Possible deviation of the sum of strengths for the double giant dipole resonance from the harmonic oscillator limit

It is shown that the part of the nuclear Hamiltonian which contains the products of $\mathrm{pp}$ and $\mathrm{hh}$ pair operators (the scattering terms) and which has been usually neglected in the calculations of the giant dipole resonance turns out to be definitely important in the study of the double giant dipole resonance (DGDR). The complete energy-weighted sum of strengths (EWSS) for the DGDR and a possible deviation from its value in the harmonic oscillators' limit are derived for the first time with the full nuclear Hamiltonian taken into account. The numerical calculations within a schematic model show an example where this deviation turns to a strong enhancement of the EWSS for the DGDR at various particle numbers.