Heisenberg evolution of quantum observables represented by unbounded operators
type: Item
instance of: scholarly article
title: Heisenberg evolution of quantum observables represented by unbounded operators
zbMATH Open document ID: 1166.81015
author: C. M. Mora
published in: Journal of Functional Analysis
publication date: 2009-02-10T00:00:00Z
full work available at URL: http://hdl.handle.net/10533/142218
review text: Consider a small quantum system with state space \(\mathfrak{h}\) weakly coupled to a heat bath. In a Markovian setting, an observable \(A\) of the small system evolves in accordance with the adjoint quantum master equation \[ \frac{d}{dt}\mathcal{T}_t(A)=\mathcal{T}_t(A)G+G^*\mathcal{T}_t(A)+\sum_{k=1}^{\infty}L_k^*\mathcal{T}_t(A)L_k, \quad \mathcal{T}_0(A)=A, \] where \(G, L_1, L_2, \dots \) are linear operators in \(\mathfrak{h}\) satisfying some certain conditions. For bounded observables \(A\) (i.e. \(A\) is a bounded operator in \(\mathfrak{h}\)), the existence and uniqueness of solutions for the equations are established in \textit{A. M. Chebotarev} [Theor. Math. Phys. 80, No. 2, 804--818 (1989); translation from Teor. Mat. Fiz. 80, No. 2, 192--211 (1989; Zbl 0694.47022)] by semigroup methods. This paper deals with the adjoint quantum master equations with initial conditions given by unbounded operators, such as the position and momentum operators of quantum oscillators and the occupation number operator in many-body particle systems. It shows the existence and uniqueness of solutions of the operator equations governing the motion of unbounded observables by developing the relation between operator evolution equations arising in quantum mechanics and stochastic evolution equations of Schrödinger type. It also explores quantum dynamical semigroup properties of the Heisenberg evolutions of unbounded observables in a general setting.
reviewed by: Miyeon Kwon
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81S25
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47D08
zbMATH DE Number: 5505593
zbMATH Keywords: master equations
zbMATH Keywords: stochastic Schrödinger equations
zbMATH Keywords: existence and uniqueness of solutions
zbMATH Keywords: unbounded observables
zbMATH Keywords: quantum dynamical semigroups
zbMATH Keywords: evolution operator equations
MaRDI profile type: Publication
OpenAlex ID: W2066607410
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DOI: 10.1016/J.JFA.2008.10.006
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description: scientific article; zbMATH DE number 5505593