A convexity theorem in the scattering theory for the Dirac operator

Abstract

The Dirac operator enters into zero curvature representation for the cubic nonlinear Schrodinger equation. We introduce and study a conformal map from the upper half-plane of the spectral parameter of the Dirac operator into itself. The action variables turn out to be limiting boundary values of the imagi- nary part of this map. We describe the image of the momentum map (convexity theorem) in the simplest case of a potential from the Schwartz class. We apply this description to the invariant manifolds for the nonlinear Schrodinger equation.