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Mat. Zametki, 2006, Volume 80, Issue 2, Pages 240–250 (Mi mz2805)  

This article is cited in 10 scientific papers (total in 10 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

DOI: https://doi.org/10.4213/mzm2805

Full text: PDF file (419 kB)
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English version:
Mathematical Notes, 2006, 80:2, 233–243

Bibliographic databases:

UDC: 517.928
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

Citation in format AMSBIB
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  • https://doi.org/10.4213/mzm2805
  • http://mi.mathnet.ru/eng/mz/v80/i2/p240

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Nesterov P. N., “Averaging method in the asymptotic integration problem for systems with oscillatory-decreasing coefficients”, Differ. Equ., 43:6 (2007), 745–756  crossref  mathscinet  zmath  isi  elib  scopus
    2. P. N. Nesterov, “Asimptoticheskoe predstavlenie reshenii sistem lineinykh raznostnykh uravnenii i metod usredneniya”, Model. i analiz inform. sistem, 14:2 (2007), 63–67  mathnet
    3. P. N. Nesterov, “Ob ustoichivosti reshenii nekotorykh uravnenii iz klassa adiabaticheskikh ostsillyatorov”, Model. i analiz inform. sistem, 15:2 (2008), 10–17  mathnet
    4. Burd V., Nesterov P., “Parametric resonance in adiabatic oscillators”, Results Math., 58:1-2 (2010), 1–15  crossref  mathscinet  zmath  isi  elib  scopus
    5. Nesterov P., “On Eigenvalues of the One-Dimensional Dirac Operator with Oscillatory Decreasing Potential”, Math. Phys. Anal. Geom., 15:3 (2012), 257–298  crossref  mathscinet  zmath  isi  elib  scopus
    6. P. N. Nesterov, “Parametricheskii rezonans v garmonicheskom ostsillyatore s peremennoi chastotoi sobstvennykh kolebanii”, Model. i analiz inform. sistem, 20:3 (2013), 5–28  mathnet
    7. Nesterov P., “Appearance of New Parametric Resonances in Time-Dependent Harmonic Oscillator”, Results Math., 64:3-4 (2013), 229–251  crossref  mathscinet  zmath  isi  scopus
    8. Davies E.B., “Singular Schrodinger Operators in One Dimension”, Mathematika, 59:1 (2013), 141–159  crossref  mathscinet  zmath  isi  elib  scopus
    9. P. N. Nesterov, “Asimptoticheskoe integrirovanie odnogo lineinogo differentsialnogo uravneniya vtorogo poryadka s zapazdyvaniem”, Model. i analiz inform. sistem, 23:5 (2016), 635–656  mathnet  crossref  mathscinet  elib
    10. Nesterov P., “Asymptotic Integration of a Certain Second-Order Linear Delay Differential Equation”, Mon.heft. Math., 182:1 (2017), 77–98  crossref  mathscinet  zmath  isi  scopus
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