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

TL;DR

This paper studies the spectral properties of the periodic Schrödinger operator under the invariant measure of the periodic KdV equation, revealing sampling sequences, concentration inequalities, and spectral curve structures.

Contribution

It introduces new spectral analysis results for Schrödinger operators with KdV-invariant measures, including sampling sequences and concentration properties of eigenvalues.

Findings

Sampling sequences form a Riesz basis in Paley-Wiener space.

Eigenvalue fluctuations satisfy Gaussian concentration inequalities.

Spectral curve divisors relate to the spectral measure distribution.

Abstract

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.