Asymptotics of eigenvalues of Sturm--Liouville problem with discrete self-similar weight

TL;DR

This paper derives asymptotic formulas for the eigenvalues of a Sturm--Liouville problem with a self-similar weight function, expanding understanding of spectral properties in such self-similar systems.

Contribution

It provides new asymptotic formulas for eigenvalues of Sturm--Liouville problems with self-similar weights, specifically for the case with spectral degree zero.

Findings

Asymptotic formulas for eigenvalues are established.

Results apply to weights derived from self-similar functions in L2.

The spectral behavior is characterized for the zero spectral degree case.

Abstract

The Sturm--Liouville problem $-y''-\lambda\rho y=0$, $y(0)=y(1)=0$, where $\rho$ is a generalized derivative of self-similar function $P\in L_2[0,1]$ with spectral degree D=0, is studied. Asymptotic formulas for eigenvalues are obtained.