In this work, the relativistic quantum oscillator under the Lorentz symmetry violation environment defined by an arbitrary constant vector field $v^{\mu }$ is analyzed. We solve the generalized Klein–Gordon oscillator with different vector field configurations and obtained the bound-states solutions. After that, we insert a static Coulomb-type scalar potential in the wave equation and solve the generalized Klein–Gordon oscillator analytically. We discuss the effects of Lorentz symmetry violation and the Coulomb-type potential on the energy spectrum of these oscillator fields.

中文翻译：

---

受相互作用势约束的固定矢量场在洛伦兹破坏效应下广义 KG 振荡器的精确解

在这项工作中，由任意常数向量场定义的洛伦兹对称破坏环境下的相对论量子振荡器$v^{\mu }$被分析。我们求解具有不同矢量场配置的广义 Klein-Gordon 振荡器并获得了束缚态解。之后，我们在波动方程中插入一个静态库仑型标量势，并解析求解广义 Klein-Gordon 振子。我们讨论了洛伦兹对称破坏和库仑势对这些振荡器场的能谱的影响。