Vibrational Modes on Bethe Lattice

The Bethe lattice (Cayley tree) with coordination number z has an exact phonon density of states given by D(ω) ∝ √(4(z-1)k²/m − ω²) / (zk/m − ω²) via the cavity method. Van Hove singularities appear at band edges, with no loops eliminating Anderson localization complications.