In this paper we consider periodic Dirac operators with skew-adjoint potentials in a large class of weighted Sobolev spaces. We characterize the smoothness of such potentials by asymptotic properties of the periodic spectrum of the corresponding Dirac operators.

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Abstract. We consider an inverse spectral problem for a class of singular AKNS operators H a , a ∈ N with an explicit singularity. We construct for each a ∈ N, a standard map λ a × κ a with spectral data λ a and some norming constant κ a . For a = 0, λ a × κ a was known to be a local coordinate system on L 2Using adapted transformation operators, we extend this result to any non-negative integer a, give a description of isospectral sets and we obtain a Borg-Levinson type theorem.

Abstract. We prove that the Birkhoff map Ω 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