Spectral properties of limit-periodic Schrödinger operators

David Damanik; Zheng Gan

doi:10.3934/cpaa.2011.10.859

Article Contents

2011, Volume 10, Issue 3: 859-871. Doi: 10.3934/cpaa.2011.10.859

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Spectral properties of limit-periodic Schrödinger operators

David Damanik^1, and
Zheng Gan^1,

1.
Department of Mathematics, Rice University, Houston, TX 77005

Received: March 31, 2009

Revised: August 31, 2009

Published: April 2011

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Abstract

We investigate the spectral properties of Schrödinger operators in $l^2(Z)$ with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling along the orbits of a minimal translation of a Cantor group. This point of view allows one to separate the base dynamics and the sampling function. We show that for any such base dynamics, the spectrum is of positive Lebesgue measure and purely absolutely continuous for a dense set of sampling functions, and it is of zero Lebesgue measure and purely singular continuous for a dense $G_\delta$ set of sampling functions.

Keywords:

Schrödinger operators,
limit-periodic potentials.

Mathematics Subject Classification: Primary: 47B36, 81Q10; Secondary: 47B80.

Citation:

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References

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