Mean energy, strength, and width of triple giant dipole resonances

We investigate the mean energy, strength, and width of triple giant dipole resonance using sum rules. We point out that the transition operator for the triple giant dipole resonance must be modified from the naive choice ${D}^{3}.$ With the modified transition operator, the strength and mean energy calculated in the Tomonaga model agree with the harmonic limit, and the width is larger than $\sqrt{3}$ times that of the giant dipole resonance. Had we adopted the naive choice, we would have obtained unphysical results for these quantities. Assuming the absence of a charge-exchange component in the nuclear interaction, we derive model-independent relationships between the mean energies and strengths of multiple giant dipole resonances, which are an extension of relations derived previously for the double giant dipole resonance.