Geometry of KDV (2): Three examples

H. P. McKean

Research output: Contribution to journalArticle

Abstract

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).

Original languageEnglish (US)
Pages (from-to)1115-1143
Number of pages29
JournalJournal of Statistical Physics
Volume46
Issue number5-6
DOIs
StatePublished - Mar 1 1987

Keywords

  • KDV manifold
  • Schrödinger operator
  • addition
  • additive class
  • unimodular isospectral class

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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