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

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

2004-08-04

arxiv.org/abs/math/0408068v2

Mathematics

Probability

Typos corrected

Scientific paper

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.

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