If one tries to solve pairwise harmonically interacting N particles (i.e) every particle interacting with every other particle via harmonic oscillator potential. For the one dimensional case one particle interacting with nearest neighbor can be decoupled using fourier transform or expanding the position and momentum coordinates in Fourier basis. However, can we apply the same logic to pairwise interacting (harmonically) interacting particle. When can we use Fourier transform to solve the problem completely ?