Asymptotics of instability zones of Hill operators with a two term potential
The authors consider the Hill operator \[ Ly:=-y''+[a\text{cos}2x+b\text{cos}4x]y, \] with \(a,b\in\mathbb{R}\), \(ab\neq 0\), and establish some estimates on the lengths \(\gamma_{n}\) corresponding to the intervals of instability zones or spectral gaps of the above operator. Using the parametrization \(a^{2}+8bt^{2}=0\), \(b=-2\alpha ^ {2}\) , \(a=-\alpha t\), they obtain the asymptotic behavior of the gaps \(\gamma_{n}(\alpha)\) both for \(\alpha \rightarrow 0\) and \(n\rightarrow\infty\).
scientific article; zbMATH DE number 2118262
scientific article