On Fractional Schrodinger Equations in sobolev spaces

TL;DR

This paper studies the fractional nonlinear Schrödinger equation in Sobolev spaces, establishing conditions for local well-posedness and ill-posedness, thereby advancing understanding of its mathematical properties relevant to quantum physics.

Contribution

It provides the first comprehensive analysis of well-posedness and ill-posedness for fractional Schrödinger equations with power nonlinearities in Sobolev spaces.

Findings

Established local well-posedness in certain Sobolev spaces.

Demonstrated ill-posedness in other Sobolev spaces.

Clarified the role of the fractional order in solution behavior.

Abstract

Let $\sigma\in(0,1)$ with $\sigma\neq\frac{1}{2}$. We investigate the fractional nonlinear Schr\"odinger equation in $\mathbb R^d$: $$i\partial_tu+(-\Delta)^\sigma u+\mu|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$$ where $(-\Delta)^\sigma$ is the Fourier multiplier of symbol $|\xi|^{2\sigma}$, and $\mu=\pm 1$. This model has been introduced by Laskin in quantum physics \cite{laskin}. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.