Forward and backward problems for abstract time-fractional Schr\"odinger equations

We investigate forward and backward problems associated with abstract time-fractional Schr\"odinger equations $\mathrm{i}^\nu \partial_t^\alpha u(t) + A u(t)=0$, $\alpha \in (0,1)\cup (1,2)$ and $\nu\in\{1,\alpha\}$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$. This kind of equation, which incorporates the Caputo time-fractional derivative of order $\alpha$, models quantum systems with memory effects and anomalous wave propagation. We first establish the well-posedness of the forward problems in two scenarios: ($\nu=1,\,$ $\alpha \in (0,1)$) and ($\nu=\alpha,\,$ $\alpha \in (0,1)\cup (1,2)$). Then, we prove well-posedness and stability results for the backward problems depending on the two cases $\nu=1$ and $\nu=\alpha$. 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.