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Izv. Akad. Nauk SSSR Ser. Mat., 1983, Volume 47, Issue 3, Pages 659–687 (Mi izv1416)  

This article is cited in 3 scientific papers (total in 3 papers)

The multidimensional Schrödinger operator with a periodic potential

M. M. Skriganov


Abstract: In this paper the author investigates the zonal structure of the spectrum of the three-dimensional Schrödinger operator with periodic potential. The main result is an estimate of the number $n(\lambda)$ of zones of the spectrum covering the real point $\lambda$. It is shown that, under certain conditions on the period lattice of the potential, $n(\lambda)>\lambda$ when $\lambda\to\infty$. From this estimate it follows that the number of lacunae in the spectrum of the Schrödinger operator is finite. It is also shown that for periodic potentials with small norm there are in general no lacunae in the spectrum. Analogous results are formulated for the Schrödinger operator in higher dimensions.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1984, 22:3, 619–645

Bibliographic databases:

UDC: 517
MSC: 35J10, 35P20

Citation: M. M. Skriganov, “The multidimensional Schrödinger operator with a periodic potential”, Izv. Akad. Nauk SSSR Ser. Mat., 47:3 (1983), 659–687; Math. USSR-Izv., 22:3 (1984), 619–645

Citation in format AMSBIB
\Bibitem{Skr83}
\by M.~M.~Skriganov
\paper The multidimensional Schr\"odinger operator with a~periodic potential
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1983
\vol 47
\issue 3
\pages 659--687
\mathnet{http://mi.mathnet.ru/izv1416}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=703598}
\zmath{https://zbmath.org/?q=an:0545.35027}
\transl
\jour Math. USSR-Izv.
\yr 1984
\vol 22
\issue 3
\pages 619--645
\crossref{https://doi.org/10.1070/IM1984v022n03ABEH001457}


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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. O. A. Veliev, “Asymptotic formulas for the eigenvalues of a periodic Schrödinger operator and the Bethe–Sommerfeld conjecture”, Funct. Anal. Appl., 21:2 (1987), 87–100  mathnet  crossref  mathscinet  zmath  isi
    2. Yu. E. Karpeshina, “Analytic perturbation theory for a periodic potential”, Math. USSR-Izv., 34:1 (1990), 43–64  mathnet  crossref  mathscinet  zmath
    3. A. S. Shamaev, V. V. Shumilova, “Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin–Voigt viscoelastic materials”, Proc. Steklov Inst. Math., 295 (2016), 202–212  mathnet  crossref  crossref  mathscinet  isi  elib  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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