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On the shape of the ground state eigenvalue density of a random Hill's equation
Author(s) -
Cambronero Santiago,
Rider Brian,
Ramírez José
Publication year - 2006
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20104
Subject(s) - mathematics , eigenvalues and eigenvectors , ground state , laplace transform , operator (biology) , white noise , probability density function , spectrum (functional analysis) , mathematical analysis , function (biology) , representation (politics) , state (computer science) , mathematical physics , quantum mechanics , statistics , physics , algorithm , evolutionary biology , biology , biochemistry , chemistry , repressor , politics , political science , transcription factor , law , gene
Consider the Hill's operator Q = − d 2 / dx 2 + q ( x ) in which q ( x ), 0 ≤ x ≤ 1, is a white noise. Denote by f (μ) the probability density function of −λ 0 ( q ), the negative of the ground state eigenvalue, at μ. We prove the detailed asymptotics$$ f(\mu) = {4 \over 3\pi} \mu \, {\rm exp} \left[-{8 \over 3}\mu^{8/3} - {1 \over 2}\mu^{1/2} \right] (1 + o(1)) $$ 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.