On the well-posedness of time-space fractional Schr\"{o}dinger equation on $\mathbb{R}^{d}$

TL;DR

This paper investigates the well-posedness of time-space fractional Schrödinger equations, establishing dispersive estimates and demonstrating local and global solutions in various function spaces, while analyzing asymptotic behavior and self-similarity.

Contribution

It introduces novel dispersive estimates for fractional Schrödinger operators, addressing derivative loss and extending well-posedness results to fractional contexts with new analytical techniques.

Findings

Established dispersive estimates for fractional Schrödinger operators.

Proved local and global well-posedness in Sobolev and Lorentz spaces.

Analyzed asymptotic behavior and existence of self-similar solutions.

Abstract

This paper considers the well-posedness of a class of time-space fractional Schr\"{o}dinger equations introduced by Naber. In contrast to the classical Schr\"{o}dinger equation, the solution operator here exhibits derivative loss and lacks the structure of a semigroup, which makes the classical Strichartz estimates inapplicable. By using harmonic analysis tools -- including the smoothing effect theory of Kenig and Ponce for Korteweg-de Vries equations \cite[\emph{Commun.~Pure Appl.~Math.}]{Kenig}, real interpolation techniques, and the Van der Corput lemma -- we establish novel dispersive estimates for the solution operator. These estimates generalize Ponce's regularity results \cite[\emph{J.~Funct.~Anal.}]{Ponce} for oscillatory integrals and enable us to address the derivative loss in the Schr\"{o}dinger kernel. For the cases $\beta<2$~(in one space dimension) and $\beta>2$~(in higher dimensions), we prove local and global well-posedness in Sobolev and Lorentz-type spaces, respectively. Additionally, we analyze the asymptotic behavior of solutions and demonstrate the existence of self-similar solutions under homogeneous initial data. The results highlight the interplay between fractional derivatives, dispersive properties, and nonlinear dynamics, extending the understanding of nonlocal evolution equations in quantum mechanics and related fields.