where S and T are integers, and ρ ρ ( ϵ ϵ ) is the density of states at a given α α . Here n counts the number of states up to the Fermi energy, and n 0 corresponds to the levels of the completely filled band (from ϵ ϵ = − − 4 to ϵ ϵ = 4 ). This equation characterizes all the solutions of Harper's equation. Most importantly, one can derive that when α α is an irrational number, there are infinitely many solution for ϵ ϵ α α .