Hamiltonian spectrum for three coupled harmonic oscillators in non-commutative space

In non-commutative spaces the invariant eigen-operator method is used to derive and calculate Hamiltonian spectra for three kinds of three coupled harmonic oscillators: no coupling, coordinate coupling and momentum coupling. According to the comparison with the results in commutative space, it is shown that when the non-commutative parameter is zero the obtained energy levels are equal to the energy levels in commutative space. Finally the effect of the coupling coefficient on Hamiltonian spectrum in non-commutative space is discussed.