  Consider the Hill's operator $Q = - d^2/dx^2 + q(x)$ in which $q(x)$, $0 \le
x \le 1$, is a White Noise. Denote by $f(\mu)$ the probability density function
of $-\lambda_0(q)$, the negative of the ground state eigenvalue, at $\mu$. We
describe the detailed asymptotics of this density as $\mu \to +\infty$. This
result is based on a precise Laplace analysis of a functional integral
representation for $f(\mu)$ established by S. Cambronero and H.P. McKean.
