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Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. (English) Zbl 1149.65062

Even solutions of the Ince equation
\[ (1+a\cos2t)y''+b(\sin2t)y'+(\lambda+d\cos2t)y=0,\tag{*} \]
where \(a,b,c\) are real with \(| a| <1\), and \(\lambda\) is regarded as a spectral parameter are considered. This equation contains the Mathieu equation, the Whittaker-Hill equation, and the Lamé equation. Let \(\sigma_n=4n^2\), \(\tau_n=Q(-n)\), \(\rho_n=Q(n-1)\) for \(n>1\), where \(Q(z)=2az^2-bz-d/2\), and \(M_n\) be the \(n\times n\) tridiagonal matrix \(M_n=\left(\begin{smallmatrix} \sigma_0&\tau_1&0&\dots&0\\ \rho_1&\sigma_1&\tau_2&\dots&0\\ 0&\rho_2&\sigma_2&\dots&0\\ \hdotsfor5\end{smallmatrix}\right)\). Consider the polynomials \(p_n (\lambda)=\det(\lambda-M_n)\). Under some assumptions, the sequence \(\{p_n\}\) is orthogonal in some sense (Theorem 1). For the polynomials \(p_n(\lambda)\), two types of results are obtained. First, if \(\lambda_{n,k}\), \(k=1,2,\dots,n\), denote the zeros of \(p_n(\lambda)\) (\(\operatorname{Re}(\lambda_{n,1}) \leq\dotsb\leq \operatorname{Re}(\lambda_{n,n})\)), then the sequence \(\lambda_{n,k}\) converges to \(\lambda_k\) – the \(k\)th eigenvalue of (*) (\(\lambda_1< \lambda_2< \dots\)), as \(n\to\infty\) (Theorem 2). Second, the interlacing properties of the zeros \(\lambda_{n,k}\) are discussed (Theorem 4). The lower and upper bounds for the eigenvalues of the Mathieu equation (Theorem 5), the Whittaker–Hill equation (Theorem 6), and the Lamé equation (Theorems 7,8) are given.

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
47A75 Eigenvalue problems for linear operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
33C47 Other special orthogonal polynomials and functions
33E10 Lamé, Mathieu, and spheroidal wave functions
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
65L70 Error bounds for numerical methods for ordinary differential equations
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References:

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