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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

$\left(1+acos2t\right){y}^{\text{'}\text{'}}+b\left(sin2t\right){y}^{\text{'}}+\left(\lambda +dcos2t\right)y=0,\phantom{\rule{2.em}{0ex}}\left(*\right)$

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 ${\sigma }_{n}=4{n}^{2}$, ${\tau }_{n}=Q\left(-n\right)$, ${\rho }_{n}=Q\left(n-1\right)$ for $n>1$, where $Q\left(z\right)=2a{z}^{2}-bz-d/2$, and ${M}_{n}$ be the $n×n$ tridiagonal matrix ${M}_{n}=\left(\begin{array}{ccccc}{\sigma }_{0}& {\tau }_{1}& 0& \cdots & 0\\ {\rho }_{1}& {\sigma }_{1}& {\tau }_{2}& \cdots & 0\\ 0& {\rho }_{2}& {\sigma }_{2}& \cdots & 0\\ \cdots & \cdots & \cdots & \cdots & \cdots \end{array}\right)$. Consider the polynomials ${p}_{n}\left(\lambda \right)=det\left(\lambda -{M}_{n}\right)$. Under some assumptions, the sequence $\left\{{p}_{n}\right\}$ is orthogonal in some sense (Theorem 1). For the polynomials ${p}_{n}\left(\lambda \right)$, two types of results are obtained. First, if ${\lambda }_{n,k}$, $k=1,2,\cdots ,n$, denote the zeros of ${p}_{n}\left(\lambda \right)$ ($\Re \left({\lambda }_{n,1}\right)\le \cdots \le \Re \left({\lambda }_{n,n}\right)$), then the sequence ${\lambda }_{n,k}$ converges to ${\lambda }_{k}$ – the $k$th eigenvalue of (*) (${\lambda }_{1}<{\lambda }_{2}<\cdots$), 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 Eigenvalue problems for ODE (numerical methods) 47A75 Eigenvalue problems (linear operators) 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators 33C47 Other special orthogonal polynomials and functions 33E10 Lamé, Mathieu, and spheroidal wave functions 34M55 Painlevé and other special equations; classification, hierarchies 65L70 Error bounds (numerical methods for ODE)
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