Hill’s Equation with a Large Potential

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 \[ \left[ - \frac{d^2 }{dx^2 } + \lambda ^2 q ( x t) \right] u = Eu. \]The potential $q ( x )$ is assumed to be periodic and to have a unique global minimum within each period, at which $q'' > 0$. 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.