Geodesic
The existence of wave operators in scattering theory for pairs of
Izvestiya. Mathematics , Tome 2 (1968) no. 5, pp. 1117-1130.

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Existence conditions are considered for wave operators for pairs of self-adjoint operators which act in different Hilbert spaces. The “operator of identification” is not assumed to be isometric. The existence criteria obtained for wave operators (local and nonlocal) are related to perturbation theory of nuclear operators. The construction of wave operators is carried out by a stationary method. 
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A. L. Belopol'skii; M. Sh. Birman. The existence of wave operators in scattering theory for pairs of. Izvestiya. Mathematics , Tome 2 (1968) no. 5, pp. 1117-1130. http://geodesic.mathdoc.fr/item/IM2_1968_2_5_a10/

[1] Kato T., Scattering theory with two Hilbert spaces, preprint, 1967 | MR

[2] Birman M. Sh., Entina S. B., “O statsionarnom podkhode v abstraktnoi teorii rasseyaniya”, Dokl. AN SSSR, 155:3 (1964), 506–508 | MR | Zbl

[3] Birman M. Sh., Entina S. B., “Statsionarnyi podkhod v abstraktnoi teorii rasseyaniya”, Izv. AN SSSR. Ser. matem., 31 (1967), 401–430 | MR | Zbl

[4] Birman M. Sh., “Lokalnyi priznak suschestvovaniya volnovykh operatorov”, Dokl. AN SSSR, 159:3 (1964), 485–488 | MR | Zbl

[5] Birman M. Sh., “Lokalnyi priznak suschestvovaniya volnovykh operatorov”, Izv. AN SSSR. Ser. matem., 32:4 (1968), 914–942 | MR | Zbl

[6] Birman M. Sh., “Ob usloviyakh suschestvovaniya volnovykh operatorov”, Izv. AN SSSR. Ser. matem., 27 (1963), 883–906 | MR | Zbl

[7] Kato T., “Wave operators and unitary equivalence”, Pacif. J. Math., 15 (1965), 171–180 | MR | Zbl

[8] Birman M. Sh., “Nekotorye prilozheniya lokalnogo priznaka suschestvovaniya volnovykh operatorov”, Dokl. AN SSSR, 1968, (v pechati)