The Mourre theory for analytically fibered operators
type: Item
instance of: scholarly article
title: The Mourre theory for analytically fibered operators
zbMATH Open document ID: 0939.47019
author: Christian Gérard
author: Francis Nier
published in: Journal of Functional Analysis
publication date: 2000-07-05T00:00:00Z
review text: The Mourre method for self-adjoint operators was devised by \textit{E. Mourre} [Commun. Math. Phys. 78, 391-408 (1981; Zbl 0489.47010)]. Let \(H_0\) be a self-adjoint operator acting on a Hilbert space \(H\). The Mourre theory for such an operator is based on the construction of another self-adjoint operator \(A\), called a conjugate operator, such that the following estimate holds: \[ 1_\Delta(H_0)[H_0, iA]1_\Delta(H_0)\geq c_0 1_\Delta(H_0)+ K,\tag{1} \] where \(1_\Delta(H_0)\) denotes the spectral projection on the interval \(\Delta\subset\mathbb{R}\) for the operator \(H_0\), \(c_0\) is a positive constant and \(K\) is a compact operator. The estimate (1) is a Mourre estimate. The main aim of the paper is the construction of the conjugate operator and to prove the strict Mourre estimate \((K=0)\) for an abstract class of self-adjoint operators which are called analytically fibered operators. Three examples of the last class include: 1. matrix valued differential operators with constant coefficients, 2. neutral two-particle systems in a magnetic field, and 3. periodic Schrödinger operators. The paper is interesting and addresses to specialists in mathematical physics.
reviewed by: J. J. Telega
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 1121259
zbMATH Keywords: Schrödinger operator
zbMATH Keywords: matrix valued differential operators with constant coefficients
zbMATH Keywords: neutral two-particle systems in a magnetic field
zbMATH Keywords: periodic Schrödinger operators
zbMATH Keywords: Mourre method
zbMATH Keywords: spectral projection
zbMATH Keywords: analytically fibered operators
MaRDI profile type: Publication
full work available at URL: jfan.1997.3154
OpenAlex ID: W2064699000
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DOI: 10.1006/JFAN.1997.3154
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description: scientific article; zbMATH DE number 1121259