On spectral asymptotics of the Sturm-Liouville problem with self-conformal singular weight

TL;DR

This paper investigates the spectral asymptotics of Sturm-Liouville problems with self-conformal singular weights, extending previous results to a broader class of measures under a strengthened bounded distortion condition.

Contribution

It introduces a stronger bounded distortion property for self-conformal measures and derives the eigenvalue counting function asymptotics, generalizing earlier self-similar measure results.

Findings

Derived the power exponent of eigenvalue asymptotics for self-conformal weights.

Extended spectral asymptotics results beyond self-similar measures.

Generalized Fujita's 1985 results to a broader class of measures.

Abstract

Spectral asymptotics of the Sturm-Liouville problem with a singular self-conformal weight measure is considered. A stronger version of the bounded distortion property is assumed for the conformal iterated function system corresponding to the weight measure. Under this restrictions the power exponent of the main term of the eigenvalue counting function asymptotics is obtained. This generalizes the result obtained by T. Fujita (Taniguchi Symp. PMMP Katata, 1985) in the case of self-similar (self-affine) measure.