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

A. A. Vladimirov ; I. A. Sheipak

Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 4, pp. 18-29.

Résumé

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).

Détail

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

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