Explicit examples of arbitrarily large analytic ergodic potentials with zero Lyapunov exponent (Q866879)

The author has introduced a large class of explicit potential functions which can be used in the study of phase transitions for one-dimensional Schrödinger equations with ergodic potentials both in the discrete case \((H_x u)_n\equiv-(u_{n+1}+u_{n-1})+\lambda V(T^nx)u_n=Eu_n\), \(n\in \mathbb{Z}\) and in the continuum one \((H_x^c u)\equiv-u''(t)+\lambda V(T^tx)u(t)=Eu(t)\), \(t\in \mathbb{R},\) where \(T\) is some ergodic homeomorphism (flow) on a compact space \(X\), \(\lambda\in \mathbb{R}.\) For certain energies \(\lambda\) these equations have zero Lyapunov exponent \(L(E,\lambda)\). He also considers the quasi-periodic case, where there is a positive Lyapunov exponent on certain regions of the spectrum and zero on other regions. An example is given, when there exists a 1-dependent random potential with zero Lyapunov exponent.