# Theoretical seminar | 29 March 2023

Online

To calculate the asymptotics we find the convenient integral representation for the Green's function $G(\bm r,\bm r'|\epsilon)$ of the quasiparticle in the external field, represent the density in the form

$$\rho_{ind}(r)=eN\int_{-\infty}^\infty\frac{d\epsilon}{2\pi}Tr\{G(\bm r,\bm r|i\epsilon)\},$$

then we perform integration, and obtain the following result:

\begin{eqnarray}

\rho_{ind}(r)=F(r)\psi(U_0,R,\lambda),\nonumber

\end{eqnarray}

where the function $F(r)$ depends on $r$ and does not depend on the potential, the function $\psi(U_0,R,\lambda)$ depends only on the parameters of the potential well. For the potential well $U(r)=-U_0\theta(R-r)$ we found simple analytical expression for $\psi(U_0,R,\lambda)$. We consider the function $\psi(U_0,R,\lambda)$ and investigate its behavior in the vicinity of the critical value of the potential well depth.

1. E. H. Wichmann and N. M. Kroll, Phys. Rev. 101, 843 (1956).

2. Ya. B. Zeldovich and V. S. Popov, Usp. Fiz. Nauk 105, 403 (1971) [Sov. Phys. Usp. 14, 673 (1972)].