# Proof that the self-energy is an inverse lifetime

My question concerns the self-energy of a diagonal propagator for a single-particle lattice problem. The context is Anderson Localisation, but really it's a problem of complex analysis. I would like to show that the imaginary part of the self-energy corresponds to a rate of loss of probability amplitude from a site $$j$$ to overlapping eigenstates at an energy $$\omega$$.

Let the site-diagonal propagator in the time and frequency domains be $$$$G_{jj}(t) = -i\Theta(t)\langle j|e^{-i\hat{H}t}|j\rangle\longleftrightarrow G_{jj}(\omega) = \frac{1}{\omega+i\eta-\epsilon_j-S_j},$$$$ where $$\omega$$ is frequency/energy, $$\epsilon_j$$ is the site energy of site $$j$$, $$\eta\equiv 0^+$$ and $$S_j$$ is the self-energy. Now, the self-energy is complex, $$$$S_j = X_j(\omega) - i\Delta_j(\omega).$$$$ In the time domain, therefore, $$$$G_{jj}(t) = \frac{1}{2\pi}\int_{-\infty}^\infty\mathrm{d}\omega\frac{e^{-i\omega t}}{\omega-\epsilon_j-X_j(\omega)+i(\eta+\Delta_j(\omega))}.$$$$ I can show that with the assumption of $$\omega$$-independence for the self-energy, $$S_j(\omega)\equiv S_j$$ $$\forall\omega$$, $$\Delta_j$$ corresponds to the rate of loss of probability amplitude for site $$j$$. That is, the reverse Fourier transform gives a factor of $$\exp(-\Delta_jt)$$. $$X_j$$ gives an overall oscillatory phase factor.

However, how does one show this connection for a general, $$\omega$$-dependent self-energy? I have seen stated in the literature that "$$\Delta_j(\omega)$$ is physically the rate of loss of probability amplitude from site $$j$$ to overlapping eigenstates at energy $$\omega$$". This I would like to show.

If it helps for background knowledge, the Hamiltonian will be of form $$$$\hat{H} = \sum_j\epsilon_j|j\rangle\langle j| + t\sum_j |j\rangle\langle j+1| + \text{h.c}$$$$ In $$1d$$, e.g., and with the site energies $$\{\epsilon_i\}$$ independent random variables, this is an Anderson problem and all states are localised.