We investigate forward and backward problems associated with abstract time-fractional Schr & ouml;dinger equations ivdta u(t) + Au(t) = 0, a E (0,1) boolean OR (1,2) and v E {1, a}, where A is a selfadjoint operator with compact resolvent on a Hilbert space H. This kind of equation, which incorporates the Caputo time-fractional derivative of order a, models quantum systems with memory effects and anomalous wave propagation. We first establish the well-posedness of the forward problems in two scenarios: (v = 1, a E (0,1)) and (v = a, a E (0,1) boolean OR (1,2)). Then, we prove wellposedness and stability results for the backward problems depending on the two cases v = 1 and v = a. Our approach employs the solution's eigenvector expansion along with the properties of the Mittag-Leffler functions, including the distribution of zeros and asymptotic expansions. Finally, we conclude with a discussion of some open problems.