Singular Behavior of Anderson Localized Wavefunctions for a Two-Site Model

We show analytically that the apparent non-analyticity 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 non-analyticity 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, non-analyticity in higher derivatives of the DOS and IPR is also proven to exist for all bounded distributions of disorder.