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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

d 2 y dx 2 +γ x + δ x-1 + ε x-ady dx+αβx-λ x(x-1)(x-a)y=0,(1)

represented by the Riemann P-symbol

P01a000αx1-γ1-δ1-εβ,

λ is a so-called accessory parameter. Suppose a[0,1]. We consider four classes of boundary value problems on [0,1], λ being an eigenvalue.

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

            

            IV:   (1), x γ-1 y is smooth at 0, (1-x) δ-1 y is smooth at 1.

Let us consider, for example, class I. Let y 0 (λ,x) be the solution of (1) satisfying y 0 (λ,0)=1, and y 1 (λ,x) be the solution of (1) satisfying y 1 (λ,1)=1.

Problem. For an eigenvalue λ n of the BVP I, estimate the norm

N n := 0 1 ω(x)[H n (x)] 2 dx,

where H n (x):=y 0 (λ n ,x), ω=x γ-1 (x-1) δ-1 (x-a) ε-1 .

Theorem.

N n =-p(x)W λ(λ n ,x)y 0 (λ n ,x) y 1 (λ n ,x),

where p(x)=x γ (x-1) δ (x-a) ε , and W is the Wronskian of y 0 and y 1 .

Note that p(w/λ) and y 0 /y 1 are independent of x. These quantities are already used during the evaluation-algorithm of the eigenvalue λ 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:
34B15Nonlinear boundary value problems for ODE
34M99Differential equations in the complex domain
34B27Green functions
33E30Functions coming from differential, difference and integral equations
65L10Boundary value problems for ODE (numerical methods)
65L15Eigenvalue problems for ODE (numerical methods)