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On the growth of convergence radii for the eigenvalues of the Mathieu equation. (English) Zbl 0912.34027

The author reconsiders the classical Mathieu equation \[ y''+ (\mu+ 2\lambda\cos 2x) y=0 \] in the restricted case where it has nontrivial \(\pi\)-periodic or \(\pi\)-antiperiodic solutions. In the first case it holds \(y(x+\pi)= y(x)\) while in the second case \(y(x+ \pi)= -y(x)\), yielding \(\mu\) as a function of \(\lambda\), say \(\mu= \mu_0(\lambda)\) \((n= 1,2,\dots)\) (eigenvalues). The aim is to find a new, sharper estimation for the radius of convergence of the power series of \(\mu_n(\lambda)\) about the point \(\lambda= 0\). The result, which seems to be the best one, is as follows: \[ \lim_{n\to\infty}\inf {\rho_n\over n^2}\geq k k' K^2= 2.041834\dots\;. \] Here, \(\rho_n\) refers the radius of convergence of power series of \(\mu_0(\lambda)\) while \(K= K(k)\) denotes the complete elliptic integral of the first kind and \(k'=(1- k^2)^{1/2}\). As to the modulus \(k\), it is determined through the relation \(2E= K\), \(E\) being the corresponding complete elliptic integral of the second kind.

MSC:

34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
33E10 Lamé, Mathieu, and spheroidal wave functions
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