Hill's spectral curves and the invariant measure of the periodic KdV equation
Abstract This paper analyses the periodic spectrum of Schrodinger's equation − f ″ + q f = λ f 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 (1994) [7] , 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 2 n + λ 2 n − 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 l 2 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 2 j ) with test function g ∈ PW ( π ) satisfy Gaussian concentration inequalities.