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TMF, 2010, Volume 164, Number 3, Pages 455–463 (Mi tmf6557)  

This article is cited in 10 scientific papers (total in 10 papers)

Averaging of quantum dynamical semigroups

V. Zh. Sakbaev

Moscow Institute for Physics and Technology, Dolgoprudnyi, Moscow Oblast, Russia

Abstract: In the framework of the elliptic regularization method, the Cauchy problem for the Schrödinger equation with discontinuous degenerating coefficients is associated with a sequence of regularized Cauchy problems and the corresponding regularized dynamical semigroups. We study a divergent sequence of quantum dynamical semigroups as a random process with values in the space of quantum states defined on a measurable space of regularization parameters with a finitely additive measure. The mathematical expectation of the considered processes determined by the Pettis integral defines a family of averaged dynamical transformations. We investigate the semigroup property and the injectivity and surjectivity of the averaged transformations. We establish the possibility of defining the process by its mathematical expectation at two different instants and propose a procedure for approximating an unknown initial state by solutions of a finite set of variational problems on compact sets.

Keywords: stochastic process, finitely additive measure, quantum state, dynamical semigroup, observability

DOI: https://doi.org/10.4213/tmf6557

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English version:
Theoretical and Mathematical Physics, 2010, 164:3, 1215–1221

Bibliographic databases:


Citation: V. Zh. Sakbaev, “Averaging of quantum dynamical semigroups”, TMF, 164:3 (2010), 455–463; Theoret. and Math. Phys., 164:3 (2010), 1215–1221

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. Zh. Sakbaev, “O dinamike mnozhestva sostoyanii kvantovoi sistemy s vyrozhdennym gamiltonianom”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2(23) (2011), 200–220  mathnet  crossref
    2. V. Z. Sakbaev, “The set of quantum states and its averaged dynamic transformations”, Russian Math. (Iz. VUZ), 55:10 (2011), 41–50  mathnet  crossref  mathscinet  elib
    3. V. Zh. Sakbaev, “Cauchy problem for degenerating linear differential equations and averaging of approximating regularizations”, Journal of Mathematical Sciences, 213:3 (2016), 287–459  mathnet  crossref  mathscinet
    4. Dzh. O. Ogun, Yu. N. Orlov, V. Zh. Sakbaev, “O preobrazovanii prostranstva nachalnykh dannykh dlya zadachi Koshi s osobennostyami resheniya tipa vzryva”, Preprinty IPM im. M. V. Keldysha, 2012, 087, 31 pp.  mathnet
    5. V. Zh. Sakbaev, “Razrushenie reshenii zadachi Koshi dlya nelineinogo uravnenii Shredingera”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 159–171  mathnet  crossref
    6. V. Sakbaev, “On dynamical properties of a one-parameter family of transformations arising in averaging of semigroups”, Journal of Mathematical Sciences, 202:6 (2014), 869–886  mathnet  crossref
    7. V. Zh. Sakbaev, “Gradient blow-up of solutions to the Cauchy problem for the Schrödinger equation”, Proc. Steklov Inst. Math., 283 (2013), 165–180  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. M. Kh. Numan Elsheikh, D. O. Ogun, Yu. N. Orlov, R. V. Pleshakov, V. Zh. Sakbaev, “Usrednenie sluchainykh polugrupp i neodnoznachnost kvantovaniya gamiltonovykh sistem”, Preprinty IPM im. M. V. Keldysha, 2014, 019, 28 pp.  mathnet
    9. Sakbaev V.Zh. Volovich I.V., “Self-Adjoint Approximations of the Degenerate Schrodinger Operator”, P-Adic Numbers Ultrametric Anal. Appl., 9:1 (2017), 39–52  crossref  mathscinet  zmath  isi  scopus
    10. I. V. Volovich, V. Zh. Sakbaev, “On quantum dynamics on $C^*$-algebras”, Proc. Steklov Inst. Math., 301 (2018), 25–38  mathnet  crossref  crossref  isi  elib  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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