Hill's Spectral Curves and the Invariant Measure of the Periodic KdV Equation

2014/09/30 by Blower, Gordon, Brett, Caroline, Doust, Ian
#FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1409.8494

Abstract

This paper analyses the periodic spectrum of Schrödinger's equation -f''+qf=λf when the potential is real, periodic, random and subject to the invariant measure ν_N^β of the periodic KdV equation. This ν_N^β is the modified canonical ensemble, as given by Bourgain (Comm. Math. Phys. 166 (1994), 1--26), and ν_N^β satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues (λ_n). For β, N>0 small, there exists a set of positive ν_N^β measure such that (± √2(λ_2n+λ_2n-1))_n=0^∞ gives a sampling sequence for Paley--Wiener space PW(π) and the reproducing kernels give a Riesz basis. Let (μ_j)_j=1^∞ be the tied spectrum; then (2√(μ_j)-j) belongs to a Hilbert cube in ℓ² and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence (√(μ_j))_j=1^∞ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics ∑_j g(√λ_2j) with test function g∈ PW(π) satisfy Gaussian concentration inequalities.

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