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

TL;DR

This paper establishes the well-posedness of fractional Schrödinger-type evolution equations on Hilbert spaces and applies these results to fractional dispersive equations, analyzing solution regularity and asymptotics.

Contribution

It introduces a novel approach using spectral theory and Mittag-Leffler functions to prove well-posedness for fractional Schrödinger equations without relying on semigroup properties.

Findings

Proved local and global well-posedness of fractional Schrödinger equations.

Analyzed regularity and asymptotic behavior of solutions.

Applied results to fractional dispersive equations with polynomial symbol conditions.

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.