Geodesic
Scattering problem for radial Schrödinger equation with a slowly decreasing potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 10 (1972) no. 2, pp. 238-248 
V. B. Matveev; M. M. Skriganov. Scattering problem for radial Schrödinger equation with a slowly decreasing potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 10 (1972) no. 2, pp. 238-248. http://geodesic.mathdoc.fr/item/TMF_1972_10_2_a8/
@article{TMF_1972_10_2_a8,
     author = {V. B. Matveev and M. M. Skriganov},
     title = {Scattering problem for radial {Schr\"odinger} equation with a slowly decreasing potential},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {238--248},
     year = {1972},
     volume = {10},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1972_10_2_a8/}
}
TY  - JOUR
AU  - V. B. Matveev
AU  - M. M. Skriganov
TI  - Scattering problem for radial Schrödinger equation with a slowly decreasing potential
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1972
SP  - 238
EP  - 248
VL  - 10
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_1972_10_2_a8/
LA  - ru
ID  - TMF_1972_10_2_a8
ER  - 
%0 Journal Article
%A V. B. Matveev
%A M. M. Skriganov
%T Scattering problem for radial Schrödinger equation with a slowly decreasing potential
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1972
%P 238-248
%V 10
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_1972_10_2_a8/
%G ru
%F TMF_1972_10_2_a8

Voir la notice de l'article provenant de la source Math-Net.Ru

The scattering problem for the Schrodinger equation with slowly decreasing potential is considered. Stationary wave operators $W_{\pm}(H,H_0)$ are constructed and their completeness is proved. It is shown that the operators $W_{\pm}(H,H_0)$ can also be defined as the limits $W_{\pm}(H,H_0)=\lim{t\to\pm\infty} \exp(itH)T_{\pm}\exp(-itH_0)$, $T_{\pm}$ being some operators, which do not depend on $t$, do not commute with $H_0$ and can be constructed explicity for the :given potential $q(x)$.The invariance principle for the wave operators $W_{\pm}$ is proved.

[1] V. S. Buslaev, V. B. Matveev, TMF, 2 (1970), 367 | MR

[2] V. S. Buslaev, Vestnik LGU, ser. matem., 1970, no. 13 | MR

[3] J. Dollard, J. Math. Phys., 5 (1964), 729 | DOI | MR

[4] W. Amrein, Ph. Martin, B. Misra, Helv. Phys. Acta, 43 (1970), 313 | MR | Zbl

[5] R. Lavine, J. funct. anal., 5 (1970), 368 | DOI | MR | Zbl

[6] R. Lavine, Commutators and scattering theory. Part I, II, preprint Cornell Univ., 1970 | MR | Zbl

[7] J. Combes, An algebraic approach to quantum scattering theory, preprint Marseille, 1969

[8] L. D. Faddeev, DAN SSSR, 152 (1963), 573 | MR

[9] K. Hepp, Theorie de renormalisation, Springer, Berlin–Heidelberg, N.-Y., 1969 | MR | Zbl

[10] T. Kato, J. funct. anal., 1:3 (1967) | DOI | MR

[11] A. L. Belopolskii, M. Sh. Birman, Izvestiya AN SSSR, ser. matem., 32 (1968), 1162 | MR

[12] M. Sh. Birman, Izvestiya AN SSSR, ser. matem., 35 (1971), 440 | MR | Zbl

[13] P. P. Kulish, L. D. Faddeev, TMF, 4 (1970), 153 | Zbl

[14] E. Ch. Titchmarsh, Razlozheniya po sobstvennym funktsiyam, t. 1, IL, 1960 | MR

[15] E. Ch. Titchmarsh, Razlozheniya po sobstvennym funktsiyam, t. 2, IL, 1961, str. 422 | MR

[16] L. D. Faddeev, UMN, 14:4 (88) (1959) | MR

[17] Z. S. Agranovich, V. A. Marchenko, Obratnaya zadacha teorii rasseyaniya, Izd-vo Kharkovskogo Gos. un-ta, 1960

[18] V. B. Matveev, M. M. Skriganov, DAN SSSR (to appear)

[19] V. G. Deich, DAN SSSR, 197 (1971), 1247 | MR | Zbl