93-98 Anton Bovier , Jean-Michel Ghez

Spectral properties of one-dimensional Schr\"odinger operators generated by substitutions (65K, TeX) Apr 27, 93

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Abstract. We investigate one-dimensional discrete Schr\"odinger operators whose potentials are invariant under a substitution rule. The spectral properties of these operators can be obtained from the analysis of a dynamical system, called the trace map. We give a careful derivation of these maps in the general case and exhibit some specific properties. Under an additional, easily verifiable hypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying Schr\"odinger operator is singular and supported on a set of zero Lebesgue measure. A condition allowing to exclude point spectrum is also given. The application of our theorems is explained on a series of examples.

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