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We develop a scattering theory for CMV matrices, similar to theFaddeev--Marchenko theory. A necessary and sufficient condition is obtained forthe uniqueness of the solution of the inverse scattering problem. We alsoobtain two sufficient conditions for the uniqueness, which are connected withthe Helson--Szeg\H o and the Strong Szeg\H o theorems. The first condition isgiven in terms of the boundedness of a transformation operator associated tothe CMV matrix. In the second case this operator has a determinant. In bothcases we characterize Verblunsky parameters of the CMV matrices, correspondingspectral measures and scattering functions.

Scattering theory for CMV matrices: uniqueness, Helson--Szeg\H{o} and Strong Szeg\H{O} theorems