we also obtain $\gamma_{22}=0.75$ MeV as the estimate of the spreading width of 2p2h states. These results will be used in Sec. \[sec:comparison\]

For the sake of comparison, let us estimate the spreading width by assuming the Fermi golden rule. The root mean square of matrix elements between 1p1h states and 2p2h states is calculated as $(\overline{ \langle {\rm 1p1h} |V_{\rm 12}|{\rm 2p2h}\rangle^2})^{1/2}=9.3\times10^{-3}$ MeV. Similarly, we calculate $(\overline{ \langle {\rm 2p2h} |V_{\rm 22}|{\rm 2p'2h'}\rangle^2})^{1/2}=1.0\times10^{-2}$ MeV. Since the level spacing of 2p2h states is $d_{\rm 2p2h}=1.2$ keV, the spreading widths $\gamma_{12}$ and $\gamma_{22}$ are approximately estimated in the Fermi golden rule as $\gamma^{\rm FG}_{12}=2\pi \overline{ \langle {\rm 1p1h} |V_{\rm 12}|{\rm 2p2h}\rangle^2}/d_{\rm 2p2h}=0.46$ MeV and $\gamma^{\rm FG}_{22}=2\pi \overline{ \langle {\rm 2p2h} |V_{\rm 22}|{\rm 2p'2h'}\rangle^2}/d_{\rm 2p2h}=0.53$ MeV, respectively, which are in approximate agreement with the direct evaluation within 30%.

Search for strongly coupled states in 2p2h states {#sec:surfacce_vib}
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In the picture of Fig. \[fig\_pic\_Ca\] 2p2h states are assumed to play a role as the chaotic background and provide the GOE fluctuation to the strength function. However, if the GQR TD state couples with not all 2p2h states equally but specific states in 2p2h states strongly, there is a possibility for this hierarchical 