The nonlinear Schr\"odinger equations with harmonic potential in modulation spaces

Abstract

We study nonlinear Schr\"odinger $i\partial_tu-Hu=F(u)$ (NLSH) equation associated to harmonic oscillator $H=-\Delta +|x|^2$ in modulation spaces $M^{p,q}.$ When $F(u)= (|x|^{-\gamma}\ast |u|^2)u, $ we prove global well-posedness for (NLSH) in modulation spaces $M^{p,p}(\mathbb R^d)$ $ (1\leq p < 2d/(d+\gamma), 0<\gamma< \min \{ 2, d/2\}).$ When $F(u)= (K\ast |u|^{2k})u$ with $K\in \mathcal{F}L^q $ (Fourier-Lebesgue spaces) or $M^{\infty,1}$ (Sj\"ostrand's class) or $M^{1, \infty},$ some local and global well-posedness for (NLSH) are obtained in some modulation spaces. When $F$ is real entire and $F(0)=0$, we prove local well-posedness for (NLSH) in $M^{1,1}.$ As a consequence, we can get local and global well-posedness for (NLSH) in a function spaces$-$which are larger than usual $L^p_s-$Sobolev spaces.