Abstract
The fractional Schr\"odinger equation is studied with a kinetic energy that slightly deviates from the usual nonrelativistic form. The harmonic oscillator and the Kepler problem are both treated in the context of small perturbations. The usual perturbation theory is used and compared with the envelope theory. The analytical results show good agreement between both methods, indicating possible future developments for many-body systems. A possible connection with experimental observations is briefly discussed.