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

Santiago V. Cambronero, Henry P. Mckean

Research output: Contribution to journalArticle

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 (equation presented) in which p̄ is the mean value ∫10pdx, and (equation presented) the left-hand side of (2) being the density for (1) and CBM0 the conditional circular Brownian measure on p̄ = 0. (1) and (2) are related by the divergence theorem in function space as suggested by the recognition of the Jacobian factor ∫10e2∫x0p x ∫10e-2∫x0 as (-2) x the outward-pointing normal component - ∫10v(x)dx 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.

Original languageEnglish (US)
Pages (from-to)1277-1294
Number of pages18
JournalCommunications on Pure and Applied Mathematics
Volume52
Issue number10
StatePublished - Oct 1999

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Hill Equation
Brownian movement
White noise
Ground state
Ground State
Eigenvalue
Initial value problems
Brownian motion
Correspondence
Divergence theorem
Finite-dimensional Approximation
Periodic Problem
Perimeter
Discriminant
Conditioning
Mean Value
Function Space
Initial Value Problem
Vector Field
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ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

The ground state eigenvalue of Hill's equation with white noise potential. / Cambronero, Santiago V.; Mckean, Henry P.

In: Communications on Pure and Applied Mathematics, Vol. 52, No. 10, 10.1999, p. 1277-1294.

Research output: Contribution to journalArticle

Cambronero, Santiago V. ; Mckean, Henry P. / The ground state eigenvalue of Hill's equation with white noise potential. In: Communications on Pure and Applied Mathematics. 1999 ; Vol. 52, No. 10. pp. 1277-1294.
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