What is line broadening phenomenon in quantum many body physics?

I have been reading Anderson's paper, "Absence of diffusion in a certain Random Lattice" and found the concept of inhomogeneous broadening. I couldn't really find a satisfactory explanation with a valid theoretical treatment of the same.
So, What is the quantum mechanical concept behind line broadening phenomenon in many-body physics?

• Anderson's paper is not about many-body physics. The paper deals with a single-particle problem. The only interaction appearing in the model is tunneling between impurity sites. – flaudemus Feb 19 at 15:56

In Anderon's case, each impurity has a random on-site energy $$E_j$$, which you can already call a model for inhomogeneous broadening. In addition, there are the coupling matrix elements $$V_{ij}$$ which, according to Anderson's model, may or may not be stochastic variables as well. All this results in an impurity band, i.e., an inhomogeneously broadened density of states.
The absence of inhomogeneous broadening in this model would mean that neither the $$E_j$$ nor the $$V_{ij}$$ are random variables.