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

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 $(\pm \sqrt{2(λ_{2n}+λ_{2n-1})})_{n=0}^\infty$ gives a sampling sequence for Paley--Wiener space $PW(π)$ and the reproducing kernels give a Riesz basis. Let $(μ_j)_{j=1}^\infty$ be the tied spectrum; then $(2\sqrt{μ_j}-j)$ belongs to a Hilbert cube in $\ell^2$ and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence $(\sqrt{μ_j})_{j=1}^\infty$ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics $\sum_j g(\sqrt{λ_{2j}})$ with test function $g\in PW(π)$ satisfy Gaussian concentration inequalities.