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 CMFD, 2013, Volume 48, Pages 93–110 (Mi cmfd242)

On dynamical properties of a one-parameter family of transformations arising in averaging of semigroups

V. Sakbaev

MIPT, Department of Higher Mathematics, Dolgoprudny, Moscow region, Russia

Abstract: To describe the dynamics of quantum systems with degenerate symmetric but not self-adjoint Hamiltonian, we consider the Naimark extension of the Hamiltonian to a self-adjoint operator in an extended Hilbert space. We relate to the symmetric Hamiltonian a one-parameter family of averaged dynamical transformations of the set of quantum states obtained from a unitary group of transformations of the extended Hilbert space by using a conditional expected value to an algebra of bounded operators acting in the original space. We establish the absence of the semigroup property and injectivity of the family of averaged dynamical transformations. We obtain a representation of trajectories of the averaged family of dynamical transformations by maximum points of functionals on the space of mappings of the time interval into the set of quantum states.

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English version:
Journal of Mathematical Sciences, 2014, 202:6, 869–886

UDC: 517.946+517.98

Citation: V. Sakbaev, “On dynamical properties of a one-parameter family of transformations arising in averaging of semigroups”, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 4, CMFD, 48, PFUR, M., 2013, 93–110; Journal of Mathematical Sciences, 202:6 (2014), 869–886

Citation in format AMSBIB
\Bibitem{Sak13} \by V.~Sakbaev \paper On dynamical properties of a~one-parameter family of transformations arising in averaging of semigroups \inbook Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14--21, 2011). Part~4 \serial CMFD \yr 2013 \vol 48 \pages 93--110 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd242} \transl \jour Journal of Mathematical Sciences \yr 2014 \vol 202 \issue 6 \pages 869--886 \crossref{https://doi.org/10.1007/s10958-014-2083-9} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84921936626} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. I. V. Volovich, V. Zh. Sakbaev, “On quantum dynamics on $C^*$-algebras”, Proc. Steklov Inst. Math., 301 (2018), 25–38
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