Eigenvalue equation for a 1--D Hamilton function in deformation quantization

TL;DR

This paper develops a potential-independent eigenvalue equation for a 1D Hamiltonian in deformation quantization, introduces a modified Fedosov construction, and solves the harmonic oscillator case with perturbation theory.

Contribution

It presents a new potential-independent eigenvalue equation and a modified Fedosov construction for deformation quantization, with solutions for the 1D harmonic oscillator.

Findings

Eigenvalue equation is potential-independent.

Solutions for 1D harmonic oscillator are obtained.

Perturbation theory in time and energy variables is developed.

Abstract

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.