The Riesz derivative is essentially the fractional Laplacian on Rn. In this talk, we only show interest in the one-dimensional case. We have found that there is difference between the truncated Riesz derivative and the fractional Laplacian. The truncated Riesz derivative (now called Riesz derivative as usual) is suitable for application. We consider the eigen-spectra of such a derivative operator defined on a bounded interval. We can obtain its analytical eigen-pairs.