Cancellation of oscillatory behaviors in incommensurate region

In several frustrating systems incommensurate behaviors are often observed. For the S=1 bilinear-biquadratic model, we show that the main oscillatory behavior, which is proportional to the free edge spins, is eliminated in the incommensurate subphase, considering the average of triplet and singlet energy spectra under open boundary conditions. In the same way, the $\pi$-mode oscillation is also removed in the commensurate subphase. Moreover, we find that higher order corrections are exponentially decaying from an analysis of small size data.