This article is cited in 18 papers

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

A. A. Vladimirov^a, I. A. Sheipak<sup>b

a</sup> Dorodnitsyn Computing Centre of the Russian Academy of Sciences, Moscow
^b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The second and third boundary value problems for the Sturm–Liouville equation in which the weight function is the generalized derivative of a Cantor-type self-similar function are considered. The oscillation properties of the eigenfunctions of these problems are studied, and on the basis of this study, known asymptotics of their spectra are substantially refined. Namely, it is proved that the function $s$ in the well-known formula
$$ N(\lambda)=\lambda^D\cdot [s(\ln\lambda)+o(1)] $$
decomposes into the product of a decreasing exponential and a nondecreasing purely singular function (and, thereby, is not constant).

Keywords: Sturm–Liouville problem, self-similar weight, Neumann boundary conditions, third-type boundary conditions, spectral periodicity.

UDC: 517.984

Received: 20.05.2011

DOI: 10.4213/faa3124

 English version: Functional Analysis and Its Applications, 2013, 47:4, 261–270

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