  The generic simplicity of the spectrum of a Schr\"odinger-type operator on
the n-dimensional torus is studied using the Rayleigh-Schr\"odinger
perturbation theory. The existence of a perturbation potential of the Laplacian
is proved and suitable conditions on the potential that guarantee the generic
simplicity of the spectrum constructed. It is also proved that with the
potential, the degeneracy of the spectrum of the Laplacian on the n-dimensional
torus splits at first order of the perturbation.
