Volkmer, Hans Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. (English) Zbl 1149.65062 J. Comput. Appl. Math. 213, No. 2, 488-500 (2008). 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. Reviewer: Valery V. Karachik (Chelyabinsk) Cited in 4 Documents 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 Keywords:Ince equation; Lamé equation; Whittaker-Hill equation; tridiagonal operators; orthogonal polynomials; eigenvalues; zeros; error bounds; convergence PDFBibTeX XMLCite \textit{H. Volkmer}, J. Comput. Appl. Math. 213, No. 2, 488--500 (2008; Zbl 1149.65062) Full Text: DOI Digital Library of Mathematical Functions: item (f) ‣ §28.34(ii) Eigenvalues ‣ §28.34 Methods of Computation ‣ Computation ‣ Chapter 28 Mathieu Functions and Hill’s Equation §28.34(ii) Eigenvalues ‣ §28.34 Methods of Computation ‣ Computation ‣ Chapter 28 Mathieu Functions and Hill’s Equation §29.20(i) Lamé Functions ‣ §29.20 Methods of Computation ‣ Computation ‣ Chapter 29 Lamé Functions References: [1] Arscott, F. M., Periodic Differential Equations (1964), Pergamon Press, MacMillan Company: Pergamon Press, MacMillan Company New York · Zbl 0121.29903 [2] J. Berezanskii, Expansions in Eigenfunctions of Self-adjoint Operators, Translations of Mathematical Monographs, vol. 17, American Mathematical Society, Providence, RI, 1968.; J. Berezanskii, Expansions in Eigenfunctions of Self-adjoint Operators, Translations of Mathematical Monographs, vol. 17, American Mathematical Society, Providence, RI, 1968. · Zbl 0157.16601 [3] Bognar, J., Indefinite Inner Product Spaces (1974), Springer: Springer New York · Zbl 0277.47024 [4] Borwein, P.; Erdélyi, T., Polynomials and Polynomials Inequalities (1995), Springer: Springer New York · Zbl 0840.26002 [5] Chihara, T., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach: Gordon and Breach New York · Zbl 0389.33008 [6] Eastham, M., The Spectral Theory of Periodic Differential Equations (1973), Scottish Academic Press: Scottish Academic Press Edinburgh, London · Zbl 0287.34016 [7] Gohberg, I.; Lancaster, P.; Rodman, L., Matrices and Indefinite Scalar Products (1983), Birkhäuser: Birkhäuser Basel · Zbl 0513.15006 [8] Horn, R.; Johnson, C., Matrix Analysis (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0576.15001 [9] Ifantis, E. K.; Panagopoulos, P. N., Limit points of eigenvalues of truncated tridiagonal operators, J. Comput. Appl. Math., 133, 413-422 (2001) · Zbl 0994.47004 [10] J.K.M. Jansen, Simple-Periodic and Non-Periodic Lamé Functions, Mathematical Centre Tract, vol. 72, Mathematisch Centrum, Amsterdam, 1977.; J.K.M. Jansen, Simple-Periodic and Non-Periodic Lamé Functions, Mathematical Centre Tract, vol. 72, Mathematisch Centrum, Amsterdam, 1977. [11] Kato, T., Perturbation Theory for Linear Operators (1980), Springer: Springer Berlin, Heidelberg, New York [12] Magnus, W.; Winkler, S., Hill’s Equation (1966), Wiley: Wiley New York · Zbl 0158.09604 [13] Volkmer, H., Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation, Constr. Approx., 20, 39-54 (2004) · Zbl 1063.33028 [14] Whittaker, E. T., On a class of differential equations whose solutions satisfy integral equations, Proc. Edinburgh Math. Soc., 33, 14-33 (1915) · JFM 45.1299.03 [15] Weinstein, A.; Stenger, W., Methods of Intermediate Problems for Eigenvalues (1972), Academic Press: Academic Press New York · Zbl 0291.49034 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.