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Hill's Equation with Random Forcing Parameters: The Limit of Delta Function Barriers

This paper considers random Hill's equations in the limit where the periodic forcing function becomes a Dirac delta function. For this class of equations, the forcing strength $q_k$, the oscillation frequency $\af_k$, and the period are allowed to vary from cycle to cycle. Such equations arise in astrophysical orbital problems in extended mass distributions, in the reheating problem for inflationary cosmologies, and in periodic Schr{ö}dinger equations. The growth rates for solutions to the periodic differential equation can be described by a matrix transformation, where the matrix elements vary from cycle to cycle. Working in the delta function limit, this paper addresses several coupled issues: We find the growth rates for the $2 \times 2$ matrices that describe the solutions. This analysis is carried out in the limiting regimes of both large $q_k \gg 1$ and small $q_k \ll 1$ forcing strength parameters. For the latter case, we present an alternate treatment of the dynamics in terms of a Fokker-Planck equation, which allows for a comparison of the two approaches. Finally, we elucidate the relationship between the fundamental parameters $(\af_k,q_k)$ appearing in the stochastic differential equation and the matrix elements that specify the corresponding discrete map. This work provides analytic -- and accurate -- expressions for the growth rates of these stochastic differential equations in both the $q_k \gg1 $ and the $q_k \ll 1$ limits.