On the Neumann problem for Sturm-Liouville equation with self-similar Cantor type weight

TL;DR

This paper investigates the spectral properties of a Sturm-Liouville problem with a self-similar Cantor-type weight, revealing detailed asymptotic behavior of eigenvalues and eigenfunctions under Neumann and mixed boundary conditions.

Contribution

It provides a more precise analysis of spectral asymptotics, showing that the oscillatory component is a product of a decreasing exponential and a nondecreasing singular function.

Findings

Eigenfunctions exhibit oscillating properties under boundary conditions.

Spectral asymptotics are refined with explicit oscillatory behavior.

The asymptotic distribution involves a non-constant oscillatory function.

Abstract

Sturm-Liouville problem with generalized derivative of self-similar Cantor type function as a weight is considered. Under Neumann and mixed boundary conditions the oscillating properties of the eigenfunctions are studied. The spectral asymptotics are made more precise then in previous papers. Namely, it is shown that for known asymptotics $N(\lambda)=\lambda^D\cdot [s(\ln\lambda)+o(1)]$ the function $s$ is a product of decreasing exponent and nondecreasing purely singular function (and hence it is not constant).