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

Abstract

Consider the Hill's operator $Q=-d^2/d{x}^2+q(x)$ 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.

Typos corrected