We have calculated the lowest energy quantized spectra of Intrinsically Localized Modes (ILMs) for the Fermi-Pasta-Ulam lattices. The quantized ILM spectra are composed of resonances in the two-phonon continuum and branches of infinitely long-lived excitations that are bound states formed from even numbers of phonons. For quartic anharmonicity and one atom per unit cell, the calculated ILMs are consistent with the results of previous calculations using the number conserving approximation. However, by contrast the ILM spectrum of the lattice with cubic interactions couples resonantly with the single-phonon spectrum and cannot be calculated within a number conserving approximation. Furthermore we argue that, by introducing a sufficiently strong cubic non-linearity, the quantized ILMs can be observed directly through the single-phonon inelastic neutron scattering spectrum. We compare our theoretical predictions with the recent experimental observation of breathers in NaI by Manley et al.
