an impurity interacting with a atom

$$H = \sum_{\langle ij \rangle } t c_i c_j + f d^+_0 d_0 + ( s c^+_0 d_0 + {\rm h.c.} )$$ (249)

If site 0 is in the central region, matrixes for the site of $c_0$ and $d_0$ are

$$G_C(\omega ^+) = \left( \begin{array}{cc} \omega -\Sigma_R(\omega ^+) - \Sigma_L(\omega ^+) & -s \\ -s & \omega -f \end{array} \right)^{-1}$$ (250)

$$= \left( \begin{array}{cc} \omega -\omega _0 + i \Gamma _0 & -s \\ -s & \omega -f \end{array} \right)^{-1}$$ (251)

$$G_C(\omega ^-) = \left( \begin{array}{cc} \omega -\omega _0 - i \Gamma _0 & -s \\ -s & \omega -f \end{array} \right)^{-1}$$ (252)

$$\Gamma _R = \left( \begin{array}{cc} \Gamma_0 & 0 \\ 0 & 0 \end{array} \right)$$ (253)

$$\Gamma _L = \left( \begin{array}{cc} \Gamma_0 & 0 \\ 0 & 0 \end{array} \right)$$ (254)

$$T = {\rm Tr}\left[ \Gamma _L G_C(e^+) \Gamma _R G_C(e^-) \right]$$ (255)

$$= \frac{ \Gamma _0^2 (\omega -f)^2 } { \left\{ (\omega -f)(\omega -\omega _0) -s^2\right\}^2 + \Gamma _0^2 (\omega -f)^2 }$$ (256)

$$= \frac{ \Gamma _0^2 (\omega -f)^2 }{ s^4 + \Gamma _0^2 (\omega -f)^2 }$$ (257)

where $\omega _0=\omega$ is used at the last equality. It is important to notice that $T=0$ at $\omega =f$, i.e. at the energy level of the impurity and the maximum is less than 1. The half-width of the minimum is $s^2/\Gamma _0$.