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Rate of convergence of Feynman approximations of semigroups generated by the oscillator Hamiltonian. (English. Russian original) Zbl 1280.81046
Theor. Math. Phys. 172, No. 1, 987-1000 (2012); translation from Teor. Mat. Fiz. 172, No. 1, 122-137 (2012).
Summary: 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 \(\tau\)-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 \(\tau\)-quantizations, the equilibrium Wigner function of a harmonic oscillator is positive definite only in the case of the Weyl quantization.

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81S05 Commutation relations and statistics as related to quantum mechanics (general)
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
47D08 Schrödinger and Feynman-Kac semigroups
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