On the spectrum of the Sturm-Liouville problem with arithmetically self-similar weight

Abstract

Spectral asymptotics of the Sturm-Liouville problem with an arithmetically self-similar singular weight is considered. Previous results by A. A. Vladimirov and I. A. Sheipak, and also by the author, rely on the spectral periodicity property, which imposes significant restrictions on the self-similarity parameters of the weight. This work introduces a new method to estimate the eigenvalue counting function. This allows to consider a much wider class of self-similar measures. The obtained asymptotics is applied to the small ball deviations problem for the Green Gaussian processes.