Weinstein, Michael I.; Keller, Joseph B. Hill’s equation with a large potential. (English) Zbl 0578.34038 SIAM J. Appl. Math. 45, 200-214 (1985). We obtain asymptotic expansions, for large values of the parameter \(\lambda\), of the stability boundaries, the stability band widths, the Floquet multipliers and the solutions of Hill’s equation \([-d^ 2/dx^ 2+\lambda^ 2q(x)]u=Eu.\) The potential q(x) is assumed to be periodic and to have a unique global minimum within each period, at which \(q''>0\). The results for the stability band widths show that they decay exponentially with \(\lambda\) as \(\lambda\) increases. These results generalize those for symmetric potentials due to Harrell, and that for the Mathieu equation due to Meixner and Schäfke. Results on the behavior of the stability intervals for large \(\lambda\) and E have been obtained by the authors in ”Asymptotic behavior of stability intervals for Hill’s equation” (to appear). Cited in 14 Documents MSC: 34E99 Asymptotic theory for ordinary differential equations 34D20 Stability of solutions to ordinary differential equations Keywords:asymptotic expansions; stability boundaries; stability band widths; Floquet multipliers; Hill’s equation; potential; Mathieu equation PDFBibTeX XMLCite \textit{M. I. Weinstein} and \textit{J. B. Keller}, SIAM J. Appl. Math. 45, 200--214 (1985; Zbl 0578.34038) Full Text: DOI Digital Library of Mathematical Functions: §29.7(ii) Lamé Functions ‣ §29.7 Asymptotic Expansions ‣ Lamé Functions ‣ Chapter 29 Lamé Functions