Singular soliton operators and indefinite metrics

Authors: Petr G Grinevich Sergey P Novikov

Publish Date: 2013/12/14

Volume: 44, Issue: 4, Pages: 809-840

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Abstract

We consider singular real second order 1D Schrödinger operators such that all local solutions to the eigenvalue problems are xmeromorphic for all λ All algebrogeometrical potentials ie “singular finitegap” 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 RFourier Transform is a partial case of this theory The potential has a pole in this case at x = 0 with asymptotics u ∼ gg + 1/x 2 Here g is the genus of spectral curve

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