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Normalization integrals of orthogonal Heun functions. (English) Zbl 0891.34023

Any second-order Fuchsian equation with four singular points is equivalent to Heun’s equation

$\frac{{d}^{2}y}{d{x}^{2}}+\left(\frac{\gamma }{x}+\frac{\delta }{x-1}+\frac{\epsilon }{x-a}\right)\frac{dy}{dx}+\frac{\alpha \beta x-\lambda }{x\left(x-1\right)\left(x-a\right)}y=0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

represented by the Riemann $P$-symbol

$P\left\{\begin{array}{cccc}0& 1& a& \infty \\ 0& 0& 0& \alpha & x\\ 1-\gamma & 1-\delta & 1-\epsilon & \beta \end{array}\right\},$

$\lambda$ is a so-called accessory parameter. Suppose $a\notin \left[0,1\right]$. We consider four classes of boundary value problems on $\left[0,1\right]$, $\lambda$ being an eigenvalue.

I:   (1), $y$ is smooth at 0, $y$ is smooth at 1.

$\cdots$

IV:   (1), ${x}^{\gamma -1}y$ is smooth at 0, ${\left(1-x\right)}^{\delta -1}y$ is smooth at 1.

Let us consider, for example, class I. Let ${y}_{0}\left(\lambda ,x\right)$ be the solution of (1) satisfying ${y}_{0}\left(\lambda ,0\right)=1$, and ${y}_{1}\left(\lambda ,x\right)$ be the solution of (1) satisfying ${y}_{1}\left(\lambda ,1\right)=1$.

Problem. For an eigenvalue ${\lambda }_{n}$ of the BVP I, estimate the norm

${N}_{n}:={\int }_{0}^{1}\omega \left(x\right){\left[{H}_{n}\left(x\right)\right]}^{2}dx,$

where ${H}_{n}\left(x\right):={y}_{0}\left({\lambda }_{n},x\right)$, $\omega ={x}^{\gamma -1}{\left(x-1\right)}^{\delta -1}{\left(x-a\right)}^{\epsilon -1}$.

Theorem.

${N}_{n}=-p\left(x\right)\frac{\partial W}{\partial \lambda }\left({\lambda }_{n},x\right)\frac{{y}_{0}\left({\lambda }_{n},x\right)}{{y}_{1}\left({\lambda }_{n},x\right)},$

where $p\left(x\right)={x}^{\gamma }{\left(x-1\right)}^{\delta }{\left(x-a\right)}^{\epsilon }$, and $W$ is the Wronskian of ${y}_{0}$ and ${y}_{1}$.

Note that $p\left(\partial w/\partial \lambda \right)$ 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.

MSC:
 34B15 Nonlinear boundary value problems for ODE 34M99 Differential equations in the complex domain 34B27 Green functions 33E30 Functions coming from differential, difference and integral equations 65L10 Boundary value problems for ODE (numerical methods) 65L15 Eigenvalue problems for ODE (numerical methods)