On the fractional abstract Schrodinger-type evolution equations on the Hilbert space and its applications to the fractional dispersive equations

Abstract

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.