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Zbl 0891.34023
Becker, Peter A.
Normalization integrals of orthogonal Heun functions.
(English)
[J] J. Math. Phys. 38, No.7, 3692-3699 (1997). ISSN 0022-2488; ISSN 1089-7658/e

Any second-order Fuchsian equation with four singular points is equivalent to Heun's equation $${d^2y\over dx^2}+ \Biggl({\gamma\over x}+ {\delta\over x-1}+{\varepsilon\over x-a}\Biggr) {dy\over dx}+ {\alpha\beta x-\lambda\over x(x-1)(x- a)} y=0,\tag1$$ represented by the Riemann $P$-symbol $$P\left\{\matrix 0 & 1 & a & \infty\\ 0 & 0 & 0 & \alpha & x\\ 1-\gamma & 1-\delta & 1-\varepsilon & \beta\endmatrix\right\},$$ $\lambda$ is a so-called accessory parameter. Suppose $a\not\in [0,1]$. We consider four classes of boundary value problems on $[0,1]$, $\lambda$ being an eigenvalue.\par\hskip17mm I:\quad (1), $y$ is smooth at $0$, $y$ is smooth at $1$.\par\hskip17mm $\cdots$\par\hskip17mm IV:\quad (1), $x^{\gamma- 1}y$ is smooth at $0$, $(1-x)^{\delta-1}y$ is smooth at $1$.\par Let us consider, for example, class I. Let $y_0(\lambda, x)$ be the solution of (1) satisfying $y_0(\lambda, 0)= 1$, and $y_1(\lambda, x)$ be the solution of (1) satisfying $y_1(\lambda, 1)= 1$.\par Problem. For an eigenvalue $\lambda_n$ of the BVP I, estimate the norm $$N_n:= \int^1_0\omega(x)[H_n(x)]^2 dx,$$ where $H_n(x):= y_0(\lambda_n, x)$, $\omega= x^{\gamma- 1}(x- 1)^{\delta-1}(x- a)^{\varepsilon- 1}$.\par Theorem. $$N_n= -p(x) {\partial W\over\partial\lambda} (\lambda_n, x) {y_0(\lambda_n, x)\over y_1(\lambda_n, x)},$$ where $p(x)= x^\gamma(x- 1)^\delta(x- a)^\varepsilon$, and $W$ is the Wronskian of $y_0$ and $y_1$.\par Note that $p(\partial w/\partial\lambda)$ and $y_0/y_1$ are independent of $x$. These quantities are already used during the evaluation-algorithm of the eigenvalue $\lambda_n$. Thus the theorem tells that the evaluation of the norm $N_n$ can be obtained as a by-product of the search for the eigenvalues, and so that this formula greatly improves the efficiency of numerical procedures involving Heun functions.
[M.Yoshida (Fukuoka)]
MSC 2000:
*34B15 Nonlinear boundary value problems of ODE
34M99 Differential equations in the complex domain
34B27 Green functions
33E30 Functions coming from diff., difference and integral equations
65L10 Boundary value problems for ODE (numerical methods)
65L15 Eigenvalue problems for ODE (numerical methods)

Keywords: second-order Fuchsian equation; singular points; Heun's equation; boundary value problems; eigenvalues; Heun functions

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