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

Abstract

We study the time of existence of the solutions of the following Schrödinger equation $$iψ_t = (-Δ)^s ψ+f(|ψ|^2)ψ, x \in \mathbb S^d, or x\in\T^d$$ where $(-Δ)^s$ 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 $s>1/2$.