Phase shifts and the modified and Møller wave operatos
type: Item
instance of: scholarly article
title: Phase shifts and the modified and Møller wave operatos
zbMATH Open document ID: 0531.35063
DOI: 10.1007/BF00403268
published in: Letters in Mathematical Physics
publication date: 1982-01-01T00:00:00Z
review text: This is an announcement of the author's results which have appeared subsequently in Trans. Am. Math. Soc. 275, 641-677 (1983). The results concern spectral and scattering theory for Schrödinger operators, \(H=- \Delta +V\) with long range central potentials V. Typical of the potentials considered are \((*)\quad V(x)=(1+r)^{- \delta}\sin(r^{\alpha})\) where \(\alpha\geq 0\), \(\delta>\min(0,1-\alpha)\) and \(r=| x|\). The spectral results for H are: essential self adjointness on \(C_ 0^{\infty}({\mathbb{R}}^ n)\), absence of singularly continuous spectrum, unitary equivalence of the continuous part of H to - \(\Delta\), and an eigenfunction expansion for H. The possibility of positive eigenvalues is also considered. We state the scattering theory results only for V as in (*): the (Dollard) modified wave operators exist and are complete if \(\delta>1/2\) or \((\delta>1/4\) and \(\delta +\alpha>1)\) or \((\delta +\alpha>5/4)\); the usual wave operators exist (and are complete) if and only if \((\delta +\alpha>1\) and \(\delta>1/2)\) or \((\delta +\alpha)>3/2\). The role of the eigenfunction expansion for H in establishing the scattering theoretical results is briefly discussed.
Mathematics Subject Classification ID: 35P25
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 81U05
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 3842392
zbMATH Keywords: phase shifts
zbMATH Keywords: wave operators
zbMATH Keywords: central oscillating potential
zbMATH Keywords: long range central potentials
zbMATH Keywords: essential self adjointness
author: Denis A. White
MaRDI profile type: Publication
cites work: Q4191141
cites work: Q4185363
cites work: Spectral and scattering theory for the adiabatic oscillator and related potentials
cites work: The existence of wave operators for oscillating potentials
cites work: Existence of the Moller wave operators for \(V(\tau)={\lambda\sin (\mu r^\alpha )\over r^\beta}\)
cites work: The existence of wave operators in scattering theory
cites work: Completeness of modified wave operators for long-range potentials
cites work: The eigenvalues of the Schrödinger operator, situated on the continuous spectrum
full work available at URL: https://doi.org/10.1007/bf00403268
OpenAlex ID: W1970489078
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description: scientific article; zbMATH DE number 3842392