Modified Prüfer and EFGP Transforms and Deterministic Models with Dense Point Spectrum

We provide a new proof of the theorem of Simon and Zhu that in the region |E|<λfor a.e. energies, −(d2/dx2)+λ cos(xα), 0<α<1 has Lyapunov behavior with a quasi-classical formula for the Lyapunov exponent. We also prove Lyapunov behavior for a.e.E∈[−2, 2] for the discrete model withV(j2)=ej,V(n)=0 ifn∉{1, 4, 9, …}. The arguments depend on a direct analysis of the equations for the norm of a solution.