Spectral Analysis of a Self-Similar Sturm-Liouville Operator

TL;DR

This paper investigates the spectral properties of a self-similar Sturm-Liouville operator, providing insights into whether its spectrum is pure point or continuous, especially in cases lacking Neumann-Dirichlet eigenfunctions.

Contribution

It offers the first spectral description of such operators on the line or half-line without Neumann-Dirichlet eigenfunctions, advancing understanding of Laplace operators on self-similar fractals.

Findings

Characterizes the spectral nature (pure point or continuous) of the operator.

Provides the first example of spectral analysis without Neumann-Dirichlet eigenfunctions.

Connects spectral properties to self-similar fractal structures.

Abstract

In this text we describe the spectral nature (pure point or continuous) of a self-similar Sturm-Liouville operator on the line or the half-line. This is motivated by the more general problem of understanding the spectrum of Laplace operators on unbounded finitely ramified self-similar sets. In this context, this furnishes the first example of a description of the spectral nature of the operator in the case where the so-called "Neumann-Dirichlet" eigenfunctions are absent.