The nonlinear Schrödinger equations with harmonic potential in modulation spaces

Abstract

We study nonlinear Schrödinger $i{\partial }_{t}u-Hu=F(u)$ (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 $M^{p,p}({\mathbb{R}}^d)$ $(1\le p<2d/(d+\gamma ),0<\gamma <\min \{2,d/2\}).$ When $F(u)=(K\ast |u{|}^{2k})u$ with $K\in \mathcal{F}{L}^q$ (Fourier-Lebesgue spaces) or $M^{\infty ,1}$ (Sjöstrand'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 $L_s^p-$Sobolev spaces.

23 pages. arXiv admin note: text overlap with arXiv:1810.04076

References in corpus (2)

• (Semi)classical limit of the Hartree equation with harmonic potential
• On the Schrödinger equation with potential in modulation spaces

Cited by in corpus (1)

• Global Cauchy problems for the Klein-Gordon, wave and fractional Schrödinger equations with Hartree nonlinearity on modulation spaces