Smooth Feshbach map and operator-theoretic renormalization group methods.
type: Item
instance of: scholarly article
title: Smooth Feshbach map and operator-theoretic renormalization group methods.
zbMATH Open document ID: 1060.47028
DOI: 10.1016/S0022-1236(03)00057-0
author: Volker Bach
author: Thomas Chen
author: Jürg Fröhlich
author: I. M. Sigal
published in: Journal of Functional Analysis
publication date: 2003-10-14T00:00:00Z
review text: Having been created by Feshbach in the 1950s as a tool of perturbation theory in Hilbert space, three of the present article's authors have, in recent years, developed the Feshbach projection method to a tool of spectral analysis of perturbed operators. Given an operator \(T\) with known spectrum, the method concerns perturbations \(H:=T+W\) of \(T\) to which an operator \(F(H,W)\) can be assigned whose main spectral properties coincide with those of \(T\) at \(0\). Since \(F\) may be considerably simpler to analyze (it may even be a finite matrix), this fact has turned out to be, apart from its mathematical appeal, of particular use in applications. The construction of every particular Feshbach map \(F_P\) is based, in the earlier work, on an appropriately chosen ``cutoff'' projection \(P\). The elements of the domain of \(F_P\) are called the Feshbach pairs associated with \(P\). The present paper generalizes the notion of Feshbach map by replacing the projection \(P\) by an arbitrary positive operator \(\chi\leq1\). The scope of the analysis are such choices of \(\chi\) that make \(F_\chi\) ``smooth'' in the sense that a new operator-theoretic renormalization group method can be established with the help of \(F_\chi\). The method is illustrated with the example of a class of Fock space Hamiltonians.
reviewed by: Bernd Kuckert
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 81T17
zbMATH DE Number: 1992838
zbMATH Keywords: Feshbach projection method
zbMATH Keywords: Feshbach map
zbMATH Keywords: perturbation theory
zbMATH Keywords: renormalization group
MaRDI profile type: Publication
cites work: Quantum electrodynamics of confined nonrelativistic particles
cites work: Renormalization group analysis of spectral problems in quantum field theory
cites work: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field
cites work: Positive commutators and the spectrum of Pauli-Fierz Hamiltonian of atoms and molecules
cites work: Spectral theory of Pauli-Fierz operators.
cites work: Unified theory of nuclear reactions
cites work: The Livsic matrix in perturbation theory
cites work: Q4110164
cites work: Positive commutators in non-equilibrium quantum statistical mechanics: Return to equilibrium
full work available at URL: s0022-1236(03)00057-0
OpenAlex ID: W1970300969
Recommended article: Q5395313
Recommended article: Fermionic renormalization group flows. Technique and theory
Recommended article: The Feshbach-Schur map and perturbation theory
Recommended article: Functional renormalization group for interacting Fermi systems
Recommended article: Renormalization group for strongly coupled maps
Recommended article: A renormalization group for Hamiltonians: numerical results
Recommended article: Functional Renormalization Group Computation of Interacting Fermi Systems
Recommended article: The renormalized Bogoliubov–Fröhlich Hamiltonian
Recommended article: Fermionic Functional Integrals and the Renormalization Group
Recommended article: Renormalization and quantum scaling of Frenkel-Kontorova models
description: scientific article; zbMATH DE number 1992838