Spectral analysis of a family of nonsymmetric fractional elliptic operators

In this work, we investigate the spectral problem {D-a+(alpha) D(b-)(beta)u = lambda u, x is an element of (a, b), u(a) = u(b) = 0, 1 < alpha + beta < 2, where the operators D-a+(alpha) and D-b-(beta) are left- and right-sided Riemann-Liouville derivatives, respectively. These operators are nonlocal and nonsymmetric, however, share certain classic elliptic properties. 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.