On the existence and uniqueness of ground states of a generalized spin-boson model
type: Item
instance of: scholarly article
title: On the existence and uniqueness of ground states of a generalized spin-boson model
zbMATH Open document ID: 0898.47048
author: Asao Arai
author: Masao Hirokawa
published in: Journal of Functional Analysis
publication date: 1998-11-02T00:00:00Z
full work available at URL: http://hdl.handle.net/2115/69116
review text: The standard spin-boson Hamiltonian \[ H_{SB}(\alpha)= {\mu\over 2} \sigma_z\otimes I+ I\otimes H_b+ \sqrt 2\alpha\sigma_x\otimes \phi(\lambda) \] acting on the Hilbert space \(\mathbb{C}^2\otimes{\mathcal F}_b\) models a system of a spin and a quantized Bose field which are coupled to each other, with a coupling parameter \(\alpha\). Here \(\sigma_x\) and \(\sigma_z\) are the conventional Pauli matrices and \(\mu\) is a positive constant. \(H_b= \int_{\mathbb{R}^\nu} dk\omega(k)a(k)^*a(k)\) is the free Hamiltonian of bosons and, with some function \(\lambda\in L^2(\mathbb{R}^\nu)\), \(\phi(\lambda)= (1/\sqrt 2)\int_{\mathbb{R}^\nu} dk(\overline{\lambda(k)} a(k)+ \lambda(k)a(k)^*)\) the field operator, both on the boson Fock space \({\mathcal F}_b\) over \(L^2(\mathbb{R}^\nu)\), where \(a(k)\) and \(a(k)^*\) stand for the annihilation and creation operators with commutation relation \([a(k), a(k')^*]= \delta(k- k')\). The function \(\omega(k)\) is nonnegative and \(m:=\inf\omega(k)\) is regarded as the mass of the boson. The paper under review considers a generalization of this model with the Hamiltonian \[ H(\alpha)= A\otimes I+ I\otimes H_b+ \alpha \sum^J_{j= 1} B_j\otimes \phi(\lambda_j) \] acting on the tensor product \({\mathcal H}\otimes{\mathcal F}_b\) of a Hilbert space \({\mathcal H}\) and the boson Fock space \({\mathcal F}_b\), where \(A\) is a selfadjoint operator and \(B_j\), \(1\leq j\leq J\), for finite \(J\), are symmetric operators in \({\mathcal H}\). Assumed is that \(A\) is bounded from below by \(\mu_0\) and for each \(j\), \(B_j\) is \((A- \mu_0)^{1/2}\)-bounded with relative bound \(a_j\) such that \(|\alpha| \sum^J_{j= 1}a_j\|\lambda_j/ \sqrt\omega\|_{L^2}< 1\), and that \(\omega(k)\) is locally Hölder-continuous with \(\lim_{| k|\to \infty} \omega(k)= \infty\) and each \(\lambda_j(k)\) is continuous. Then the main result is: If \(A\) has a compact resolvent, there exists a ground state of \(H(\alpha)\), not only in the massive case \((m> 0)\…
reviewed by: Takashi Ichinose
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 81Q10
zbMATH DE Number: 1117827
zbMATH Keywords: ground state
zbMATH Keywords: spin-boson Hamiltonian
zbMATH Keywords: constructive quantum field theory
MaRDI profile type: Publication
OpenAlex ID: W2063458690
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DOI: 10.1006/JFAN.1997.3140
description: scientific article; zbMATH DE number 1117827