Phonon-particle coupling effects in single-particle energies of semi-magic nuclei

A method is presented to evaluate the particle-phonon coupling (PC) corrections to the single-particle energies (SPEs) in semi-magic nuclei. In such nuclei always there is a collective low-lying $2^+$ phonon, and a strong mixture of single-particle and particle-phonon states often occurs. As in magic nuclei, the so-called $g^2_L$ approximation, where $g_L$ is the vertex of the $L$-phonon creation, can be used for finding the PC correction $\delta \Sigma^{\rm PC}(\varepsilon)$ to the initial mass operator $\Sigma_0$. In addition to the usual pole diagram, the phonon "tadpole" diagram is also taken into account. In semi-magic nuclei, the perturbation theory in $\delta \Sigma^{\rm PC}(\varepsilon)$ with respect to $\Sigma_0$ is often invalid for finding the PC corrected SPEs. Instead, the Dyson equation with the mass operator $\Sigma(\varepsilon){=}\Sigma_0{+}\delta \Sigma^{\rm PC}(\varepsilon)$ is solved directly, without any use of the perturbation theory. Results for a chain of semi-magic Pb isotopes are presented.