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

Abstract

We consider the spectral asymptotics of the Sturm–Liouville problem with an arithmetically self-similar singular weight. In previous papers, A. A. Vladimirov and I. A. Sheĭpak, as well as the author, relied on the spectral periodicity property, which places major constraints on the self-similarity parameters of the weight. In this study, a different approach to estimation of the eigenvalue counting function is presented. As a result, a significantly wider class of self-similar measures can be considered. The obtained asymptotics are applied to the problem of small ball deviations for the Green Gaussian processes.