Hill's Equation with a Large Potential

摘要

We obtained 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(squared)/(dx-squared)) + (lambda - squared)q(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 Schaefke. (Reprints)