- 94-232 Hof A., Knill O., Simon.B.
- Singular Continuous Spectrum for Palindromic Schr\"odinger Operators
Jul 14, 94
(auto. generated ps),
of related papers
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 $J$.