×

On Hill’s equation with a discontinuous coefficient. (English) Zbl 1027.34032

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.

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
PDFBibTeX XMLCite
Full Text: DOI EuDML