The nonlinear Fourier transform for two-dimensional subcritical potentials

Michael Music

doi:10.3934/ipi.2014.8.1151

Article Contents

2014, Volume 8, Issue 4: 1151-1167. Doi: 10.3934/ipi.2014.8.1151

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The nonlinear Fourier transform for two-dimensional subcritical potentials

Michael Music^1,

1.
Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027

Received: October 31, 2013

Revised: May 31, 2014

Published: October 2014

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Abstract

The inverse scattering method for the Novikov-Veselov equation is studied for a larger class of Schrödinger potentials than could be handled previously. Previous work concerns so-called conductivity type potentials, which have a bounded positive solution at zero energy and are a nowhere dense set of potentials. We relax the conductivity type assumption to include logarithmically growing positive solutions at zero energy. These potentials are stable under perturbations. Assuming only that the potential is subcritical and has two weak derivatives in a weighted Sobolev space, we prove that the associated scattering transform can be inverted, and the original potential is recovered from the scattering data.

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Mathematics Subject Classification: Primary: 37K15; Secondary: 35J10.

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