The time fractional Schrödinger equation on Hilbert space

We study the linear fractional Schrödinger equation on a Hilbert space, with a fractional time derivative of order $0<α<1,$ and a self-adjoint generator $A.$ 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 $\{U_α(t)\}_{t\geq 0}$. Moreover, we prove that the solution family $U_α(t)$ converges strongly to the family of unitary operators $e^{-itA},$ as $α$ approaches to $1$.

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