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Spectral and dynamical properties of sparse one-dimensional continuous Schrödinger and Dirac operators

Published online by Cambridge University Press:  28 June 2013

Silas L. Carvalho
Affiliation:
Instituto de Ciência e Technologia – UNIFESP, São José dos Campos, SP 12231–280, Brazil ([email protected])
César R. de Oliveira
Affiliation:
Departamento de Matemática – UFSCar, São Carlos, SP 13560–970, Brazil ([email protected])
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Abstract

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Spectral and dynamical properties of some one-dimensional continuous Schrödinger and Dirac operators with a class of sparse potentials (which take non-zero values only at some sparse and suitably randomly distributed positions) are studied. By adapting and extending to the continuous setting some of the techniques developed for the corresponding discrete operator cases, the Hausdorff dimension of their spectral measures and lower dynamical bounds for transport exponents are determined. Furthermore, it is found that the condition for the spectral Hausdorff dimension to be positive is the same for the existence of a singular continuous spectrum.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2013 

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