We shall examine the fractional generalization of the eigenvalue problem of Schrodinger’s equation for one dimensional problems in connection with Levy stable probability distributions. The corresponding Sturm–Liouville (SL) problem for the fractional Schrodinger equation is formulated and solved on \(\mathbb {R}\) satisfying natural Dirichlet boundary conditions. The eigenvalues and eigenfunctions are computed in a numerical Sinc approximation applied to the Riesz–Feller representation of Schrodinger’s generalized equation. We demonstrate that the eigenvalues for a fractional operator approach deliver the well known eigenvalues of the integer order Schrodinger equation and are consistent with analytic WKB estimations. We can also confirm the conjecture that only for skewness parameters \(\theta =0\) the eigenvalues are real quantities and thus relevant in quantum mechanics. However, for skewness parameters \(\theta \ne 0\), the Sinc approach yields complex eigenvalues with related complex eigenfunctions, and a fortiori, real probability densities.