Dispersive estimates for time and space fractional Schr\"odinger equations

TL;DR

This paper establishes sharp dispersive decay estimates for solutions to fractional Schrödinger equations involving time and space fractional derivatives, advancing understanding of their long-term behavior.

Contribution

It provides the first sharp decay rate estimates for fractional Schrödinger equations with both time and space fractional derivatives.

Findings

Proved sharp decay rates for fractional Schrödinger solutions

Established dispersive estimates for equations with fractional derivatives

Enhanced understanding of fractional quantum dynamics

Abstract

In this paper, we consider the Cauchy problem for the fractional Schr\"odinger equation $i D_t^\alpha u + (-\Delta)^{\frac{\beta}{2}} u =0$ with $0<\alpha<1$, $\beta>0$. We establish the dispersive estimates for the solutions. In particular, we prove that the decay rates are sharp.