Comparing exact energy solutions of quartic eigenvalue polynomials in commutative, non-commutative and non-commutative phase frameworks for boson π−

Abstract

In this paper, the behavior of a Duffin–Kemmer–Petiau (DKP) boson particle in the presence of a harmonic energy-dependent interaction, under the influence of an external magnetic field is precisely studied. In order to exactly solve all equations in commutative (C), non-commutative (NC) and non-commutative phase (NCP) frameworks, the Nikiforov–Uvarov (NU) powerful exact approach is employed. All these attempts end up with solving their quartic equations, trying to find and discuss on their discriminant function [Formula: see text], in a unique way which has never been discussed for any boson in any other research, especially for the boson [Formula: see text] on which, we have been exclusively concerned. We finally succeeded to obtain the exact energy spectrums and wave functions under the effects of NC and NCP parameters and energy-dependent interaction on energy eigenvalues. In this step, we analyze the behaviors of their quartic energy eigenvalue polynomials in three sections and accurately compare all achieved physical-admissible roots one by one. This comparison surprisingly shows that the NC and NCP effects on the other hand, and the assumed harmonic energy-dependent interaction on the other hand, have almost the same order of perturbation effects for limited amounts of the magnetic field in a system of DKP bosons. Furthermore, through some calculations within this paper, we came up with a very crucial point about the NU method which was mistakenly being used in many papers by several researchers and improved it to be used safely.