Geodesic
Schroedinger operator with weakly accelerating potential
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 17, Tome 147 (1985), pp. 10-12 
M. V. Buslaeva. Schroedinger operator with weakly accelerating potential. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 17, Tome 147 (1985), pp. 10-12. http://geodesic.mathdoc.fr/item/ZNSL_1985_147_a1/
@article{ZNSL_1985_147_a1,
     author = {M. V. Buslaeva},
     title = {Schroedinger operator with weakly accelerating potential},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {10--12},
     year = {1985},
     volume = {147},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_147_a1/}
}
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TI  - Schroedinger operator with weakly accelerating potential
JO  - Zapiski Nauchnykh Seminarov POMI
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VL  - 147
UR  - http://geodesic.mathdoc.fr/item/ZNSL_1985_147_a1/
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ID  - ZNSL_1985_147_a1
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%G ru
%F ZNSL_1985_147_a1

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Conceptions of the scattering theory were used for construction of an unitary operator, which realized the equivalence of the operator $-id/d\xi$ on $L_2(\mathbb{R})$ and the Schroedinger operator on simi-axis with the potential $v(x)$, admitting the estimate $-v_-x^{2d}\leq v(x)\leq-v_+x^{2d}$, $v_+>0$, $0<\alpha<1$.