Geodesic
Feynman--Chernoff iterations and their applications in quantum dynamics
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 209-218.

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The notion of Chernoff equivalence for operator-valued functions is generalized to the solutions of quantum evolution equations with respect to the density matrix. A semigroup is constructed that is Chernoff equivalent to the operator function arising as the mean value of random semigroups. As applied to the problems of quantum optics, an operator is constructed that is Chernoff equivalent to a translation operator generating coherent states.
Keywords: Feynman formulas, Chernoff equivalence, averaging of quantum semigroups, coherent states.
Mots-clés : Liouville equation 
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Yu. N. Orlov; V. Zh. Sakbaev. Feynman--Chernoff iterations and their applications in quantum dynamics. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 209-218. http://geodesic.mathdoc.fr/item/TM_2018_301_a14/

[1] F. A. Berezin, “Non-Wiener functional integrals”, Theor. Math. Phys., 6:2 (1971), 141–155 | DOI | MR | Zbl

[2] O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1: $C^*$- and $W^*$-Algebras, Symmetry Groups, Decomposition of States, Springer, New York, 1979 | MR | Zbl

[3] Chernoff P. R., “Note on product formulas for operator semigroups”, J. Funct. Anal., 2:2 (1968), 238–242 | DOI | MR | Zbl

[4] Engel K.-J., Nagel R., One-parameter semigroups for linear evolution equations, Springer, Berlin, 2000 | MR | Zbl

[5] Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Rate of convergence of Feynman approximations of semigroups generated by the oscillator Hamiltonian”, Theor. Math. Phys., 172:1 (2012), 987–1000 | DOI | DOI | MR | Zbl

[6] 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 | DOI | DOI | MR | Zbl

[7] Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Unbounded random operators and Feynman formulae”, Izv. Math., 80:6 (2016), 1131–1158 | DOI | DOI | MR

[8] J. Peřina, Quantum Statistics of Linear and Nonlinear Optical Phenomena, D. Reidel, Dordrecht, 1984 | MR | MR

[9] Smolyanov O. G., Tokarev A. G., Truman A., “Hamiltonian Feynman path integrals via the Chernoff formula”, J. Math. Phys., 43:10 (2002), 5161–5171 | DOI | MR | Zbl

[10] Smolyanov O. G., von Weizsäcker H., Wittich O., “Chernoff's theorem and discrete time approximations of Brownian motion on manifolds”, Potential Anal., 26 (2007), 1–29 | DOI | MR | Zbl