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Zap. Nauchn. Sem. POMI, 2014, Volume 425, Pages 86–98 (Mi znsl6022)  

This article is cited in 5 scientific papers (total in 5 papers)

On spectral asymptotics of the Neumann problem for the Sturm–Liouville equation with self-similar generalized Cantor type weight

N. V. Rastegaevab

a St. Petersburg State University, St. Petersburg, Russia
b St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia

Abstract: Spectral asymptotics of the weighted Neumann problem for the Sturm–Liouville equation is considered. The weight is assumed to be the distributional derivative of a self-similar generalized Cantor type function. The spectrum is shown to have a periodicity property for a wide class of Cantor type self-similar functions. The weaker “quasi-periodicity” property is demonstrated under certain mixed boundary value conditions. This allows for a more precise description of the main term of the eigenvalue counting function asymptotics. Previous results by A. A. Vladimirov and I. A. Sheipak are generalized.

Key words and phrases: self-similar measures, spectral asymptotics, spectral periodicity, spectral quasi-periodicity.

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English version:
Journal of Mathematical Sciences (New York), 2015, 210:6, 814–821

UDC: 517
Received: 05.08.2014

Citation: N. V. Rastegaev, “On spectral asymptotics of the Neumann problem for the Sturm–Liouville equation with self-similar generalized Cantor type weight”, Boundary-value problems of mathematical physics and related problems of function theory. Part 44, Zap. Nauchn. Sem. POMI, 425, POMI, St. Petersburg, 2014, 86–98; J. Math. Sci. (N. Y.), 210:6 (2015), 814–821

Citation in format AMSBIB
\Bibitem{Ras14}
\by N.~V.~Rastegaev
\paper On spectral asymptotics of the Neumann problem for the Sturm--Liouville equation with self-similar generalized Cantor type weight
\inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~44
\serial Zap. Nauchn. Sem. POMI
\yr 2014
\vol 425
\pages 86--98
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6022}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2015
\vol 210
\issue 6
\pages 814--821
\crossref{https://doi.org/10.1007/s10958-015-2592-1}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84944711903}


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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. J. V. Tikhonov, I. A. Sheipak, “On the string equation with a singular weight belonging to the space of multipliers in Sobolev spaces with negative index of smoothness”, Izv. Math., 80:6 (2016), 1242–1256  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. N. V. Rastegaev, “Ob asimptotike spektra tenzornogo proizvedeniya operatorov s pochti regulyarnymi marginalnymi asimptotikami”, Algebra i analiz, 29:6 (2017), 197–229  mathnet  elib
    3. N. V. Rastegaev, “On Spectral Asymptotics of the Neumann Problem for the Sturm–Liouville Equation with Arithmetically Self-Similar Weight of a Generalized Cantor Type”, Funct. Anal. Appl., 52:1 (2018), 70–73  mathnet  crossref  crossref  isi  elib
    4. I. A. Ibragimov, M. A. Lifshits, A. I. Nazarov, D. N. Zaporozhets, “On the history of St. Petersburg school of probability and mathematical statistics: II. Random processes and dependent variables”, Vestn. St Petersb. Univ.-Math., 51:3 (2018), 213–236  crossref  crossref  isi  scopus
    5. N. V. Rastegaev, U. R. Freiberg, “On spectral asymptotics of the Sturm–Liouville problem with self-conformal singular weight with strong bounded distortion property”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii. 47, K 85-letiyu Vsevoloda Alekseevicha SOLONNIKOVA, Zap. nauchn. sem. POMI, 477, POMI, SPb., 2018, 129–135  mathnet
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