Analyzing the dependence of finite-fold approximations of the harmonic oscillator equilibrium density matrix and of the Wigner function on the quantization prescription

For an arbitrary statistical mixture of $\tau$-quantizations, we obtain an explicit expression for the leading parts of the equilibrium density matrix and for the corresponding Wigner function of a harmonic oscillator in the approach of Feynman approximations using the Chernoff theorem. Taking the oscillator Hamiltonian as an example, we determine the convergence rate for approximations of means of operators of observables depending on the approximation order and depending on the quantization rule. We demonstrate that the convergence rate of approximations of the mean of the energy operator is not uniform with respect to the Gibbs parameter.