SINGULAR CONTINUOUS-SPECTRUM FOR PALINDROMIC SCHRODINGER-OPERATORS

We give new examples of discrete Schrodinger operators with potentials taking finitely many values that have purely singular continuous spec trum. If the hull X of the potential is strictly ergodic, then the exi stence of just one potential x in X for which the operator has no eige nvalues implies that there is a generic set in X for which the operato r has purely singular continuous spectrum. A sufficient condition for the existence of such an x is that there is a z epsilon X that contain s arbitrarily long palindromes. Thus we can define a large class of pr imitive substitutions for which the operators are purely singularly co ntinuous for a generic subset in X. The class includes well-known subs titutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Piso t and ternary non-Pisot. We also show that the operator has no absolut ely continuous spectrum for all x epsilon X if X derives from a primit ive substitution. For potentials defined by circle maps, x(n) = 1(J)(t heta(0) + n alpha), we show that the operator has purely singular cont inuous spectrum for a generic subset in X fur all irrational alpha and every half-open interval J.