Fractional Sturm-Liouville eigenvalue problems, II

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 $1\mathrm{/}2$, 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.