On the symplectic phase space of KdV

TL;DR

This paper extends the Birkhoff map for KdV to interpolate between different Sobolev spaces, enabling a symplectic phase space description at $H^{1/2}_0$, and relates potential regularity to spectral gap decay.

Contribution

It demonstrates the interpolation of the Birkhoff map for KdV between $H^{-1}_0$ and $L^2_0$, and characterizes potential regularity via spectral gap decay.

Findings

Birkhoff map can be interpolated between $H^{-1}_0$ and $L^2_0$

Symplectic phase space $H^{1/2}_0$ described in Birkhoff coordinates

Potential regularity characterized by spectral gap decay

Abstract

We prove that the Birkhoff map $\Om$ for KdV constructed on $H^{-1}_0(\T)$ can be interpolated between $H^{-1}_0(\T)$ and $L^2_0(\T)$. In particular, the symplectic phase space $H^{1/2}_0(\T)$ can be described in terms of Birkhoff coordinates. As an application, we characterize the regularity of a potential $q\in H^{-1}(\T)$ in terms of the decay of the gap lengths of the periodic spectrum of Hill's operator on the interval $[0,2]$.