On the symplectic phase space of KdV

We prove that the Birkhoff map $\text{Om}$ for KdV constructed on $H_0^{-1}(\text{T})$ can be interpolated between $H_0^{-1}(\text{T})$ and $L_0^2(\text{T})$. In particular, the symplectic phase space $H_0^{1\mathrm{/}2}(\text{T})$ can be described in terms of Birkhoff coordinates. As an application, we characterize the regularity of a potential $q\in H^{-1}(\text{T})$ in terms of the decay of the gap lengths of the periodic spectrum of Hill’s operator on the interval $[0,2]$.