Smoothing effect for some Schrödinger equations (Q1823402)

We study the Cauchy problem for the Schrödinger equation in \({\mathbb{R}}^ n:\) \[ (S)\quad i\partial_ tu+(1/2)\Delta u=V_ 1u+(V_ 2*| u|^ 2)u,\quad (t,x)\in {\mathbb{R}}\times {\mathbb{R}}^ n;\quad u(0)\neq \emptyset,\quad x\in {\mathbb{R}}^ n, \] where \(V_ 1=V_ 1(x)=\lambda_ 1| x|^{-\gamma_ 1}\), \(V_ 2=V_ 2(x)=\sum^{3}_{k=2}\lambda_ k| x|^{-\lambda_ k}\), \(\lambda_ k\in {\mathbb{R}}\) (1\(\leq k\leq 3)\), \(0<\gamma_ 2,\gamma_ 3<\min (2,n).\) We prove the existence, uniqueness, and smoothing effect of global solutions of (2) with \(\phi\) not necessarily in the Sobolev space \(H^ 1({\mathbb{R}}^ n)\).