Spectral analysis of a family of nonsymmetric fractional elliptic operators

TL;DR

This paper analyzes the spectral properties of a class of nonlocal, nonsymmetric fractional elliptic operators involving Riemann-Liouville derivatives, establishing eigenvalue existence, distribution, and eigenfunction behavior.

Contribution

It develops new analytical tools for nonlocal fractional operators, proving eigenvalue existence and characterizing their distribution in the complex plane.

Findings

Existence of real eigenvalues is proven.

Eigenvalues are characterized within a specific complex range.

Numerical analysis illustrates eigenvalue distribution and eigenfunction behavior.

Abstract

In this work, we investigate the spectral problem $Au = {\lambda}u$ where $A$ is a fractional elliptic operator involving left- and right-sided Riemann-Liouville derivatives. These operators are nonlocal and nonsymmetric, however, share certain classic elliptic properties. The eigenvalues correspond to the roots of a class of certain special functions. Compared with classic Sturm-Liouville problems, the most challenging part is to set up the framework for analyzing these nonlocal operators, which requires developing new tools. We prove the existence of the real eigenvalues, find the range for all possible complex eigenvalues, explore the graphs of eigenfunctions, and show numerical findings on the distribution of eigenvalues on the complex plane.