Forward and backward problems for abstract time-fractional Schrödinger equations

Abstract

We investigate forward and backward problems associated with abstract time-fractional Schrödinger equations ${\mathrm{i}}^{\hat{I}}½{\partial }_t^{\hat{I}}\pm u(t)+Au(t)=0$, $\hat{I}\pm \in (0,1)\cup (1,2)$ and $\hat{I}½\in \{1,\hat{I}\pm \}$, 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 $\hat{I}\pm$, models quantum systems with memory effects and anomalous wave propagation. We first establish the well-posedness of the forward problems in two scenarios: ($\hat{I}½=1,$ $\hat{I}\pm \in (0,1)$) and ($\hat{I}½=\hat{I}\pm ,$ $\hat{I}\pm \in (0,1)\cup (1,2)$). Then, we prove well-posedness and stability results for the backward problems depending on the two cases $\hat{I}½=1$ and $\hat{I}½=\hat{I}\pm$. 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.