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Mat. Sb., 2007, Volume 198, Number 8, Pages 3–34 (Mi msb1986)  

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

The spectrum of a self-adjoint differential operator with rapidly oscillating coefficients on the axis

D. I. Borisovab, R. R. Gadyl'shina

a Bashkir State Pedagogical University
b Nuclear Physics Institute, Academy of Sciences of the Czech Republic

Abstract: The asymptotic behaviour of the spectrum of a self-adjoint second-order differential operator on the axis is investigated. The coefficients of this operator depend on rapid and slow variables and are periodic in the rapid variable. The period of oscillations in the rapid variable is a small parameter. The dependence of the coefficients on the rapid variable is localized, and they stop depending on it at infinity. Asymptotic expansions for the eigenvalues and the eigenfunctions of the operator in question are constructed. It is shown that, apart from eigenvalues convergent to eigenvalues of the homogenized operator as the small parameter converges to zero, the perturbed operator can also have an eigenvalue convergent to the boundary of the continuous spectrum. Necessary and sufficient conditions for the existence of such an eigenvalue are obtained.
Bibliography: 22 titles.

DOI: https://doi.org/10.4213/sm1986

Full text: PDF file (722 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2007, 198:8, 1063–1093

Bibliographic databases:

UDC: 517.956
MSC: 34L20, 47E05
Received: 18.07.2006 and 19.03.2007

Citation: D. I. Borisov, R. R. Gadyl'shin, “The spectrum of a self-adjoint differential operator with rapidly oscillating coefficients on the axis”, Mat. Sb., 198:8 (2007), 3–34; Sb. Math., 198:8 (2007), 1063–1093

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    Erratum
    • Errata
      D. I. Borisov, R. R. Gadyl'shin
      Mat. Sb., 2008, 199:3, 160


    This publication is cited in the following articles:
    1. D. I. Borisov, “Asymptotics of the eigenvalues of elliptic systems with fast oscillating coefficients”, Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S35–S45  mathnet  crossref  elib
    2. A. Yu. Trynin, “On inverse nodal problem for Sturm-Liouville operator”, Ufa Math. J., 5:4 (2013), 112–124  mathnet  crossref  elib
    3. Duchene V., Vukicevic I., Weinstein M.I., “Scattering and Localization Properties of Highly Oscillatory Potentials”, Commun. Pure Appl. Math., 67:1 (2014), 83–128  crossref  mathscinet  zmath  isi  scopus
    4. A. Yu. Trynin, “On some properties of sinc approximations of continuous functions on the interval”, Ufa Math. J., 7:4 (2015), 111–126  mathnet  crossref  isi  elib
    5. Dimassi M., “Semi-classical Asymptotics for Schrödinger Operator with Oscillating Decaying Potential”, Can. Math. Bul.-Bul. Can. Math., 59:4 (2016), 734–747  crossref  mathscinet  zmath  isi  scopus
    6. Duchene V., Raymond N., “Spectral Asymptotics For the Schrodinger Operator on the Line With Spreading and Oscillating Potentials”, Doc. Math., 23 (2018), 599–636  mathscinet  zmath  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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