The dispersion law of a one-dimensional diatomic lattice with on-site potential cross on its dispersion relation is solved under the harmonic approximation with quantized invariant eigen-operator method (IEO) and the influences of on-site potential and force constant are discussed. It is shown that due to the existence of on-site potential the lattice vibration spectra induced are shifted. Under certain conditions, the point of intersection exist between optical branch and acoustic branch in the boundary of Brillioun zone, which means the frequency gap between optical branch and acoustic branch is zero.