On the shape of the ground state eigenvalue density of a random Hill's equation

Consider the Hill's operator Q = − d 2 / dx 2 + q ( x ) in which q ( x ), 0 ≤ x ≤ 1, is a white noise. Denote by f (μ) the probability density function of −λ 0 ( q ), the negative of the ground state eigenvalue, at μ. We prove the detailed asymptotics$$ f(\mu) = {4 \over 3\pi} \mu \, {\rm exp} \left[-{8 \over 3}\mu^{8/3} - {1 \over 2}\mu^{1/2} \right] (1 + o(1)) $$ as μ → + ∞. This result is based on a precise Laplace analysis of a functional integral representation for f (μ) established by S. Cambronero and H. P. McKean in 5. © 2005 Wiley Periodicals, Inc.