Odd-parity negative modes of Einstein-Yang-Mills black holes and sphalerons

An analytical proof of the existence of negative modes in the odd-parity perturbation sector is given for all known non-Abelian Einstein-Yang-Mills black holes. The significance of the normalizability condition in the functional stability analysis is emphasized. The role of the odd-parity negative modes in the sphaleron interpretation of the Bartnik-McKinnon solutions is discussed.