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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 $\vert a\vert <1$, 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 $M_n=\left(\smallmatrix \sigma_0&\tau_1&0&\dots&0\\ \rho_1&\sigma_1&\tau_2&\dots&0\\ 0&\rho_2&\sigma_2&\dots&0\\ \hdotsfor5\endsmallmatrix\right)$. 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 $p_n(\lambda)$ ($\Re(\lambda_{n,1}) \le\dotsb\le \Re(\lambda_{n,n})$), 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.
65L15Eigenvalue problems for ODE (numerical methods)
47A75Eigenvalue problems (linear operators)
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
33C47Other special orthogonal polynomials and functions
33E10Lamé, Mathieu, and spheroidal wave functions
34M55Painlevé and other special equations; classification, hierarchies
65L70Error bounds (numerical methods for ODE)
Full Text: DOI
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