Extended States in a Lifshitz Tail Regime for Random Schrödinger Operators on Trees

We resolve an existing question concerning the location of the mobility edge for operators with a hopping term and a random potential on the Bethe lattice. The model has been among the earliest studied for Anderson localization, and it continues to attract attention because of analogies which have been suggested with localization issues for many particle systems. We find that extended states appear through disorder enabled resonances well beyond the energy band of the operator’s hopping term. For weak disorder this includes a Lifshitz tail regime of very low density of states.

Figure 1

A sketch of the previously known parts of the phase diagram. The outer region is of proven localization; the smaller hatched region is of proven delocalization. The new result extends the latter up to the outer curve defined by ${\phi }_{\lambda }(1;E)=-\log K$, assuming equality holds only along a line. The depiction is schematic; however, the points at which the curve meets the energy axis are stated exactly.