Hill's spectral curves and the invariant measure of the periodic KdV equation

This paper analyses the periodic spectrum of Schr\"odinger's equation $-f''+qf=\lambda f$ when the potential is real, periodic, random and subject to the invariant measure $\nu_N^\beta$ of the periodic KdV equation. This $\nu_N^\beta$ is the modified canonical ensemble, as given by Bourgain ({Comm. Math. Phys.} {166} (1994), 1--26), and $\nu_N^\beta$ satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues $(\lambda_n)$. For $\beta, N>0$ small, there exists a set of positive $\nu_N^\beta$ measure such that $(\pm \sqrt{2(\lambda_{2n}+\lambda_{2n-1})})_{n=0}^\infty$ gives a sampling sequence for Paley--Wiener space $PW(\pi )$ and the reproducing kernels give a Riesz basis. Let $(\mu_j)_{j=1}^\infty$ be the tied spectrum; then $(2\sqrt{\mu_j}-j)$ 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 $(\sqrt{\mu_j})_{j=1}^\infty$ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics $\sum_j g(\sqrt{\lambda_{2j}})$ with test function $g\in PW(\pi)$ satisfy Gaussian concentration inequalities.\pa