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?  
 A: The term inhomogeneous broadening refers to the broadening of a resonant transition or an energy level. If you consider an ensemble of objects, each having in principle the same transition energy (or energy level) if isolated, but each of them experiences a different local environment, which shifts the individual transition energies (or energy levels) in an uncontrolled random way, then the measured resonance (or energy level) of the ensemble will  be broadened. This is the inhomogeneous broadening.
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.
