Exceptional circles of radial potentials

Abstract

A nonlinear scattering transform is studied for the two-dimensional Schrödinger 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.

A test for the existence of exceptional points in the Faddeev scattering problem

“…Indeed, operator S 0 k is an elliptic PDO of order −1 on the compact manifold ∂O, and therefore it has zero index. Then the same is true for the operator S k , since function G k − G 0 k is infinitely smooth in (z, k) and analytic in k 1 , k 2 where k = k 1 + ik 2 = 0 (eg [10]). Thus S k is a Fredholm family of operators analytic in k 1 , k 2 = (0, 0).…”

“…The first result is the absence of exceptional points in a neighborhood of the origin k = 0 in the case of absorbing potentials. The second one is a generalisation of the results of [10], [18] on sign definite perturbations of conductive potentials to non spherically symmetrical problems. We do not assume the radial symmetry of either the underlying conductive potential or its perturbation.…”

“…The case of so-called subcritical potentials was studied in [11]. Exceptional points for sign definite perturbations of conductive potentials were studied in [10], [18] under the condition that the potential and its perturbation are spherically symmetric. The existence or non-existence of exceptional points in this case depends on the sign of the perturbation.…”

A data-driven edge-preserving D-bar method for electrical impedance tomography

“…(Other kinds of potentials may have exceptional points, or k values with no unique ψ( · , k). See [32] for more details. )…”

The D-Bar Method for Diffuse Optical Tomography: A Computational Study

“…Let us fix E = −1. In the numerical tests that follow, for the DN-and S λ -matrices we use N = 16, see the definition (20). We add gaussian noise to each element using equation (21) so that the relative matrix norm between the original DN-matrix and the noisy DN-matrix is 0.005%.…”