On Hill's equation with a discontinuous coefficient (Q1811932)

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.