Singular Soliton Operators and Indefinite Metrics

TL;DR

This paper develops a spectral theory for singular 1D Schrödinger operators with potentials that produce meromorphic solutions, exploring their properties in periodic and rapidly decreasing cases with indefinite metrics.

Contribution

It introduces a spectral framework for singular finite-gap potentials with indefinite inner products, extending Fourier analysis to Riemann surfaces.

Findings

Spectral theory for singular potentials with indefinite metrics.

Preservation of negative squares under KdV hierarchy.

Development of a Fourier-like transform on Riemann surfaces.

Abstract

The singular real second order 1D Schrodinger operators are considered here with such potentials that all local solutions near singularities to the eigenvalue problem are meromorphic for all values of the spectral parameter. All algebro-geometrical or "singular finite-gap" potentials satisfy to this condition. A Spectral Theory is constructed here for the periodic and rapidly decreasing cases in the special classes of functions with singularities and indefinite inner product. It has a finite number of negative squares if the unimodular Bloch multipliers are fixed in the periodic case, and in the rapidly decreasing case. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform for the Riemann Surfaces preserving remarkable multiplicative properties of the ordinary (i.e. genus zero) Fourier Transform based on the standard exponential basis, leads to such operators as it was shown in our previous works.