Fractional Sturm-Liouville eigenvalue problems, II

TL;DR

This paper investigates the spectral properties of a non self-adjoint fractional Sturm-Liouville problem, revealing how the number of real eigenvalues varies with the fractional order and approaches classical behavior as the order nears one.

Contribution

It extends the understanding of fractional Sturm-Liouville problems by analyzing eigenvalue existence, asymptotics, and the transition to classical differential operators as fractional order approaches one.

Findings

Finite real eigenvalues for certain fractional orders

Potential absence of real eigenvalues near 

Infinite eigenvalues as fractional order approaches 1

Abstract

We continue the study of a non self-adjoint fractional three-term Sturm-Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left-Riemann-Liouville fractional integral under {\it Dirichlet type} boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter $\alpha$, $0<\alpha<1$, there is a finite set of real eigenvalues and that, for $\alpha$ near $1/2$, there may be none at all. As $\alpha \to 1^-$ we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm-Liouville problem with the composition of the operators becoming the operator of second order differentiation.