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 Trudy Mat. Inst. Steklova, 2018, Volume 301, Pages 33–47 (Mi tm3904)

On quantum dynamics on $C^*$-algebras

I. V. Volovicha, V. Zh. Sakbaevb

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

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.

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

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 301, 25–38

Bibliographic databases:

UDC: 517.98

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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3904
• https://doi.org/10.1134/S0371968518020036
• http://mi.mathnet.ru/eng/tm/v301/p33

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This publication is cited in the following articles:
1. 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
2. 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
3. B. O. Volkov, “Levy Laplacians and instantons on manifolds”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 23:2 (2020), 2050008
4. V. Zh. Sakbaev, N. V. Tsoi, “Analogue of Chernoff Theorem For Cylindrical Pseudomeasures”, Lobachevskii J. Math., 41:12, SI (2020), 2369–2382
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