Abstract
Let Q be a 1-dimensional Schroedinger operator with spectrum bounded from - infinity. By addition I mean a map of the form Q ..-->.. Q' = Q - 2D/sup 2/1g e with Qe = lambdae, lambda to the left of spec Q, and either integral/sub - infinity//sup 0/ e/sup 2/ or integral/sub 0//sup infinity/ e/sup 2/ 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 x 2 spectral weight dF by fixing (a) the measure class of dF, and (b) the value of ..sqrt..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).
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