97-368 Last Y., Simon B.

Modified Pr\"ufer and EFGP Transforms and deterministic models with dense point spectrum (38K, AMSTeX) Jun 23, 97

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Abstract. We provide a new proof of the theorem of Simon and Zhu that in the region $|E| < \lambda$ for a.e.~energies, $-\frac{d^2}{dx^2}+\lambda \cos (x^\alpha)$, $0<\alpha <1$ has Lyapunov behavior with a quasi-classical formula for the Lyapunov exponent. We also prove Lyapunov behavior for a.e.~$E \in [-2,2]$ for the discrete model with $V(j^2) = e^j$, $V(n)=0$ if $n\notin \{1,4,9,\dots \}$. The arguments depend on a direct analysis of the equations for the norm of a solution.

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