We characterize the real eigenvalues of the spectral problem involving Caputo fractional derivative \(\begin{aligned} {\left\{ \begin{array}{ll} {^C\mathcal {D}_{0+}^{\alpha }}u(t)+\lambda u(t)=0, t\in (0,1) \\ u(0)=u(1)=0, 1<\alpha <2 \end{array}\right. } \end{aligned}\) and furthermore show that it has no real eigenvalues at all when \(\alpha \) is close to 1 (therefore, the conventional principal eigenvalue does not exist). This is in sharp contrast with the case of fractional Sturm-Liouville problem with Riemann-Liouville derivative, where the real eigenvalues and principal eigenvalue always exist for \(1<\alpha <2\) . This work reveals a major difference between the classic Sturm-Liouville theory and the fractional counterpart regarding the principal eigenvalue. Some other related results are also presented.
