Time dependent representations of the stationary wave operators for ''oscillating'' long-range potentials
type: Item
instance of: scholarly article
title: Time dependent representations of the stationary wave operators for ''oscillating'' long-range potentials
zbMATH Open document ID: 0549.47003
author: K. Mochizuki
author: Jun Uchiyama
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1982-01-01T00:00:00Z
review text: Assume that \(V:{\mathbb{R}}^ n\to {\mathbb{R}}\) is a potential that can be split into a short-range part and a long-range term \(V_ L\) which may oscillate in a certain way. Let c be a real number which is specified below. The unique self-adjoint extensions of \(-\Delta +c\) and \(-\Delta +V\) in \(L^ 2({\mathbb{R}}^ n)\) are denoted by \(H_ 0\) and H, respectively, and \(E_ 0(I)\) is the spectral projection of \(H_ 0\) for the interval \(I\subset\subset {\mathbb{R}}\). With the help of functions \(\delta_+\) and \(\delta_-\) which satisfy a certain eikonal inequality, and using the spectral representation of \(H_ o\), the authors introduce identification operators \(J_{\pm}(I)\). They prove that \[ W_{\pm}(t):=e^{itH}J_{\pm}(I)e^{-itH_ 0}E_ 0(I)\quad (t\in {\mathbb{R}}) \] are bounded operators by following a method of \textit{A. P. Calderón} and \textit{R.Vaillancourt} [Proc. Nat. Acad. Sci. USA 69, 1185- 1187 (1972; Zbl 0244.35074)] to show the \(L^ 2\)-boundedness of pseudo- differential operators. Next, using the stationary phase method, they prove that \(W_{\pm}(t)\) converge strongly as \(t\to\pm \infty\) to wave opertors \(U_{\pm}(I)^{\pm}\) the existence and completeness of which the authors had established by means of a time-independent method in an earlier paper [J. Math. Kyoto Univ. 19, 47-70 (1979; Zbl 0414.47013)]. The position of the interval I is determined by the number c which is explicitly given, essentially in terms of the asymptotic behaviour of \[ r \partial V_ L/\partial r+const.\quad V_ L. \] The authors mention that earlier modified wave operators coincide modulo simple unitary operators with the usual Møller wave operators when V is entirely short-range, but that it is an open problem whether this continues to hold for long-range potentials for which the latter exist, for example \(V=V_ L=a \sin br/r.\)
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 3876005
zbMATH Keywords: time dependent representations of the stationary wave operators
zbMATH Keywords: oscillating long-range potentials
zbMATH Keywords: short-range part
zbMATH Keywords: spectral projection
zbMATH Keywords: eikonal inequality
zbMATH Keywords: spectral representation
zbMATH Keywords: \(L^ 2\)-boundedness of pseudo-differential operators
zbMATH Keywords: stationary phase method
zbMATH Keywords: time-independent method
zbMATH Keywords: modified wave operators
reviewed by: Hubert Kalf
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195183288
OpenAlex ID: W2086449730
cites work: Q5679323
cites work: Q5592185
cites work: Q4186815
cites work: Spectral and scattering theory for the adiabatic oscillator and related potentials
cites work: A Class of Bounded Pseudo-Differential Operators
cites work: The existence of wave operators in scattering theory
cites work: A stationary approach to the existence and completeness of long-range wave operators
cites work: Q4157198
cites work: Scattering theory with two Hilbert spaces
cites work: Scattering theory for Schrödinger operators with long-range potentials. II: Spectral and scattering theory
cites work: Radiation conditions and spectral theory for 2-body Schrödinger operators with ''oscillating'' long-range potentials. I: The principle of limiting absorption
cites work: Radiation conditions and spectral theory for 2-body Schrödinger operators with ''oscillating'' long-range potentials. II: Spectral representation
cites work: Q5332803
DOI: 10.2977/PRIMS/1195183288
Recommended article: Q4336948
Recommended article: Remarks on our paper ''A scattering theory for time-dependent long-range potentials''
Recommended article: Q4724190
Recommended article: Semi-classical pseudo-differential operators
Recommended article: Spectral distributions for long range perturbations
Recommended article: The scattering amplitude for the Schrödinger equation with a long-range potential
Recommended article: A time-dependent, two Hilbert space approach to long-range quantum scattering
Recommended article: Q4022610
Recommended article: Schrodinger Operators with Rapidly Oscillating Central Potentials
Recommended article: The scattering of fractional Schrödinger operators with short range potentials
description: scientific article; zbMATH DE number 3876005