Almost global existence for a fractional Schrodinger equation on spheres and tori

TL;DR

This paper proves almost global existence of solutions for a fractional Schrödinger equation on spheres and tori, for a broad class of fractional powers greater than one-half, extending understanding of long-term behavior in such systems.

Contribution

It establishes almost global existence results for fractional Schrödinger equations on spheres and tori for almost all fractional powers greater than one-half, a significant extension in the field.

Findings

Almost global existence for fractional Schrödinger equations with s>1/2.

Results hold for solutions on spheres and tori.

Applicable to a broad class of smooth nonlinearities.

Abstract

We study the time of existence of the solutions of the following Schr\"odinger equation $$i\psi_t = (-\Delta)^s \psi +f(|\psi|^2)\psi, x \in \mathbb S^d, or x\in\T^d$$ where $(-\Delta)^s$ stands for the spectrally defined fractional Laplacian with $s>1/2$ and $f$ a smooth function. We prove an almost global existence result for almost all $s>1/2$.