2}}f"-(30)^{2_{x}2}f+(2\nu+1)f=0$ (9) Number of significant digits $\nu=0$ $k_{0}=6$ $K=2 \lfloor\frac{7(n-k_{0})}{16}\rfloor+k_{0}$ $J=2 \lfloor\frac{15(n-k_{0})}{32}\rfloor+k_{0}$ $5$. 6 $N+1=30000$ 8783 $10^{2}$ 2599 (8). $.\cdot\cdot$ $\rho$ 1 Fuchs $ODE$ Legendre $(1-x^{2})f"-2xf +( \nu(\nu+1)-\frac{\mu^{2}}{1-x^{2}})f=0(10)$ $10^{1}$.. $\neg$ $\cdot$ $ \cdot\cdot--$ $I0^{2} 10^{3} 10^{4}$ 3: $ODE$ (7) $L2fo$ 3 $(-\infty, -1)$ $(-1,1)$ $(1, \infty)$ $\{C\cdot 1_{[-1,1]}(x)$. $(1-x^{2})^{L^{I}}2L_{\nu}^{\mu}(x) C\in \mathbb{c}\}$ ( $1_{I}(x)$ $(1-x^{2}){\} L_{\nu}^{\mu}(x)I$ Legendre ) $ODE$ 7 $(-1,1)$ $(-1,1)$ $0$ $(-1,1)$ $\mu$ $ODE$ $\nu$ $N+1$ $xf (x)+f (x)+(- \frac{x}{4}+(\nu+\frac{\mu+1}{2})-\mu_{-}^{2}x)f(x)$ $=0(11)$ Laguerre $L_{\nu}^{\mu}(x)$ 4: (8) $\rho$ ( $ODE$ (7) ) $N+1$ $f(x)=\{\begin{array}{ll}cx^{\mu/2}e^{-x/2}l_{\nu}^{\mu}(x) (x\geq 0)0 (x<0)\end{array}$ $(C\in \mathbb{r})$. $\mu=4$ $\nu=$ $N+1=200,500_{\tau}K=2 \lfloor\frac{3n}{8}\rfloor+k_{0}$

11 $\ovalbox{\tt\small REJECT}_{f0}^{130}$ $\Delta$ $\Delta$ 142 7: $ODE$ (10) $ff_{b}^{j}$ 5: $ODE$ (9) 1 8: $ODE$ (11) 0.5 $0$ $1\cup^{-}$ $1\cup^{-}$ $1\cup$ $N+1$ 6: (8) ( $\rho$ $ODE$ (9) ) Number of significant digits of (9)$/f(4)$ $10^{3}ODE(11.3), va\dot{n}able:us.t.x=900u^{2}$ $\pi 9yf\langle 4\rangle=g\langle 3130yg(2l30)$.. $10^{2}$.. $\Delta $10^{\{}$ $10^{0}$ $\Delta\Delta$ $\Delta\Delta ODE(11.2)$. \Delta A$ variable:x $10^{3} 10^{4}$ $N*1$ 9: $f(9)/f(4)$ ( $ODE$ (11) $ODE$ (12))

12 143 $J=2 \lfloor\frac{7n}{16}\rfloor+$ 8 $x<0$ $x\geq 0$ $((10)$ $x=\pm 1$ $(11)$ $x=0)$ $ODE$(II) $xarrow u$ ( $x=cu^{2}c$ ) $u^{2}g"(u)+ug (u)$ $+(-c^{4}u^{4}+c^{2}(4\nu+2\mu+2)u^{2}-\mu^{2})g(u)=0(12)$ $x=0$ 10: $c=30$ $ODE$ (11) $N$ ( 3 ) $(O(N^{3}(\log N)^{2}))$ [2] $O(N^{2}(\log N)^{2}))$ [2] $\bullet$ ( (9) ) $\cross$ 11:64 (9)

13 $\prime$ $\prime\backslash,$ $\blacksquare$ $\Delta$ $\bullet$ $6\bullet\bullet^{\bullet}$ $\bullet$ $\bullet^{\bullet}$ $\bullet$ $arrow$ $\bullet^{\bullet}$ $\bullet;f"+(x^{2}+1)f=0,$ : $Ce^{-x^{2}/1}$, : $*_{f0}^{1}=e^{-\}}.$ $\triangle:(x^{9}+12x^{5}-20x^{3}+15x)f" $ $+ (x^{8}+4x^{4}+4x^{2}. 1)f"$ $+(x^{7}+x^{5}-x^{3^{-}}-x)f $ $+(x^{6}+3x^{4}+3x^{2}+1)f=0,$ $C_{\frac{e^{-x^{2}}}{x+1}}$ :, $\frac{f(3)}{f\langle 0)}=\frac{e^{-9}}{10}.$ : $10^{4}$ $8*10^{2}\#\mathbb{R}e$ 6 $QOx\Delta_{V}Q_{\Delta}^{x}6\Delta:v$ $6_{A}^{B_{A}^{6}}{\}^{\#}o^{0}$ $0 9^{8}\bullet$ $\Delta$ $:f - \frac{4x^{10}+14x^{8}-4x^{7}-6x^{8}-39x^{4}+16x^{3}-14x^{2}+2x+3}{(x^{2}+1)^{4}}f$ $=0,$ $aee$ $Q$ $g$ $g^{v}$ $6v$ $Ce^{-\frac{x^{4}+x+1}{x^{\ell}+1}}$ :, : $*_{!0}^{2}=e^{-\#}.$ $:f + \frac{24x^{3}+36x^{2}-6}{(x^{2}+x+1)^{3}}f=0$ : $C_{\frac{1}{x+x+1}}$ $*_{f0}^{2}= : \frac{1}{7}$ $X$ ; $(_{dx}=-x+1)(_{\overline{d}x}d^{2}\nabla^{-x^{2}+5)f=0}$ : $C_{1}e^{-\not\simeq^{2}}+C_{2}(4x^{2}-2)e^{-*^{2}}$ : $f(2)=-3e^{-2}f(0)+4e\s f(1)$ $10^{0}$ $M_{10^{10}}\#\Xi\perp$ $10^{2} 10^{3} 10^{4}$ $N+1$ $v\s^{h^{\delta}}\#_{\bullet}$ $9_{\bullet}ve^{Q}v\bullet^{\bullet}$ : $\bullet$ $vv_{6}^{v}$ $\cross$ 5 64 $N$ 3 ( 3 5 ) $\sim$ $10^{5}10^{2}$ $v$ $v^{v_{b}}$ $\theta^{q}$ $\bullet^{\bullet^{\bullet}}$ $10^{3}$ $\vee\epsilon^{9}$ $10^{4}$ $N+1$ $\bullet\triangle$ 12: $\cross$

$145 6 $P(x, \frac{d}{dx})f=$ $0$ $P(x, \frac{d}{dx})f=g$ 1 $(g \frac{d}{dx}-g )P(x, \frac{d}{dx})f=0$ [1] [2] [3] $V\backslash (V\cap\ell^{2}(Z^{+})))$ $x arrow z=\frac{x-i}{x+i}$ ( $z=1$ 3 5 )$

14 145 6 $P(x, \frac{d}{dx})f=$ $0$ $P(x, \frac{d}{dx})f=g$ 1 $(g \frac{d}{dx}-g )P(x, \frac{d}{dx})f=0$ [1] [2] [3] $V\backslash (V\cap\ell^{2}(Z^{+})))$ $x arrow z=\frac{x-i}{x+i}$ ( $z=1$ 3 5 ) $\sim$ $[$5] [6] ( [7]) [1] F. Sakaguchi and M. Hayashi, General theory for integer-type algorithm for higher or- $B_{P}$ der differential equations, Numerical Functional Analysis and optimization, 32(5), (2011). [2] $id.$, Practical implementation and error bound of integer-type algorithm for higher-order differential equations, ibid., 32(12), (2011). $\psi_{k,\ddot{n}}(x)$ $\psi_{k,\ddot{m}}(x)\psi_{\kappa,\ddot{n}}(x)=\psi_{k+\kappa+1,\ddot{m}+\ddot{n}}(x)$ [3] $id.$, Differentiability of eigenfunctions of the (5) closures of differential operators with rational $(k+1$ $)$ 1 coefficient functions, arxiv: (2009, 2010).

15 146 [4] $id.$, Coherent states and annihilation-creation operators associated with the irreducible unitary representations of, $\epsilon u(1,1) $ Journal of Mathematical Physics, 43(5), (2002). [5] $id.$, Integer-type algorithm for eigenfunction/ eigenvalue problem of self-adjoint operators and its application to Schrodinger operators (in preparation). [6] 2010 (2010). [7] $-\rfloor$, (2012).

14 6. $P179$ 1984 r ( 2 $arrow$ $arrow$ F 7. $P181$ 2011 f ( 1 418[? [ 8. $P243$ ( $\cdot P260$ 2824 F ( 1 151? 10. $P292

$14 6. $P179$ 1984 r ( 2 $arrow$ $arrow$ F 7. $P181$ 2011 f ( 1 418[? [ 8. $P243$ ( $\cdot P260$ 2824 F ( 1 151? 10. $P292$ 1130 2000 13-28 13 USJC (Yasukuni Shimoura I. [ ]. ( 56 1. 78 $0753$ [ ( 1 352[ 2. 78 $0754$ [ ( 1 348 3. 88 $0880$ F ( 3 422 4. 93 $0942$ 1 ( ( 1 5. $P121$ 1281 F ( 1 278 [ 14 6. $P179$ 1984 r ( 2 $arrow$