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

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.