Abstract.  The existence of eigenfunctions of fractional Laplacian with homogeneous zero boundary condition has been well-known a long time ago, however, how to write out their analytic expressions is a challenging task. In this presentation, I will focus on the 1-dimensional case only and provide a potential way to obtain the analytic eigenfunctions. We will show the spectral problem of 1-D fractional Laplacian is equivalent to the spectrum problem of double-sided Riemann-Liouville fractional diffusion derivatives. By using the latter, we turn the fractional differential equation into the singular integral equation involving the Hilbert transform, from which the eigenfunctions can be solved informally. Lastly, some open questions of this ongoing work will be shared as well.