Singular perturbations in the interaction representation, II
type: Item
instance of: scholarly article
title: Singular perturbations in the interaction representation, II
zbMATH Open document ID: 0531.47016
DOI: 10.1016/0022-1236(82)90071-4
author: Rhonda J. Hughes
published in: Journal of Functional Analysis
publication date: 1982-01-01T00:00:00Z
review text: This paper is a sequel and generalization of a previous paper [J. Funct. Anal. 38, 71-98 (1980; Zbl 0454.47014)] by the same author and \textit{I. E. Segal} and is devoted to the definition and construction of the unitary one-parameter group U(t) in \(L^ 2({\mathbb{R}})\) (more generally \(L^ 2({\mathbb{R}}^ k))\) formally generated by the operator \(H=H_ 0+V\) where \(H_ 0=(-id/dx)^ m\) in \(L^ 2({\mathbb{R}})\) (more generally \(H_ 0=(- \Delta)^{m/2}\), \(m\geq 2\), in \(L^ 2({\mathbb{R}}^ k))\), and V is a real distribution in \({\mathcal S}'({\mathbb{R}})\) (or \({\mathcal S}'({\mathbb{R}}^ k))\) with Fourier transform \(\hat V\in L^{\infty}_{loc}\) and prescribed polynomial growth at infinity in moment space. The method is the same as in the previous paper and consists in proving the norm convergence for small time intervals of the perturbation series for the two parameter unitary group \[ U(t;s)=\exp(it H_ 0)\quad \exp(-i(t-s)H)\exp(is H_ 0), \] generated by H in the so called interaction representation familiar in Quantum Mechanics (see the abstract of the first paper for more detail). In dimension 1, the result holds under the growth condition on \^V: \(| \hat V(\xi)| \leq M(1+| \xi |^{\alpha})\quad with\quad \alpha =(m-(1+\epsilon))/2,\quad 0<\epsilon \leq m-1.\) In addition, U is continuous as a function of V in the uniform operator topology for \(\hat V\) converging uniformly in \({\mathbb{R}}\), and in the strong operator topology for \(\hat V\) converging pointwise almost everywhere and satisfying the previous estimate with fixed M and \(\alpha\). The analogous results in dimension \(k>1\) are briefly mentioned in the last section of the paper. The proof consists in momentum space estimates of the integral kernels of the operator \[ \int^{t}_{0}ds \exp(is H_ 0)V \exp(-is H_ 0) \] and suitable iterates thereof, which constitute the terms in the perturbation series for U(t,s).
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 81Q15
Mathematics Subject Classification ID: 35G10
zbMATH DE Number: 3842537
zbMATH Keywords: singular perturbation
zbMATH Keywords: unitary one-parameter group
zbMATH Keywords: interaction representation
reviewed by: Jean Ginibre
MaRDI profile type: Publication
full work available at URL: 0022-1236(82)90071-4
OpenAlex ID: W4249863003
cites work: Singular Perturbations and Nonstandard Analysis
cites work: Q5726865
cites work: Perturbations of the Schrödinger equation by potentials with small support
cites work: Singular perturbations in the interaction representation
cites work: Q4110164
description: scientific article; zbMATH DE number 3842537