Effects of a harmonic-type potential on a scalar field in a background of CPT-odd Lorentz symmetry violation

We have investigated the relativistic quantum dynamics of a scalar field in an anisotropic background governed by the Lorentz symmetry violation. This analysis is done through a non-minimal coupling in the Klein–Gordon equation, which is inspired by the CPT-odd gauge sector of the standard model extension. From the elaboration of a vector and electromagnetic field configuration as a possible Lorentz break scenario, we induced an electric field which influences the scalar field. In this scenario, we show that it is possible to determine two energy profiles for the system, that is, a description of the relativistic quantum dynamics of the scalar field in the xy-plane and in all space-time