On the instability intervals of the Mathieu--Hill operator
The author obtains the precise asymptotic formulas for the widths \(\gamma _{k}\) of the instability intervals of the operator \(L\) considered as the Mathieu--Hill operator defined by \(Ly=-y''+(2h\cos 2x)y,\) \(-\infty <x<+\infty \) in \(L_{2}(\mathbb R)\), where \(h\in\mathbb R\setminus\{0\}\). The formula states the isolated terms of arbitrary number in the asymptotics of the sequence \(\gamma_{k}\) for large \(k\) and verifies the results of \textit{E.\,M.\thinspace Harrell} [in: Contributions to analysis and geometry, Suppl.\ Am.\ J.\ Math., 139--150 (1981; Zbl 0544.34014)] and \textit{J.\,Avron} and \textit{B.\,Simon} [Ann.\ Phys.\ 134, 76--84 (1981; Zbl 0464.34020)].
scientific article; zbMATH DE number 5544972
scientific article