Exact removal of the center-of-mass spurious states from level density

We present recursive formulae to exactly remove the contribution of the ceter-of-mass spurious states from the fixed-J and parity shell model nuclear level density, if one knows the level density for restricted classes of shell model configurations. The method is valid for a large class of problems that use a harmonic oscillator shell model basis and translational invariant shell model Hamiltonians. Using our methods for calculating nuclear level densities based on fixed-J configuration centroids and widths for restricted classes of shell model configuration (see e.g. M. Horoi et al, Phys. Rev. C {\bf 69}, 041307(R) (2004)), such as $N\hbar \omega$ excitations, one can calculate very accurately nuclear level densities free of center-of-mass spurious states.