 94232 Hof A., Knill O., Simon.B.
 Singular Continuous Spectrum for Palindromic Schr\"odinger Operators
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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 wellknown substitutions like Fibonacci, ThueMorse,
Period Doubling, binary nonPisot and ternary nonPisot.
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
halfopen interval $J$.
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