Singularities of the Green function of the nonstationary Schrödinger equation
type: Item
instance of: scholarly article
title: Singularities of the Green function of the nonstationary Schrödinger equation
zbMATH Open document ID: 0920.35008
DOI: 10.1007/BF02482598
author: M. V. Buslaeva
author: V. S. Buslaev
publication date: 1999-09-13T00:00:00Z
review text: The Green function \(\psi= \psi(x,y,t)\) of the nonstationary Schrödinger equation is determined by the relations \[ ih\psi_t= -\textstyle{{1\over 2}} h^2\Delta\psi+ v(x)\psi,\quad \psi(x,y,0)= \delta(x- y),\;t\geq 0,\;y\in\mathbb{R}^d. \] In what follows, we assume that \(v\) is a smooth real function. For a chosen \(t>0\), the function \(\psi\) can have singularities with respect to \(x\). These singularities depend on the rate of growth of \(v(x)\) as \(x\to\infty\). If \(v\) satisfies the condition \[ | v(x)|\leq c| x|^{2\gamma},\quad \gamma<1, \] for sufficiently large \(x\), then \(\psi\) is smooth. This assertion, as well as several subsequent assertions, is given here in a simplified form. The main objective of the present paper is to describe the formula mechanism responsible for the form of the singularities of \(\psi\) for \(\gamma>1\) in the case \(d=1\). Under certain restrictions, we shall be able to say something about the character of the singularities of \(\psi\) for \(d>1\).
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 1267936
zbMATH Keywords: Green function
zbMATH Keywords: nonstationary Schrödinger equation
zbMATH Keywords: singularities
zbMATH Keywords: formula mechanism
MaRDI profile type: Publication
cites work: Reconstruction of singularities for solutions of Schrödinger's equation
cites work: Q4880958
cites work: On the fundamental solution of a perturbed harmonic oscillator
cites work: Q3896271
full work available at URL: https://doi.org/10.1007/bf02482598
OpenAlex ID: W2076175096
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1267936