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Influence of the Nambu-Goldstone (NG) mode on the energy-weighted sum (EWS)of the excitation strengths is analyzed, within the random-phase approximation(RPA). When a certain symmetry is broken at the mean-field level, a NG modeemerges in the RPA, which can be represented by canonical variables forming atwo-dimensional Jordan block. A general formula is rederived which separatesout the NG-mode contribution to the EWS, via the projection on the subspacedirected by the NG mode. As examples, the formula is applied to the $E1$excitation and the rotational excitations in nuclei, further confirmingtheoretical consistency of the RPA.

Influence of Nambu-Goldstone mode on energy-weighted sum of excitation strengths in random-phase approximation