Continuous-time random walk as a guide to fractional Schrödinger equation

We argue that the continuous-time random walk approach may be a useful guide to extend the Schrödinger equation in order to incorporate nonlocal effects, avoiding the inconsistencies raised by Jeng et al. [J. Math. Phys. 51, 062102 (2010)]. As an application, we work out a free particle in a half space, obtaining the time dependent solution by considering an arbitrary initial condition.