Recent studies have shown that the Sturm-Liouville problem \(-Au=\lambda u\) with zero boundary conditions on an interval has no principal eigenvalue when A is the Caputo fractional derivative , provided that \(\alpha \) is sufficiently close to 1. It has been shown that the principal eigenvalue, denoted by \(\lambda (\alpha ),\) always exists when A is the Riemann-Liouville fractional derivative \({\mathcal {D}_{a+}^{\alpha }}\) for any \(1<\alpha <2\) , highlighting a striking contrast with the Caputo case. We introduce appropriate spaces to develop a systematic study of \(\lambda (\alpha )\) as a function of the real parameter \(\alpha \) and show, for example, that although \(\lambda (\alpha )\) is non-monotonic, the quotient \(\frac{\Gamma (\alpha -1)}{(b-a)^{\alpha -1}}\frac{1}{\lambda (\alpha )}\) is strictly decreasing with respect to \(\alpha \) . For the sake of completeness, we shall again show the existence of a principal eigenvalue for each \(\alpha \in (1,2)\) using the techniques developed here. Finally, we extend our techniques and results to the more general weighted spectral problem \(-Au=\lambda p(x) u\) for a class of coefficient functions p.
