Karaca, Ilkay Yaslan On Hill’s equation with a discontinuous coefficient. (English) Zbl 1027.34032 Int. J. Math. Math. Sci. 2003, No. 25, 1599-1614 (2003). Given the following second-order equation \[ y''(x) + \{ \lambda r(x) - q(x)\}y(x) = 0, \quad -\infty < x < \infty,\tag{1} \] where \(q\) and \(r\) are periodic functions with period \(a\), \(q\) is piecewise continuous, \(r''\) is piecewise continuous in \((0,b)\) and \((b,a)\), where \(0 < b < a\) and \(r(x) \geq r_0 > 0\). The author studies the asymptotic formula of the lengths of the instability intervals for (1) using a method based on Rouche’s theorem on roots of analytic functions. Reviewer: Emil Minchev (Chiba) MSC: 34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) 34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators 47E05 General theory of ordinary differential operators Keywords:Hill equation; discontinuous coefficients; instability intervals PDFBibTeX XMLCite \textit{I. Y. Karaca}, Int. J. Math. Math. Sci. 2003, No. 25, 1599--1614 (2003; Zbl 1027.34032) Full Text: DOI EuDML