Essential spectrum of a self-adjoint operator on an abstract Hilbert space of Fock type and applications to quantum field Hamiltonians
type: Item
instance of: scholarly article
title: Essential spectrum of a self-adjoint operator on an abstract Hilbert space of Fock type and applications to quantum field Hamiltonians
zbMATH Open document ID: 0965.47015
author: Asao Arai
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-08-27T00:00:00Z
full work available at URL: http://hdl.handle.net/2115/69195
review text: Let \({\mathcal H}\) and \({\mathcal K}\) be Hilbert spaces and \({\mathcal F}\) the tensor product \({\mathcal H}\otimes {\mathcal F}_b({\mathcal K})\), where \({\mathcal F}_b({\mathcal K})\) be the Bosonic Fock space over \({\mathcal K}\). Let \(A\) be a self-adjoint opertor on \({\mathcal H}\) which is bounded below, \(S\) a nonnegative injective self-adjoint operator on \({\mathcal K}\), \(d\Gamma(S)\) its second quantization, \(H_0= A\otimes I+ I\otimes d\Gamma(S)\) and \(H_I\) a symmetric operator on \({\mathcal F}\), with \({\mathcal D}(H_0)\cap{\mathcal D}(H_I)\) dense in \({\mathcal F}\). The paper is concerned with the symmetric operator \(H= H_0+ H_I\). In the case when \(H\) is self-adjoint, theorems are established on the location of the essential spectrum \(\sigma_{\text{ess}}(H)= [E_0(H),\infty)\), where \(E_0(H)= \inf\sigma(H)\). Applications to quantum field models are given.
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 1496351
zbMATH Keywords: second quantization
zbMATH Keywords: spin-boson model
zbMATH Keywords: Pauli-Fierz model
zbMATH Keywords: tensor product
zbMATH Keywords: Bosonic Fock space
zbMATH Keywords: injective self-adjoint operator
zbMATH Keywords: symmetric operator
zbMATH Keywords: location of the essential spectrum
zbMATH Keywords: quantum field models
reviewed by: W. D. Evans
MaRDI profile type: Publication
OpenAlex ID: W2056059098
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DOI: 10.1006/JMAA.2000.6782
description: scientific article; zbMATH DE number 1496351