Smoothing effect for some Schrödinger equations
type: Item
instance of: scholarly article
title: Smoothing effect for some Schrödinger equations
zbMATH Open document ID: 0681.35079
DOI: 10.1016/0022-1236(89)90039-6
author: Nakao Hayashi
author: Tohru Ozawa
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: 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 \(H^ 1({\mathbb{R}}^ n)\).
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 4115167
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: smoothing effect
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90039-6
OpenAlex ID: W2063162794
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description: scientific article; zbMATH DE number 4115167