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Zbl 0578.34038
Weinstein, Michael I.; Keller, Joseph B.
Hill's equation with a large potential. (English)
[J] SIAM J. Appl. Math. 45, 200-214 (1985). ISSN 0036-1399; ISSN 1095-712X/e

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 $[-d\sp 2/dx\sp 2+\lambda\sp 2q(x)]u=Eu.$ 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. \par 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).

MSC 2000:: *34E99 Asymptotic theory of ODE
34D20 Lyapunov stability of ODE

Keywords: asymptotic expansions; stability boundaries; stability band widths; Floquet multipliers; Hill's equation; potential; Mathieu equation

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