contain the enhancement of the coupling with the surface vibrations in the damping of the GQR. Therefore we consider in the following the 2p2h states as background states which do not have specific structures.

                \#    $\overline{V_{c\alpha}^2}$ (MeV$^2$)   $\Gamma_2^{\rm FG}$ (MeV)    $\Gamma_2$ (MeV)
  ----------- ------ -------------------------------------- --------------------------- ------------------
  s.v.+1p1h    909            $0.65\times 10^{-4}$                     0.092                         0.074
  2p2h         3342           $0.72\times 10^{-4}$                     0.38                           0.41

  :  Averaged value of squared coupling matrix elements $\overline{V_{c\alpha}^2}$ between the GQR TD state and surface vibration plus 1p1h states or the whole 2p2h states(third column), the associated spreading width $\Gamma_2^{\rm FG}$ of the GQR TD state evaluated by the Fermi golden rule (fourth column), and the spreading width $\Gamma_2$ estimated by the FWHM of the strength function based on Eq. (\[doorwaystr\]) (fifth column). Second column shows the number of states considered. The second row shows the results obtained by including only the s.v.+1p1h states while the third row shows those for the case of the whole 2p2h states. []{data-label="table1"}

Physical origin of the difference between $^{40}$Ca and $^{208}$Pb {#sec:comparison}
------------------------------------------------------------------

In the above subsections, we have evaluated the physical quantities such as the various spreading widths, with which we have discussed the damping process of $^{40}$Ca and $^{208}$Pb, especially the 