  Various sum rules for multiple giant dipole resonance states are derived. For
the triple giant dipole resonance states, the energy-weighted sum of the
transition strengths requires a model to be related to those of the single and
double giant dipole resonance states. It is also shown that the non-diagonal
matrix elements of the double commutator between the dipole operator and the
nuclear Hamiltonian give useful identities for the excitation energy and
transition strength of each excited state. Using those identities, the
relationship between width of the single dipole state and those of the multiple
ones is qualitatively discussed.
