Geodesic
On the scattering on the line. I. Existence of wave operators and asymptotic completeness in periodic case
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 20, Tome 186 (1990), pp. 136-141

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Existence the of wave operators is shown for the scattering on the line in $\mathbb{R}^3$. These wave operators are asymptotically complete and $S$-operator in unitary though the completeness of the wave operators is broken-down. 
@article{ZNSL_1990_186_a13,
     author = {Ya. V. Kurylev},
     title = {On the scattering on the line. {I.} {Existence} of wave operators and asymptotic completeness in periodic case},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {136--141},
     publisher = {mathdoc},
     volume = {186},
     year = {1990},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1990_186_a13/}
}
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Ya. V. Kurylev. On the scattering on the line. I. Existence of wave operators and asymptotic completeness in periodic case. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 20, Tome 186 (1990), pp. 136-141. http://geodesic.mathdoc.fr/item/ZNSL_1990_186_a13/