Fractional Sturm-Liouville problem on metric graphs

TL;DR

This work formulates a fractional Sturm–Liouville problem on a metric graph using a symmetric positive operator built from the left Riemann–Liouville and right Caputo derivatives. It develops a variational framework on graphs and proves the spectrum is discrete and unbounded, with eigenpairs forming an orthonormal basis in $L_2(\mathcal{G})$; it also establishes the convergence of the reciprocal-eigenvalue series $\sum_{k=1}^{\infty} 1/\lambda_k$ and provides eigenfunction bounds. These results extend spectral theory for graphs to the fractional setting and improve related estimates over prior work. The findings have potential implications for space-time fractional diffusion and spectral problems on network-like domains.

Abstract

In the present paper, we investigate the fractional analog of the Sturm-Liouville problem on a metric graph using a combination of left Riemann-Liouville and right Caputo fractional derivatives. This combination creates a symmetric and positive analog of the Sturm-Liouville operator. We demonstrated that the operator has a countable number of eigenvalues converging to infinity and analyzed the convergence of the series of the reciprocal eigenvalues, providing estimates for the eigenfunctions.

Figures (1)

• Figure 1: Metric graph

Theorems & Definitions (3)

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