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On Spectral Asymptotics of the Neumann Problem for the Sturm–Liouville Equation with Self-Similar Weight of Generalized Cantor Type

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. A weaker “quasiperiodicity” property is established 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. Bibliography: 17 titles.

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Correspondence to N. V. Rastegaev.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 425, 2014, pp. 86–98.

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Rastegaev, N.V. On Spectral Asymptotics of the Neumann Problem for the Sturm–Liouville Equation with Self-Similar Weight of Generalized Cantor Type. J Math Sci 210, 814–821 (2015). https://doi.org/10.1007/s10958-015-2592-1

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  • DOI: https://doi.org/10.1007/s10958-015-2592-1

Keywords

  • Neumann Problem
  • Liouville Equation
  • Counting Function
  • Liouville Problem
  • Power Exponent