Singular behavior of Anderson-localized wave functions for a two-site model

We show analytically that the apparent nonanalyticity discovered recently in the inverse participation ratio (IPR) of the eigenstates in Anderson's model of localization is also present in a simple two-site model, along with a concurrent nonanalyticity in the density of states (DOS) at the same energy. We demonstrate its evolution from two sites to the thermodynamic limit by numerical methods. For the two-site model, nonanalyticity in higher derivatives of the DOS and IPR is also proven to exist for all bounded distributions of disorder.