Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions |
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Authors: | Anton Bovier Jean-Michel Ghez |
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Institution: | (1) Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, D-10117 Berlin, Germany;(2) Centre de Physique Théorique, Luminy Case 907, F-132 88 Marseille Cedex 9, France;(3) PHYMAT, Département de Mathématiques, Université de Toulon et du Var, B.P. 132, F-83957 La Garde Cedex, France |
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Abstract: | We investigate one-dimensional discrete Schrödinger 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 ypothesis 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 Schrödinger 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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