Wigner function of a quantum system with polynomial potential

The Moyal equation for the Wigner function was obtained under the assumption that the potential is an analytic function. The polynomial form of the potential is a natural approximation of the analytical potential with any necessary accuracy. The simplest quantum system with a second-order polynomial potential is a quantum harmonic oscillator. In this paper, for a quantum system with a polynomial potential of arbitrary order, explicit expressions are obtained for the matrix elements of the kernel operator in the basis of the eigenfunctions of the harmonic oscillator. Using the explicit representation for the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems with polynomial potentials. The connection of the modified Vlasov equation with the Moyal equation for the Wigner function is shown. Examples of effective numerical algorithms for finding Wigner functions with high accuracy are given.