The Absolutely Continuous Spectrum of One-Dimensional Schrödinger Operators with Decaying Potentials (original) (raw)
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The spectral theory of the one-dimensional Schrödinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy E belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrödinger cocycle has Cantor spectrum.
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We consider 1-D Schrödinger operators on L 2 (R + ) with slowly decaying potentials. Under some conditions on the potential, related to the first integrals of the KdV equation, we prove that the a.c. spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable. Examples show that for special classes of sparse potentials these results can not be improved.
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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 speci c properties. Under an additional, easily veri able 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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We consider a Schr\"odinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive half-line. We determine series of trace formulas. Here we have the new term: a singular measure, which is absent for real potentials. Moreover, we estimate of sum of Im part of eigenvalues plus singular measure in terms of the norm of potentials. The proof is based on classical results about the Hardy spaces.