Geodesic
Calculating losses in scattering problems
Sbornik. Mathematics, Tome 26 (1975) no. 1, pp. 71-87 
B. S. Pavlov. Calculating losses in scattering problems. Sbornik. Mathematics, Tome 26 (1975) no. 1, pp. 71-87. http://geodesic.mathdoc.fr/item/SM_1975_26_1_a3/
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     title = {Calculating losses in scattering problems},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1975_26_1_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this article we solve the problem about the calculation of losses in a scattering problem with “Lax” and “non-Lax” channels. For the initial scattering matrix we consider the scattering matrix of the basic operator of the problem with respect to a simple unperturbed operator, which acts in a distinguished subspace (a Lax channel) that is the orthogonal sum of the incoming and outgoing subspaces. It turns out that this scattering matrix is nonunitary when the basic space contains other channels besides the distinguished one, including non-Lax channels. The concept of losses is connected with the fact that the scattering matrix is nonunitary. We calculate the losses by constructing in the orthogonal complement of a Lax channel a new selfadjoint operator, which with the original unperturbed operator forms a modified unperturbed operator. The latter has a unitary scattering matrix with respect to the basic operator of the problem. We explain the significance of the elements of the new scattering matrix that include the original matrix. Bibliography: 9 titles.

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[4] B. S. Pavlov, “Ob odnomernom rasseyanii ploskikh voln na proizvolnom potentsiale”, Teor. i matem. fizika, 16:1 (1973), 105–115 | Zbl

[5] B. S. Pavlov, “O nepreryvnom spektre rezonansov na nefizicheskom liste”, DAN SSSR, 206:2 (1972), 1301–1304 | MR | Zbl

[6] B. Sekefalvi-Nad, Ch. Foiash, Garmonicheskii analiz operatorov v gilbertovom prostranstve, izd-vo «Mir», Moskva, 1970 | MR

[7] B. S. Pavlov, “Ob usloviyakh otdelimosti spektralnykh komponent dissipativnogo operatora”, Izv. AN SSSR, seriya matem., 39 (1975), 123–148 | MR | Zbl

[8] B. S. Pavlov, “K spektralnomu analizu dissipativnykh operatorov”, DAN SSSR, 212:2 (1973), 298–301 | MR | Zbl

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