Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Rate of convergence of Feynman approximations of semigroups generated by the oscillator Hamiltonian”, TMF, 172:1 (2012), 122–137; Theoret. and Math. Phys., 172:1 (2012), 987

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Teoreticheskaya i Matematicheskaya Fizika, 2012, Volume 172, Number 1, Pages 122–137
DOI: https://doi.org/10.4213/tmf6936 (Mi tmf6936)

This article is cited in 16 scientific papers (total in 16 papers)

Rate of convergence of Feynman approximations of semigroups generated by the oscillator Hamiltonian

Yu. N. Orlov^a, V. Zh. Sakbaev^b, O. G. Smolyanov^c

^a Keldysh Institute of Applied Mathematics, RAS, Moscow, Russia
^b Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast, Russia
^c Lomonosov Moscow State University, Moscow, Russia

Full-text PDF (494 kB) Citations (16)

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DOI: https://doi.org/10.4213/tmf6936

Abstract: We determine the rate with which finitely multiple approximations in the Feynman formula converge to the exact expression for the equilibrium density operator of a harmonic oscillator in the linear

τ

$\tau$ -quantization. We obtain an explicit analytic expression for a finitely multiple approximation of the equilibrium density operator and the related Wigner function. We show that in the class of

τ

$\tau$ -quantizations, the equilibrium Wigner function of a harmonic oscillator is positive definite only in the case of the Weyl quantization.

Keywords: finitely multiple approximation, Feynman formula, Chernoff theorem, linear quantization, harmonic oscillator, Wigner function.

Received: 09.08.2011

English version:
Theoretical and Mathematical Physics, 2012, Volume 172, Issue 1, Pages 987–1000
DOI: https://doi.org/10.1007/s11232-012-0090-x

Bibliographic databases:

Language: Russian

Citation: Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Rate of convergence of Feynman approximations of semigroups generated by the oscillator Hamiltonian”, TMF, 172:1 (2012), 122–137; Theoret. and Math. Phys., 172:1 (2012), 987–1000

Citation in format AMSBIB

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https://www.mathnet.ru/eng/tmf6936

https://doi.org/10.4213/tmf6936

https://www.mathnet.ru/eng/tmf/v172/i1/p122

This publication is cited in the following 16 articles:

Oleg E. Galkin, Ivan D. Remizov, “Upper and lower estimates for rate of convergence in the Chernoff product formula for semigroups of operators”, Isr. J. Math., 2024
K. A. Dragunova, N. Nikbakht, I. D. Remizov, “Chislennoe issledovanie skorosti skhodimosti chernovskikh approksimatsii k resheniyam uravneniya teploprovodnosti”, Zhurnal SVMO, 25:4 (2023), 255–272
O. E. Galkin, I. D. Remizov, “Rate of Convergence of Chernoff Approximations of Operator $C_0$-Semigroups”, Math. Notes, 111:2 (2022), 305–307
A. V. Vedenin, “Bystro skhodyaschiesya chernovskie approksimatsii k resheniyu uravneniya teploprovodnosti s peremennym koeffitsientom teploprovodnosti”, Zhurnal SVMO, 24:3 (2022), 280–288
A. V. Vedenin, V. S. Voevodkin, V. D. Galkin, E. Yu. Karatetskaya, I. D. Remizov, “Speed of Convergence of Chernoff Approximations to Solutions of Evolution Equations”, Math. Notes, 108:3 (2020), 451–456
Borisov L.A. Orlov Yu.N. Sakbaev V.Zh., “Feynman Averaging of Semigroups Generated By Schrodinger Operators”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 21:2 (2018), 1850010
Chen L., Jakobsen E.R., Naess A., “On Numerical Density Approximations of Solutions of SDEs With Unbounded Coefficients”, Adv. Comput. Math., 44:3 (2018), 693–721
Yu. N. Orlov, V. Zh. Sakbaev, “Feynman–Chernoff iterations and their applications in quantum dynamics”, Proc. Steklov Inst. Math., 301 (2018), 197–206
Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Unbounded random operators and Feynman formulae”, Izv. Math., 80:6 (2016), 1131–1158
Butko Ya.A. Grothaus M. Smolyanov O.G., “Feynman formulae and phase space Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton functions”, J. Math. Phys., 57:2 (2016), 023508
L. A. Borisov, Yu. N. Orlov, “Analyzing the dependence of finite-fold approximations of the harmonic oscillator equilibrium density matrix and of the Wigner function on the quantization prescription”, Theoret. and Math. Phys., 184:1 (2015), 986–995
L. A. Borisov, Yu. N. Orlov, V. Zh. Sakbaev, “Formuly Feinmana dlya usredneniya polugrupp, porozhdaemykh operatorami tipa Shredingera”, Preprinty IPM im. M. V. Keldysha, 2015, 057, 23 pp.
L. A. Borisov, Yu. N. Orlov, V. Zh. Sakbaev, “Ekvivalentnost po Chernovu primenitelno k uravneniyam evolyutsii matritsy plotnosti i funktsii Vignera dlya lineinogo kvantovaniya”, Preprinty IPM im. M. V. Keldysha, 2015, 066, 28 pp.
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.
Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Feynman formulas as a method of averaging random Hamiltonians”, Proc. Steklov Inst. Math., 285 (2014), 222–232
Yana Butko, “Feynman formulae for evolution semigroups”, S&E BMSTU, 14:03 (2014)

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

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