From a model that goes beyond the Standard Model, we study Lorentz symmetry breaking effects around a cylindrical cavity at a low-energy regime. We consider a background of the Lorentz symmetry violation determined by a fixed vector field, then, we study the influence of a cut-off point on a harmonic potential induced by the effects of the Lorentz symmetry violation. We show that the cut-off point modifies the spectrum of energy of the harmonic-type potential. One aspect observed is the infinity degeneracy of the energy levels. Besides, we discuss a lower bound for the Lorentz symmetry violation parameters. [ABSTRACT FROM AUTHOR]