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Applicable Analysis
An International Journal
Volume 86, 2007 - Issue 10
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Original Articles

Sturm–Liouville operators in the half axis with shifted potentials

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Pages 1211-1221 | Received 21 Mar 2007, Accepted 29 May 2007, Published online: 21 Sep 2010
 

Abstract

We consider Sturm–Liouville operators in the half axis generated by shifts of the potential and prove that Lebesgue measure is equivalent to a measure defined as an average of spectral measures which correspond to these operators. This is then used to obtain results on stability of spectral types under change of parameters such as boundary conditions, local perturbations, and shifts. In particular if for a set of shifts of positive measure the corresponding operators have α-singular, singular continuous and (or) point spectrum in a fixed interval, then this set of shifts has to be unbounded. Moreover, there are large sets of boundary conditions and local perturbations for which the corresponding operators enjoy the same property.

Acknowledgements

Rafael del Rio was partially supported by Project PAPIIT IN111906–3 and Project CONACYT 37444–E. Carmen A. Martínez was partially supported by Scholarship CONACYT 200317 and Project CONACYT 37444–E.

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