### 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 D^{2} lg e with Qe=λe, λ to the left of spec Q, and either ∫_{-∞}^{0}e^{2} or ∫_{0}^{∞} e^{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 2×2 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 language | English (US) |
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Pages (from-to) | 1115-1143 |

Number of pages | 29 |

Journal | Journal of Statistical Physics |

Volume | 46 |

Issue number | 5-6 |

DOIs | |

State | Published - 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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## Cite this

*Journal of Statistical Physics*,

*46*(5-6), 1115-1143. https://doi.org/10.1007/BF01011159