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

Abstract

AbstractConsider the Hill's operator Q = −d2/dx2 + q(x) in which q(x), 0 ≤ x ≤ 1, is a white noise. Denote by f(μ) the probability density function of −λ0(q), the negative of the ground state eigenvalue, at μ. We prove the detailed asymptotics 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.