Forward and backward problems for abstract time-fractional Schrödinger equations

We investigate forward and backward problems associated with abstract time-fractional Schrödinger equations i^ν∂ₜ^αu(t) + A u(t)=0, α∈ (0,1)∪ (1,2) and ν∈\1,α\, 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 $α$, models quantum systems with memory effects and anomalous wave propagation. We first establish the well-posedness of the forward problems in two scenarios: ($ν=1,\,$ α∈ (0,1)) and ($ν=α,\,$ α∈ (0,1)∪ (1,2)). Then, we prove well-posedness and stability results for the backward problems depending on the two cases $ν=1$ and $ν=α$. 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.