The ground state eigenvalue of Hill's equation with white noise potential

Abstract

The distribution of the ground state eigenvalue λ0(Q) of Hill's operator Q = −d2/dx2 + q(x) on the circle of perimeter 1 is expressed in two different ways in case the potential q is standard white noise. Let WN be the associated white noise measure, and let CBM be the measure for circular Brownian motion p(x), 0 ≤ x < 1, formed from the standard Brownian motion b(x), 0 ≤ x ≤ 1, starting at b(0) = a, by conditioning so that b(1) = a, and distributing this common level over the line according to the measure da. The connection is based upon the Ricatti correspondence q(x) = λ + p′ (x) + p2(x). The two versions of the distribution are (1) in which \documentclass{minimal}\begin{document}$\overline{p}$\end{document} is the mean value ∫ pdx, and (2) the left-hand side of (2) being the density for (1) and CBM0 the conditional circular Brownian measure on \documentclass{minimal}\begin{document}$\overline{p}$\end{document} = 0. (1) and (2) are related by the divergence theorem in function space as suggested by the recognition of the Jacobian factor the outward-pointing normal component of the vector field v(x) = ∂Δ(λ)/∂q(x), 0 ≤ x < 1, Δ being the Hill's discriminant for Q. The Ricatti correspondence prompts the idea that (1) and (2) are instances of the Cameron-Martin formula, but it is not so: The latter has to do with the initial value problem for Ricatti, but it is the periodic problem that figures here, so the proof must be done by hand, by finite-dimensional approximation. The adaptation of 1 and 2 to potentials of Ornstein-Uhlenbeck type is reported without details. © 1999 John Wiley & Sons, Inc.