Abstract
LetQ be a 1-dimensional Schrödinger operator with spectrum bounded from −∞. Byaddition I mean a map of the formQ→Q′=Q−2D 2 lge withQe=λe, λ to the left of specQ, and either ∫ 0−∞ e 2 or ∫ ∞0 e2 finite. Theadditive class ofQ 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 ofQ is the same as itsunimodular spectral class defined in terms of the 2×2 spectral weightdF by fixing (a) the measure class ofdF, and (b) the value of √detdF. 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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This paper is dedicated to the memory of Mark Kac by a grateful student. Courant Institute of Mathematical Sciences, New York, New York.
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McKean, H.P. Geometry of KDV (2): Three examples. J Stat Phys 46, 1115–1143 (1987). https://doi.org/10.1007/BF01011159
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DOI: https://doi.org/10.1007/BF01011159