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May  2011, 10(3): 859-871. doi: 10.3934/cpaa.2011.10.859

Spectral properties of limit-periodic Schrödinger operators

David Damanik 1, and  Zheng Gan 1,

1. 

Department of Mathematics, Rice University, Houston, TX 77005, USA Government

Received  April 2009 Revised  September 2009 Published  December 2010

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: limit-periodic potentials., Schrödinger operators.
Mathematics Subject Classification: Primary: 47B36, 81Q10; Secondary: 47B8.
Citation: David Damanik, Zheng Gan. Spectral properties of limit-periodic Schrödinger operators. Communications on Pure & Applied Analysis, 2011, 10 (3) : 859-871. doi: 10.3934/cpaa.2011.10.859
References:
[1]

A. Avila, On the spectrum and Lyapunov exponent of limit periodic Schrödinger operators, Commun. Math. Phys., 288 (2009), 907-918.  Google Scholar

[2]

J. Avron and B. Simon, Almost periodic Schrödinger operators. I. Limit periodic potentials, Commun. Math. Phys., 82 (1981), 101-120.  Google Scholar

[3]

V. Chulaevsky, Perturbations of a Schrödinger operator with periodic potential, Uspekhi Mat. Nauk, 36 (1981), 203-204.  Google Scholar

[4]

V. Chulaevsky, "Almost Periodic Operators and Related Nonlinear Integrable Systems," Manchester University Press, Manchester, 1989.  Google Scholar

[5]

D. Damanik and Z. Gan, Limit-periodic Schrödinger operators in the regime of positive Lyapunov exponents, J. Funct. Anal., 258 (2010), 4010-4025.  Google Scholar

[6]

F. Delyon and D. Petritis, Absence of localization in a class of Schrödinger operators with quasiperiodic potential, Commun. Math. Phys., 103 (1986), 441-444.  Google Scholar

[7]

A. Gordon, The point spectrum of the one-dimensional Schrödinger operator, Usp. Math. Nauk., 31 (1976), 257-258.  Google Scholar

[8]

Y. Last, On the measure of gaps and spectra for discrete $1$D Schrödinger operators, Commun. Math. Phys., 149 (1992), 347-360.  Google Scholar

[9]

S. Molchanov and V. Chulaevsky, The structure of a spectrum of the lacunary-limit-periodic Schrödinger operator, Functional Anal. Appl., 18 (1984), 343-344.  Google Scholar

[10]

J. Moser, An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helv., 56 (1981), 198-224.  Google Scholar

[11]

J. Pöschel, Examples of discrete Schrödinger operators with pure point spectrum, Commun. Math. Phys., 88 (1983), 447-463.  Google Scholar

[12]

L. Ribes and P. Zalesskii, "Profinite Groups," Springer-Verlag, Berlin, 2000.  Google Scholar

[13]

B. Simon, Szegös theorem and its descendants: spectral theory for $l^2$ perturbations of orthogonal polynomials, Princeton University Press, 2010 Google Scholar

[14]

G. Teschl, "Jacobi Operators and Completely Integrable Nonlinear Lattices," Mathematical Surveys and Monographs 72, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[15]

M. Toda, "Theory of Nonlinear Lattices," 2nd edition, Springer Series in Solid-State Sciences 20, Springer-Verlag, Berlin, 1989.  Google Scholar

[16]

J. Wilson, "Profinite Groups," Oxford University Press, New York, 1998.  Google Scholar

show all references

References:
[1]

A. Avila, On the spectrum and Lyapunov exponent of limit periodic Schrödinger operators, Commun. Math. Phys., 288 (2009), 907-918.  Google Scholar

[2]

J. Avron and B. Simon, Almost periodic Schrödinger operators. I. Limit periodic potentials, Commun. Math. Phys., 82 (1981), 101-120.  Google Scholar

[3]

V. Chulaevsky, Perturbations of a Schrödinger operator with periodic potential, Uspekhi Mat. Nauk, 36 (1981), 203-204.  Google Scholar

[4]

V. Chulaevsky, "Almost Periodic Operators and Related Nonlinear Integrable Systems," Manchester University Press, Manchester, 1989.  Google Scholar

[5]

D. Damanik and Z. Gan, Limit-periodic Schrödinger operators in the regime of positive Lyapunov exponents, J. Funct. Anal., 258 (2010), 4010-4025.  Google Scholar

[6]

F. Delyon and D. Petritis, Absence of localization in a class of Schrödinger operators with quasiperiodic potential, Commun. Math. Phys., 103 (1986), 441-444.  Google Scholar

[7]

A. Gordon, The point spectrum of the one-dimensional Schrödinger operator, Usp. Math. Nauk., 31 (1976), 257-258.  Google Scholar

[8]

Y. Last, On the measure of gaps and spectra for discrete $1$D Schrödinger operators, Commun. Math. Phys., 149 (1992), 347-360.  Google Scholar

[9]

S. Molchanov and V. Chulaevsky, The structure of a spectrum of the lacunary-limit-periodic Schrödinger operator, Functional Anal. Appl., 18 (1984), 343-344.  Google Scholar

[10]

J. Moser, An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helv., 56 (1981), 198-224.  Google Scholar

[11]

J. Pöschel, Examples of discrete Schrödinger operators with pure point spectrum, Commun. Math. Phys., 88 (1983), 447-463.  Google Scholar

[12]

L. Ribes and P. Zalesskii, "Profinite Groups," Springer-Verlag, Berlin, 2000.  Google Scholar

[13]

B. Simon, Szegös theorem and its descendants: spectral theory for $l^2$ perturbations of orthogonal polynomials, Princeton University Press, 2010 Google Scholar

[14]

G. Teschl, "Jacobi Operators and Completely Integrable Nonlinear Lattices," Mathematical Surveys and Monographs 72, American Mathematical Society, Providence, RI, 2000.  Google Scholar

[15]

M. Toda, "Theory of Nonlinear Lattices," 2nd edition, Springer Series in Solid-State Sciences 20, Springer-Verlag, Berlin, 1989.  Google Scholar

[16]

J. Wilson, "Profinite Groups," Oxford University Press, New York, 1998.  Google Scholar

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