Singular Continuous Spectrum for Palindromic Schrr Odinger Operators

We give new examples of discrete Schrr odinger operators with potentials taking nitely 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 suucient condition for the existence of such an x is that there is a z 2 X that contains arbitrarily long palin-dromes. Thus we can deene 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 2 X if X derives from a primitive substitution. For potentials deened by circle maps, xn = 1J(0 + nn), we show that the operator has purely singular continuous spectrum for a generic subset in X for all irrational and every half-open interval J.