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Zh. Vychisl. Mat. Mat. Fiz., 2010, Volume 50, Number 4, Pages 679–698 (Mi zvmmf4861)  

This article is cited in 6 scientific papers (total in 6 papers)

A perturbed boundary eigenvalue problem for the Schrödinger operator on an interval

I. Kh. Khusnullin

Bashkir State Pedagogical University, ul. Oktyabr'skoi revolyutsii 3a, Ufa, 450000 Bashkortostan, Russia

Abstract: A perturbed two-parameter boundary value problem is considered for a second-order differential operator on an interval with Dirichlet conditions. The perturbation is described by the potential $\mu^{-1}V((x-x_0)\varepsilon^{-1})$, where $0<\varepsilon\ll1$ and $\mu$ is an arbitrary parameter such that there exists $\delta>0$ for which $\varepsilon/\mu=o(\varepsilon^\delta)$. It is shown that the eigenvalues of this operator converge, as $\varepsilon\to0$, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed.

Key words: second-order differential operator, singular perturbation, eigenvalue, asymptotics.

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English version:
Computational Mathematics and Mathematical Physics, 2010, 50:4, 646–664

Bibliographic databases:

UDC: 519.634
Received: 10.09.2009

Citation: I. Kh. Khusnullin, “A perturbed boundary eigenvalue problem for the Schrödinger operator on an interval”, Zh. Vychisl. Mat. Mat. Fiz., 50:4 (2010), 679–698; Comput. Math. Math. Phys., 50:4 (2010), 646–664

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. R. Gadyl'shin, I. Kh. Khusnullin, “Schrödinger operator on the axis with potentials depending on two parameters”, St. Petersburg Math. J., 22:6 (2011), 883–894  mathnet  crossref  mathscinet  zmath  isi
    2. R. R. Gadylshin, I. Kh. Khusnullin, “Vozmuschenie operatora Shredingera uzkim potentsialom”, Ufimsk. matem. zhurn., 3:3 (2011), 55–66  mathnet  zmath
    3. R. R. Gadyl'shin, I. Kh. Khusnullin, “Perturbation of a periodic operator by a narrow potential”, Theoret. and Math. Phys., 173:1 (2012), 1438–1444  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    4. A. R. Bikmetov, R. R. Gadylshin, “Vozmuschenie ellipticheskogo operatora uzkim potentsialom v $n$-mernoi oblasti”, Ufimsk. matem. zhurn., 4:2 (2012), 28–64  mathnet  mathscinet
    5. A. R. Bikmetov, V. F. Vil'danova, I. Kh. Khusnullin, “On perturbation of a Schrödinger operator on axis by narrow potentials”, Ufa Math. J., 7:4 (2015), 24–31  mathnet  crossref  isi  elib
    6. A. R. Bikmetov, I. Kh. Khusnullin, “Perturbation of Hill operator by narrow potentials”, Russian Math. (Iz. VUZ), 61:7 (2017), 1–10  mathnet  crossref  isi
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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