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