  In this paper we prove the local and global well-posedness of the time
fractional abstract Schr\"odinger type evolution equation on the Hilbert space
and as an application, we prove the local and global well-posedness of the
fractional dispersive equation with static potential under the only assumption
that the symbol of P(D) behaves like a polynomial of highest degree m at
infinity. In appendix, we also give the Holder regularities and the asymptotic
behaviors of the mild solution to the linear time fractional abstract
Schr\"odinger type equation. Because of the lack of the semigroup properties of
the solution operators, we employ a strategy of proof based on the spectral
theorem of the self-adjoint operators and the asymptotic behaviors of the
Mittag-Leffler functions.
