The time fractional Schr\"odinger equation on Hilbert space

We study the linear fractional Schr"odinger equation on a Hilbert space, with a fractional time derivative of order $0<\alpha <1,$ and a self-adjoint generator $A\mathrm{.}$ Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family $\{U_{\alpha }(t){\}}_{t\ge 0}$. Moreover, we prove that the solution family $U_{\alpha }(t)$ converges strongly to the family of unitary operators $e^{-itA},$ as $\alpha$ approaches to $1$.