Geodesic
Construction of the Asymptotics of the Solutions
Matematičeskie zametki, Tome 80 (2006) no. 2, pp. 240-250.

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

We obtain asymptotic formulas for the solutions of the one-dimensional Schrödinger equation $-y''+q(x)y=\nobreak 0$ with oscillating potential $q(x)=x^\beta P(x^{1+\alpha})+cx^{-2}$ as $x\to+\nobreak \infty$. The real parameters $\alpha$ and $\beta$ satisfy the inequalities $\beta-\alpha\ge\nobreak -1$, $2\alpha-\beta>\nobreak 0$ and $c$ is an arbitrary real constant. The real function $P(x)$ is either periodic with period $T$, or a trigonometric polynomial. To construct the asymptotics, we apply the ideas of the averaging method and use Levinson's fundamental theorem.
Keywords: Schrödinger equation, averaging method, oscillating potential, Levinson's theorem. 
@article{MZM_2006_80_2_a9,
     author = {P. N. Nesterov},
     title = {Construction of the {Asymptotics} of the {Solutions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {240--250},
     publisher = {mathdoc},
     volume = {80},
     number = {2},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_2_a9/}
}
TY  - JOUR
AU  - P. N. Nesterov
TI  - Construction of the Asymptotics of the Solutions
JO  - Matematičeskie zametki
PY  - 2006
SP  - 240
EP  - 250
VL  - 80
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2006_80_2_a9/
LA  - ru
ID  - MZM_2006_80_2_a9
ER  - 
%0 Journal Article
%A P. N. Nesterov
%T Construction of the Asymptotics of the Solutions
%J Matematičeskie zametki
%D 2006
%P 240-250
%V 80
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2006_80_2_a9/
%G ru
%F MZM_2006_80_2_a9
P. N. Nesterov. Construction of the Asymptotics of the Solutions. Matematičeskie zametki, Tome 80 (2006) no. 2, pp. 240-250. http://geodesic.mathdoc.fr/item/MZM_2006_80_2_a9/

[1] A. R. Its, “Asimptoticheskoe povedenie reshenii radialnogo uravneniya Shrëdingera s ostsilliruyuschim potentsialom pri nulevoi energii”, Problemy matematicheskoi fiziki, Sb. statei, no. 9, Izd-vo Leningradskogo un-ta, Leningrad, 1979, 30–41 | MR

[2] N. N. Bogolyubov, Yu. A. Mitropolskii, Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974 | MR

[3] V. Sh. Burd, V. A. Karakulin, “Asimptoticheskoe integrirovanie sistem lineinykh differentsialnykh uravnenii s kolebatelno ubyvayuschimi koeffitsientami”, Matem. zametki, 64:5 (1998), 658–666 | MR | Zbl

[4] B. P. Demidovich, Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967 | MR

[5] E. A. Koddington, N. Levinson, Teoriya obyknovennykh differentsialnykh uravnenii, IL, M., 1958

[6] I. Z. Shtokalo, Lineinye differentsialnye uravneniya s peremennymi koeffitsientami, Izd-vo AN USSR, Kiev, 1960

[7] P. N. Nesterov, Asimptoticheskoe povedenie reshenii odnomernogo uravneniya Shrëdingera s bystro ostsilliruyuschim potentsialom, Ruk. dep. v VINITI RAN 29.04.2005, No 640-V, 2005