Resolvent estimates for a sum of tensor products with applications to the spectral theory of differential operators
type: Item
instance of: scholarly article
title: Resolvent estimates for a sum of tensor products with applications to the spectral theory of differential operators
zbMATH Open document ID: 0591.47034
DOI: 10.1007/BF02790185
author: Matania Ben-Artzi
author: Allen Devinatz
published in: Journal d'Analyse Mathématique
publication date: 1984-01-01T00:00:00Z
review text: Let H be a self-adjoint operator in a Hilbert space \({\mathcal H}\), \(\sigma\) (H) its spectrum and \(R(z)=(H-z)^{-1}\), Im \(z\neq 0\), its resolvent. Let \(K\subseteq \sigma (H)\). H is said to satisfy a limiting absorption principle (l.a.p. for short) in K, if the limits \(R^{\pm}(\lambda)=\lim_{\epsilon \to 0^+}R(\lambda \pm i\epsilon)\), \(\lambda\in K\) exist in some suitable operator topology. (*) \(H=H_ 1\otimes I_ 2+I_ 1\otimes H_ 2\) where \(H_ 1\) and \(H_ 2\) are self-adjoint operators on Hilbert spaces \({\mathcal H}_ 1\) and \({\mathcal H}_ 2\), and \(I_ 1\) and \(I_ 2\) are the identity operators of these Hilbert spaces, respectively. Clearly H acts on \({\mathcal H}={\mathcal H}_ 1\otimes {\mathcal H}_ 2.\) The paper revolves around the question: Given that \(H_ 1\) and \(H_ 2\) satisfy a l.a.p. in \({\mathcal H}_ 1,{\mathcal H}_ 2\) respectively. What are the conditions needed to guarantee that H defined by (*), satisfies a l.a.p. in \({\mathcal H}?\) After developing some elementary expressions for the resolvent of H, the authors relate the validity of the l.a.p. for H with the Hölder continuity properties of \(A_ 1(\lambda)\) i.e. \(\frac{d}{d\lambda}E_ 1(\lambda)\), where \(E_ 1(\lambda)\) is the spectral family of \(H_ 1\). The succeed in obtaining the l.a.p. for a class of operators including the Hamiltonian of a uniform electric field. Later they replace Hölder continuity properties by other assumptions on \(H_ 1,H_ 2\). Several concrete examples of differential operators are outlined. There are two appendices one containing the review of some basic properties of the spectral measure associated with a self-adjoint constant coefficient differential operator in \({\mathbb{R}}^ n\), and the l.a.p. for some second order ordinary differential operators is discussed in the second.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47L90
Mathematics Subject Classification ID: 47A25
zbMATH DE Number: 3949094
zbMATH Keywords: spectrum
zbMATH Keywords: resolvent
zbMATH Keywords: limiting absorption principle
zbMATH Keywords: Hamiltonian of a uniform electric field
zbMATH Keywords: Hölder continuity
zbMATH Keywords: spectral measure
zbMATH Keywords: self- adjoint constant coefficient differential operator
zbMATH Keywords: second order ordinary differential operators
MaRDI profile type: Publication
cites work: Q4076370
cites work: Asymptotic properties of solutions of differential equations with simple characteristics
cites work: An Application of Asymptotic Techniques to Certain Problems of Spectral and Scattering Theory of Stark-Like Hamiltonians
cites work: Unitary equivalence and scattering theory for Stark-like Hamiltonians
cites work: Q5539254
cites work: Q3280274
cites work: Q3663745
cites work: THE PRINCIPLE OF LIMIT AMPLITUDE
cites work: Q4069306
cites work: Unitary equivalence of Stark Hamiltonians
cites work: Banach space techniques in the perturbation theory of self-adjoint operators with continuous spectra
cites work: The abstract theory of scattering
cites work: Spectral properties of operators arising in acoustic wave propagation in an ocean of variable depth
cites work: Spectral analysis of N-body Schrödinger operators
cites work: Q5653193
cites work: Q4066072
cites work: Spectral representations for Schrödinger operators with long-range potentials
cites work: Q5631480
cites work: Spectral analysis of the Pekeris operator in the theory of acoustic wave propagation in shallow water
cites work: Q3866506
full work available at URL: https://doi.org/10.1007/bf02790185
OpenAlex ID: W2014370083
description: scientific article; zbMATH DE number 3949094