The Time Fractional Schrödinger Equation on Hilbert Space

We study the linear fractional Schrodinger equation on a Hilbert space, with a fractional time derivative of order (0u003calpha u003c1,) 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_{alpha }(t)}_{tge 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.