Geodesic
Scattering problems for differential operators with perturbation of the space
Izvestiya. Mathematics , Tome 5 (1971) no. 2, pp. 459-474.

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For differential operators with constant coefficients in $R^m$ we study the existence and completeness of wave operators corresponding to perturbation of the metric of the original Hilbert space. The results are applied, in particular, to scattering problems of Maxwell's system in an anisotropic medium and for the wave equation. The study is based on abstract criteria (of “nuclear type”) of existence of complete wave operators. 
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M. Sh. Birman. Scattering problems for differential operators with perturbation of the space. Izvestiya. Mathematics , Tome 5 (1971) no. 2, pp. 459-474. http://geodesic.mathdoc.fr/item/IM2_1971_5_2_a9/

[1] Birman M. Sh., “Nekotorye prilozheniya lokalnogo priznaka suschestvovaniya volnovykh operatorov”, Dokl. AN SSSR, 185:4 (1969), 735–738 | MR | Zbl

[2] Birman M. Sh., “Zadachi rasseyaniya dlya differentsialnykh operatorov s postoyannymi koeffitsientami”, Funkts. analiz, 3:3 (1969), 1–16 | MR | Zbl

[3] Kato T., “Scattering theory with two Hilbert spaces”, J. Funct. Anal., 1:3 (1967), 342–369 | DOI | MR | Zbl

[4] Belopolskii A. L., Birman M. Sh., “Suschestvovanie volnovykh operatorov v teorii rasseyaniya dlya pary prostranstv”, Izv. AN SSSR. Ser. matem., 32:5 (1968), 1162–1175 | MR

[5] Birman M. Sh., “Priznak suschestvovaniya polnykh volnovykh operatorov v teorii rasseyaniya dlya pary prostranstv”, Problemy matem. fiziki, v. 4, Len. Gos. un-t, 1970, 22–26 | MR

[6] Wilcox C., “Wave operators and asymptotic solutions of wave propagation problems of classical physics”, Arch. Rational Mech. Anal., 22:1 (1966), 37–78 | DOI | MR | Zbl

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

[8] Berezanskii Yu. M., Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Naukova dumka, Kiev, 1965 | MR

[9] Mikhlin S. G., Mnogomernye singulyarnye integraly i integralnye uravneniya, Fizmatgiz, M., 1962 | MR | Zbl

[10] Birman M. Sh., “Ob usloviyakh suschestvovaniya volnovykh operatorov”, Izv. AN SSSR. Ser. matem., 27:4 (1963), 883–906 | MR | Zbl

[11] Mochizuki K., “Spectral and scattering theory for symmetric hyperbolic systems in an exterior domain”, Publ. RIMS Kyoto Univ. (A), 5:2 (1969), 219–258 | DOI | MR | Zbl

[12] Lax P., Phillips R., Scattering Theory, Academic Press, New York, 1967 | Zbl

[13] Avila G., “Spectral theory for acoustic problem”, J. Math. Anal. Applic., 23:1 (1968), 213–224 | DOI | MR | Zbl