The eigenvalue equation for a 1-D Hamilton function in deformation quantization

► We propose a method of solution of the ⁎-genvalue equation for a 1-D Hamiltonian by the change of coordinates (q,p)→(time,energy). ► A covariant form of the ⁎-genvalue equation for Hamiltonian is presented. ► Continuity of a Wigner function of a pure state with respect to q and p has been proved. ► Three criteria for an arbitrary function to be a Wigner function of a pure state are presented and proved. ► A perturbation theory based on the variables time and energy is elaborated.