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

Part of: General mathematical topics and methods in quantum theory Elliptic equations and systems Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  28 June 2013

Silas L. Carvalho  and
César R. de Oliveira
Silas L. Carvalho
Affiliation:
Instituto de Ciência e Technologia – UNIFESP, São José dos Campos, SP 12231–280, Brazil (slcarvalho@unifesp.br)
César R. de Oliveira
Affiliation:
Departamento de Matemática – UFSCar, São Carlos, SP 13560–970, Brazil (oliveira@dm.ufscar.br)
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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.

Keywords

spectral theorysparse potentialSchrödinger operatorsDirac operators

MSC classification

Secondary: 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 35P05: General topics in linear spectral theory 35J10: Schrödinger operator
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 3 , October 2013 , pp. 667 - 700
Copyright
Copyright © Edinburgh Mathematical Society 2013 

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