I. V. Volovich, V. Zh. Sakbaev, “On quantum dynamics on $C^*$-algebras”, Complex analysis, mathematical physics, and applications, Collected papers, Trudy Mat. Inst. Steklova, 301, MAIK Nauka/Interperiodica, Moscow, 2018, 33–47; Proc. Steklov Inst. Math., 301 (2018), 25

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2018, Volume 301, Pages 33–47
DOI: https://doi.org/10.1134/S0371968518020036 (Mi tm3904)

This article is cited in 11 scientific papers (total in 11 papers)

On quantum dynamics on

$C^*$ -algebras

I. V. Volovich^a, V. Zh. Sakbaev^b

^a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
^b Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701 Russia

Full-text PDF (255 kB) Citations (11)

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DOI: https://doi.org/10.1134/S0371968518020036

Abstract: We consider the problem of constructing quantum dynamics for symmetric Hamiltonian operators that have no self-adjoint extensions. For an earlier studied model, it was found that an elliptic self-adjoint regularization of a symmetric Hamiltonian operator allows one to construct quantum dynamics for vector states on certain

$C^*$ -subalgebras of the algebra of bounded operators in a Hilbert space. In the present study, we prove that one can extend the dynamics to arbitrary states on these

$C^*$ -subalgebras while preserving the continuity and convexity. We show that the obtained extension of the dynamics of the set of states on

$C^*$ -subalgebras is the limit of a sequence of regularized dynamics under removal of the elliptic regularization. We also analyze the properties of the limit dynamics of the set of states on the

$C^*$ -subalgebras.

Funding agency	Grant number
Ministry of Education and Science of the Russian Federation	5-100
The second author was supported by the Moscow Institute of Physics and Technology within the Russian Academic Excellence Project 5-100.

Received: September 26, 2017

English version:
Proceedings of the Steklov Institute of Mathematics, 2018, Volume 301, Pages 25–38
DOI: https://doi.org/10.1134/S008154381804003X

Bibliographic databases:

Document Type: Article

UDC: 517.98

Language: Russian

Citation: I. V. Volovich, V. Zh. Sakbaev, “On quantum dynamics on

$C^*$ -algebras”, Complex analysis, mathematical physics, and applications, Collected papers, Trudy Mat. Inst. Steklova, 301, MAIK Nauka/Interperiodica, Moscow, 2018, 33–47; Proc. Steklov Inst. Math., 301 (2018), 25–38

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Linking options:

https://www.mathnet.ru/eng/tm3904

https://doi.org/10.1134/S0371968518020036

https://www.mathnet.ru/eng/tm/v301/p33

This publication is cited in the following 11 articles:

V. Zh. Sakbaev, A. D. Shiryaeva, “Nonlinear Schrödinger equation with delay and its regularization”, Lobachevskii J. Math., 44:3 (2023), 936
Yu. N. Orlov, V. Zh. Sakbaev, E. V. Shmidt, “Compositions of random processes in a Hilbert space and its limit distribution”, Lobachevskii J. Math., 44:4 (2023), 1432
V. Zh. Sakbaev, E. V. Shmidt, V. Shmidt, “Limit distribution for compositions of random operators”, Lobachevskii J. Math., 43:7 (2022), 1740
V. Zh. Sakbaev, A. D. Shiryaeva, “Blow-up of states in the dynamics given by the Schrödinger equation with a power-law nonlinearity in the potential”, Diff. Equat., 58:4 (2022), 497
Yu. N. Orlov, V. Zh. Sakbaev, E. V. Shmidt, “Operator approach to weak convergence of measures and limit theorems for random operators”, Lobachevskii J. Math., 42:10, SI (2021), 2413–2426
V. M. Busovikov, V. Zh. Sakbaev, “Sobolev spaces of functions on a Hilbert space endowed with a translation-invariant measure and approximations of semigroups”, Izv. Math., 84:4 (2020), 694–721
B. O. Volkov, “Levy Laplacians and instantons on manifolds”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 23:2 (2020), 2050008
V. Zh. Sakbaev, N. V. Tsoi, “Analogue of Chernoff Theorem For Cylindrical Pseudomeasures”, Lobachevskii J. Math., 41:12, SI (2020), 2369–2382
A. D. Grekhneva, V. Zh. Sakbaev, “Dynamics of a set of quantum states generated by a nonlinear Liouville–von Neumann equation”, Comput. Math. Math. Phys., 60:8 (2020), 1337–1347
K. Yu. Zamana, V. Zh. Sakbaev, O. G. Smolyanov, “Stochastic processes on the group of orthogonal matrices and evolution equations describing them”, Comput. Math. Math. Phys., 60:10 (2020), 1686–1700
L. S. Efremova, A. D. Grekhneva, V. Zh. Sakbaev, “Phase flows generated by Cauchy problem for nonlinear Schrodinger equation and dynamical mappings of quantum states”, Lobachevskii J. Math., 40:10, SI (2019), 1455–1469

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Труды Математического института имени В. А. Стеклова

Proceedings of the Steklov Institute of Mathematics

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