Exceptional circles of radial potentials

Abstract

A nonlinear scattering transform is studied for the two-dimensional Schrodinger equation at zero energy with a radial potential. First explicit examples are presented, both theoretically and computationally, of potentials with nontrivial singularities in the scattering transform. The singularities arise from non-uniqueness of the complex geometric optics solutions that define the scattering transform. The values of the complex spectral parameter at which the singularities appear are called exceptional points. The singularity formation is closely related to the fact that potentials of conductivity type are critical in the sense of Murata.

References in corpus (4)

• Two-dimensional rational solitons and their blow-up via the Moutard transformation
• Miura Maps and Inverse Scattering for the Novikov-Veselov Equation
• The Novikov-Veselov Equation and the Inverse Scattering Method, Part I: Analysis
• Faddeev eigenfunctions for point potentials in two dimensions

Cited by in corpus (6)

• Miura Maps and Inverse Scattering for the Novikov-Veselov Equation
• A Data-Driven Edge-Preserving D-bar Method for Electrical Impedance Tomography
• Nonlinear Fourier analysis for discontinuous conductivities: computational results
• Positive-energy D-bar method for acoustic tomography: a computational study
• Faddeev eigenfunctions for two-dimensional Schrodinger operators via the Moutard transformation
• Exceptional points in Faddeev scattering problem