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Singular soliton operators and indefinite metrics

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Abstract

We consider singular real second order 1D Schrödinger operators such that all local solutions to the eigenvalue problems are x-meromorphic for all λ. All algebrogeometrical potentials (i.e. “singular finite-gap” and “singular solitons”) satisfy to this condition. A Spectral Theory is constructed for the periodic and rapidly decreasing potentials in the classes of functionswith singularities: The corresponding operators are symmetric with respect to some natural indefinite inner product as it was discovered by the present authors. It has a finite number of negative squares in the both (periodic and rapidly decreasing) cases. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform on Riemann Surfaces with good multiplicative properties (the R-Fourier Transform) is a partial case of this theory. The potential has a pole in this case at x = 0 with asymptotics u ∼ g(g + 1)/x 2. Here g is the genus of spectral curve.

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Authors and Affiliations

  1. Landau Institute for Theoretical Physics, Moscow, Russia

    Petr G. Grinevich & Sergey P. Novikov

  2. University of Maryland, College Park, USA

    Sergey P. Novikov

Authors
  1. Petr G. Grinevich
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  2. Sergey P. Novikov
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Correspondence to Sergey P. Novikov.

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Grinevich, P.G., Novikov, S.P. Singular soliton operators and indefinite metrics. Bull Braz Math Soc, New Series 44, 809–840 (2013). https://doi.org/10.1007/s00574-013-0035-5

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  • DOI: https://doi.org/10.1007/s00574-013-0035-5

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