Scattering Theory for CMV Matrices: Uniqueness, Helson–Szegő and Strong Szegő Theorems

Abstract

We develop a scattering theory for CMV matrices, similar to the Faddeev-Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for uniqueness, which are connected with the Helson-Szegő and the strong Szegő theorems. The first condition is given in terms of the boundedness of a transformation operator associated with the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions.Mathematics Subject Classification (2010). 30H15, 30H10, 47B36.

The inverse scattering transform for weak Wigner–von Neumann type potentials *

“…In the conclusion of this section we mention that the formulation of Faddeev-Marchenko inverse scattering theory in terms of Hankel operators defined as in definition 2.1 (adjusted to the unit circle) and the techniques stemming from this theory appeared only in the present century in [15,35]. However, the inverse scattering problem was studied therein only for Jacobi operators and not in the context of integrable systems.…”

The inverse scattering transform for weak Wigner-von Neumann type potentials

“…In the conclusion of this section we mention that the formulation of Faddeev-Marchenko inverse scattering theory in terms of Hankel operators defined as in Definition 2.1 (adjusted to the unit circle) and the techniques stemming from this theory appeared only in the present century in [17,39]. However, the inverse scattering problem was studied therein only for Jacobi operators and not in the context of integrable systems.…”

Abstract Interpolation Problem and Some Applications. II: Coefficient Matrices

“…A discussion on this matter was given in [22]. An application to uniqueness of the inverse scattering for CMV matrices was given in [25,11].…”