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

The eigenvalue equation has been found for a Hamilton function in a form independent of the choice of a potential. This Letter proposes a modified Fedosov construction on a flat symplectic manifold. Necessary and sufficient conditions for solutions of an eigenvalue equation to be Wigner functions of pure states are presented. The 1-D harmonic oscillator eigenvalue equation in the coordinates time and energy is solved. A perturbation theory based on the variables time and energy is elaborated. -- Highlights: ► 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.