On the symplectic phase space of KdV

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 $[0,2]$