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On the double points of a Mathieu equation. (English) Zbl 0944.34063

The authors discuss the eigenvalues for the Mathieu equation

$\frac{{d}^{2}y}{d{x}^{2}}+\left(\lambda -2qcos2x\right)y=0\phantom{\rule{2.em}{0ex}}\left(1\right)$

with the boundary conditions $y\left(0\right)=y\left(\frac{\pi }{2}\right)=0$.

The interest of the authors is in the case when two consecutive eigenvalues merge and become equal for some values of parameter $q$. This pair of merging points is called a double point of (1) for that value of $q$.

Here, the authors discuss only real double points. The eigenvalues can be regarded as functions of the parameter $q$. The authors find values of $q$ when adjacent eigenvalues of the same type become equal yielding double points of (1). The problem reduces to an equivalent eigenvalue problem of an infinite linear algebraic system with an infinite tridiagonal matrix. A method is developed to locate the first double eigenvalue to any required degree of accuracy when $q$ is an imaginary number. Computational results are given to illustrate the theory for the first double eigenvalue. Numerical results are given for some subsequent double points.

Reviewer: D.M.Bors (Iaşi)
##### MSC:
 34L05 General spectral theory for OD operators 34B30 Special ODE (Mathieu, Hill, Bessel, etc.) 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators 34L16 Numerical approximation of eigenvalues and of other parts of the spectrum