Transience and recurrence of quantum Markov semigroups
type: Item
instance of: scholarly article
title: Transience and recurrence of quantum Markov semigroups
zbMATH Open document ID: 1024.60031
publication date: 2003-08-20T00:00:00Z
review text: The authors introduce the notion of a potential for a quantum Markov semigroup, and classes of transient and recurrent quantum Markov semigroups. It is shown that an irreducible semigroup is either transient or recurrent. Transient semigroups are characterized in terms of the existence of non-trivial superharmonic operators. As examples, the quantum Brownian motion and the Jaynes-Cummings micromaster model of quantum optics are considered. Applications to scattering theory are given. In particular, it is shown that for a transient quantum semigroup \(T_t\) any state \(\varphi\) is scattering, that is for any finite-dimensional projection \(p\), \[ \lim_{t\to \infty}\varphi T_t(p)=\lim_{t\to \infty}T_{*t}(\varphi)(p)=0 . \]
reviewed by: Anatoly N. Kochubei
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 37A50
Mathematics Subject Classification ID: 47A40
Mathematics Subject Classification ID: 81S25
zbMATH DE Number: 1968523
zbMATH Keywords: quantum Markov semigroup
zbMATH Keywords: potential
zbMATH Keywords: transience
zbMATH Keywords: recurrence
author: Franco Fagnola
author: Rolando Rebolledo
MaRDI profile type: Publication
full work available at URL: s00440-003-0268-0
OpenAlex ID: W2092527261
DOI: 10.1007/S00440-003-0268-0
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1968523