The translation invariant massive Nelson model. III: asymptotic completeness below the two-boson threshold
type: Item
instance of: scholarly article
title: The translation invariant massive Nelson model. III: asymptotic completeness below the two-boson threshold
zbMATH Open document ID: 1329.81355
author: W. Dybalski
author: Jacob Schach Møller
published in: Annales Henri Poincaré
publication date: 2015-11-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1210.6645
review text: Based on results in Part II of this series [the second author and \textit{M. G. Rasmussen}, ibid. 14, No. 4, 793--852 (2013; Zbl 1272.81077)] and Mourre estimate [\textit{E. Mourre}, Commun. Math. Phys. 78, 391--408 (1981; Zbl 0489.47010)], asymptotic completeness of two body scattering for a class of translation invariant models describing a single quantum particle (the electron) linearly coupled to a massive scalar field (bosons) is shown (Theorem 2.3. Corollary 2.4). The models cover wide classes such as massive \textit{E. Nelson} model [``Interaction of nonrelativistic particles with a quantum scalar filed'', J. Math. Phys. 5, 1190--1197 (1964)] and the \textit{H. Frölich} polaron model [``Electron in lattice filed'', Adv. Phys. 3. 325--362 (1954)]. Precisely, the Hilbert space of the system is \(\mathcal{H}=\mathcal{K}\otimes\mathcal{F}\). Here \(\mathcal{K}=L^2(\mathbb{R}_y^\nu)\), is the Hilbert space of a quantum mechanical particle moving in \(\mathbb{R}^\nu\), whose position is denoted by \(y\) and whose moment is \(D_y=-\mathrm{i}\nabla_y\) and \(\mathcal{F}\) is the Fock space \(\Gamma(\mathfrak{h})=\bigoplus^\infty_{n=0}\Gamma^{(n)}(\mathfrak{h})\), \(\Gamma^{(n)}(\mathfrak{h})=\mathfrak{h}^{\otimes_s n}\), where \(\mathfrak{h}=L^2(\mathbb{R}^\nu_k)\) is the Hilbert space of a single boson, whose dispersion relation is denoted by \(\omega(k)\). The total energy and momentum operators of bosons are given by \(H_{\mathrm{ph}}=d\Gamma(\omega)\) and \(P_{\mathrm{ph}}=d\Gamma(k)\). The Hamiltonian, which governs the dynamics of the system, is \[ H=\Omega(D_y)\otimes 1+1\otimes H_{\mathrm{ph}}+\phi(G_y) \] (\S2. (2.4).). In Part II, it is shown under minimal conditions (described as Condition 1 in \S2. The authors remark that this conditions are a little stronger than those of in Part II), \(H\) is essentially self-adjoint on \(C_0^\infty(\mathbb{R}^\nu)\otimes\mathcal{C}\) (detailed definition of \(\mathcal{C}=\Gamma_{\mathrm{fin}}(C_0^\infty(\mathbb{R}^\nu)…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 81U05
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 47A40
zbMATH DE Number: 6507224
zbMATH Keywords: asymptotic completeness
zbMATH Keywords: two-body scattering
zbMATH Keywords: translation invariant model, massive Nelson model
zbMATH Keywords: polaron model
zbMATH Keywords: Mourre estimate
zbMATH Keywords: wave operator
MaRDI profile type: Publication
OpenAlex ID: W2078152750
arXiv ID: 1210.6645
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DOI: 10.1007/S00023-014-0384-7
description: scientific article