Geodesic
The future and past wave operators on the singular spectrum
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 5-20

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We consider averaged wave operators constructed for singular unitary operators $U_1$, $U_2$ and a bounded identification operator $A$. In the case of rank-two commutator $AU_1-U_2A$, we show that averaged wave operators of past and future exist or do not exist simultaneously, and if they exist, they must coincide. As a consequence, we obtain some results concerning the boundary behavior of Cauchy-type integrals. 
@article{ZNSL_2011_389_a0,
     author = {R. V. Bessonov},
     title = {The future and past wave operators on the singular spectrum},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--20},
     publisher = {mathdoc},
     volume = {389},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a0/}
}
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R. V. Bessonov. The future and past wave operators on the singular spectrum. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 5-20. http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a0/