Effects of a background scalar field induced by the Lorentz symmetry violation on Non Relativistic Quantum Mechanics

The extension of Standard Model of the Fundamental Particles which consider the Lorentz Symmetry Violation governed by a background tensor field presented by Colladay and Kostelecky consider the hypothesis of having a privileged direction in the space-time. Here we investigated a extension of the Schr"odinger Equation inspired on that model taking into account a background scalar field. We analyze the behavior of this modified Schr"odinger equation on a particle trapped in a well and quantum harmonic oscillator.