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 paper 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.