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

Abstract

This paper analyses the periodic spectrum of Schrödinger's equation $-f^{\prime \prime }+qf=\hat{I}»f$ when the potential is real, periodic, random and subject to the invariant measure $\hat{I}{½}_N^2$ of the periodic KdV equation. This $\hat{I}{½}_N^2$ is the modified canonical ensemble, as given by Bourgain ({Comm. Math. Phys.} {166} (1994), 1--26), and $\hat{I}{½}_N^2$ satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues $(\hat{I}{»}_n)$. For ${\hat{I}}^2,N>0$ small, there exists a set of positive $\hat{I}{½}_N^2$ measure such that $(\pm \sqrt{2(\hat{I}{»}_{2n}+\hat{I}{»}_{2n-1})}{)}_{n=0}^{\infty }$ gives a sampling sequence for Paley--Wiener space $PW(\ddot{I}€)$ and the reproducing kernels give a Riesz basis. Let $(\hat{I}{¼}_j{)}_{j=1}^{\infty }$ be the tied spectrum; then $(2\sqrt{\hat{I}{¼}_j}-j)$ 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 $(\sqrt{\hat{I}{¼}_j}{)}_{j=1}^{\infty }$ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics ${\sum }_{j}g(\sqrt{\hat{I}{»}_{2j}})$ with test function $g\in PW(\ddot{I}€)$ satisfy Gaussian concentration inequalities.

34 pages