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

Abstract

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 $f(μ)$ the probability density function of $-λ_0(q)$, the negative of the ground state eigenvalue, at $μ$. We describe the detailed asymptotics of this density as $μ\to +\infty$. 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.