Geodesic
Wave operators for the Schrödinger equation with a slowly decreasing potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 2 (1970) no. 3, pp. 367-376 
V. S. Buslaev; V. B. Matveev. Wave operators for the Schrödinger equation with a slowly decreasing potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 2 (1970) no. 3, pp. 367-376. http://geodesic.mathdoc.fr/item/TMF_1970_2_3_a11/
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     author = {V. S. Buslaev and V. B. Matveev},
     title = {Wave operators for the {Schr\"odinger} equation with a~slowly decreasing potential},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {367--376},
     year = {1970},
     volume = {2},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1970_2_3_a11/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The present article is devoted to the study in space $L_2(R^n)$ of the energy operator $\displaystyle H_q=-\frac 1{2m}\Delta+q(x)$, where the function $q(x)$ decreases slower that $|x|^{-\alpha}$, $\alpha>0$, as $|x|\to\infty$. An explicit “regularizing” operator $U_q(t)$ is constructed and the existence of generalized wave operators $$ W_{\pm}(H_q, H_0)=\mathop{\textrm{s-lim}}_{t\to\pm\infty}\exp\{-itH_q\}\exp\{itH_0\}U_q(t) $$ is proved.

[1] J. D. Dollard, J. Math. Phys., 5:6 (1964) | DOI | MR

[2] L. A. Sakhnovich, Tr. Mosk. matem. ob-va, 29, 1968

[3] V. S. Buslaev, Vestn. LGU, ser. matem., 1970 (to appear)

[4] M. V. Fedoryuk, ZhVM i MF, 2 (1962), 145 | Zbl

[5] A. Erdeii, Asimptoticheskie razlozheniya, Fizmatgiz, 1962

[6] N. I. Akhiezer, I. M. Glazman, Teoriya lineinykh operatorov v gilbertovom prostranstve, «Nauka», 1966 | MR | Zbl