|
This article is cited in 16 scientific papers (total in 16 papers)
Construction of the Asymptotics of the Solutions
of the One-Dimensional Schrödinger Equation
with Rapidly Oscillating Potential
P. N. Nesterov P. G. Demidov Yaroslavl State University
Abstract:
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.
Received: 02.08.2005
Citation:
P. N. Nesterov, “Construction of the Asymptotics of the Solutions
of the One-Dimensional Schrödinger Equation
with Rapidly Oscillating Potential”, Mat. Zametki, 80:2 (2006), 240–250; Math. Notes, 80:2 (2006), 233–243
Linking options:
https://www.mathnet.ru/eng/mzm2805https://doi.org/10.4213/mzm2805 https://www.mathnet.ru/eng/mzm/v80/i2/p240
|
Statistics & downloads: |
Abstract page: | 424 | Full-text PDF : | 211 | References: | 33 | First page: | 1 |
|