A Fractal Dirac Eigenvalue Problem: Spectral Properties and Numerical Examples

TL;DR

This work develops a fractal Dirac eigenvalue problem using the $F^\alpha$-derivative on a finite interval $[0,\pi]$. It proves that the associated operator is self-adjoint in $\mathcal{L}^\alpha_2(0,\pi)$ with a real, simple spectrum, characterized by a scalar characteristic function $\Delta(\lambda)$ whose zeros give eigenvalues and by weight numbers $\{\alpha_n\}$ satisfying $D_F^\alpha(\lambda_n)=\beta_n\,\alpha_n$. A numerical framework based on the fractal Runge–Kutta method is used to compute eigenvalues for $\alpha\in(0,1]$, with results collapsing to the classical case as $\alpha\to 1$ and illustrating spectral behavior for different fractal supports. The work contributes a tractable approach to spectral problems on fractal domains, with potential applications in physics and fractal dynamics.

Abstract

In this paper, we study a Dirac boundary value problem where the operator is considered with a derivative of order $α\in (0, 1]$, known as the $F^α$-derivative. We prove some spectral properties of eigenvalues and eigenfunctions and present numerical examples to demonstrate the practical implications of our approach.

Figures (6)

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Theorems & Definitions (14)

• Theorem 1
• proof
• Lemma 2
• proof
• Corollary 3
• Theorem 4
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• Lemma 5
• proof
• Corollary 6