Geometry of KDV (2): Three examples

Let Q be a 1-dimensional Schrödinger operator with spectrum bounded from -∞. By addition I mean a map of the form Q→Q′=Q-2 D² lg e with Qe=λe, λ to the left of spec Q, and either ∫_-∞⁰e² or ∫₀^∞ e² finite. The additive class of Q is obtained by composite addition and a subsequent closure; it is a substitute for the KDV invariant manifold even if the individual KDV flows have no existence. KDV(1) = McKean [1987] suggested that the additive class of Q is the same as its unimodular spectral class defined in terms of the 2×2 spectral weight dF by fixing (a) the measure class of dF, and (b) the value of √det dF. The present paper verifies this for (1) the scattering case, (2) Hill's case, and (3) when the additive class is finite-dimensional (Neumann case).