The relativistic quantum dynamics of the generalized Klein–Gordon (KG) oscillator having position-dependent mass in the Gödel-type space–time is investigated. We have presented the generalized KG oscillator in this space–time, and discussed the effect of Cornell potential and linear potential for our considered system. The modification from the parameters of position-dependent mass and characterizing the space–time for the energy spectrums are presented.

In this work, we investigate the spin-half relativistic particles described by the Dirac equation on the topological defect background induced by the cosmic string and torsion with an internal magnetic field. We derive the general expression of the generalized Dirac oscillator on the topological defect background and analyze the analogue of the Aharonov-Bohm effect for the Dirac oscillator with function considered as the Cornell potential, and we explain the influence of related parameters on the energy levels of the studied system.

In this paper, we investigate the Dirac equation with the Killingbeck potential under the external magnetic field in non-commutative space. Corresponding to the expressions of the energy level and wave functions in spin symmetry limit and pseudo-spin symmetry limit are derived by using the Bethe ansatz method. The parameter B associated with the external magnetic field and non-commutative parameter θ make to modify the energy level for considered systems.

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This paper proposes a new high-order generalized uncertainty principle, which can modify the momentum operator and position operator simultaneously. Moreover, the new form of GUP is consistent with the viewpoint of the existence of the minimum length uncertainty and the maximum observable momentum proposed by the mainstream quantum gravity theory. By using the new GUP, the maximum localization state and position eigenfunction are discussed, and the corresponding conclusions are compared with the existing literature. The harmonic oscillator is further discussed at the end of this article as an example.