the Fermi surface, and included all 1p1h states and 2p2h states whose unperturbed energies are less than 15MeV. Resultant number of 1p1h states and 2p2h states are 39 and 8032, respectively. We diagonalized the Hamiltonian within this model space and obtained the strength function for the isoscalar quadrupole operator.

![ Calculated strength function of the IS GQR in$^{208}$Pb. Dotted curve shows the smooth strength function by means of the Strutinsky method with the smoothing width 0.2 MeV. []{data-label="fig_strfun"}](fig1.eps){width="6cm"}

Figure  \[fig\_strfun\] shows the calculated strength function. The average of the excitation energy weighted by the strength is about 10.5 MeV, and the standard deviation around the average is about 2.6 MeV, where all levels are considered. The peak position lies at the same value as the average. These values are consistent with the (p,p’) experimental data [@shevchenko]. Moreover, the agreement of the global shape with the experimental data is also good. The dotted curve in Fig. \[fig\_strfun\] represents the smooth strength function by means of the Strutinsky method [@ring-schuck] with the smoothing width 0.2 MeV. The value of the FWHM of this smooth strength function is 0.63 MeV. In order to quantitatively characterize the spreading of the strength function around the 