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Hill’s equation with a large potential. (English) Zbl 0578.34038

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 $\left[-{d}^{2}/d{x}^{2}+{\lambda }^{2}q\left(x\right)\right]u=Eu·$ The potential q(x) is assumed to be periodic and to have a unique global minimum within each period, at which ${q}^{\text{'}\text{'}}>0$. 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).

MSC:
 34E99 Asymptotic theory of ODE 34D20 Stability of ODE