  System of a two-flavor mixture of ultra-cold fermions confined in a
one-dimensional harmonic trap is studied in the frame of the center of mass. We
present a numerical method of obtaining energetic spectra in this frame for an
arbitrary mass ratio of fermionic species. We identify a specific invariant
encoded in many-body correlations which enable one to determine an eigenstate
of the Hamiltonian and to label excitations of the center of mass. The tool
presented may be particularly useful in experimental analysis of the
interparticle interactions which do not affect the center of mass excitations
in a harmonic potential.
