94-232 Hof A., Knill O., Simon.B.

Singular Continuous Spectrum for Palindromic Schr\"odinger Operators (37K, LaTeX) Jul 14, 94

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Abstract. We give new examples of discrete Schr\"odinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull $X$ of the potential is strictly ergodic, then the existence of just one potential $x$ in $X$ for which the operator has no eigenvalues implies that there is a generic set in $X$ for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such an $x$ is that there is a $z\in X$ that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset in $X$. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for all $x\in X$ if $X$ derives from a primitive substitution. For potentials defined by circle maps, $x_n = 1_J (\theta_0+ n\alpha)$, we show that the operator has purely singular continuous spectrum for a generic subset in $X$ for all irrational $\alpha$ and every half-open interval $J$.

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