Limit-periodic Schrödinger operators with uniformly localized eigenfunctions

Damanik, David; Guan, Zheng

doi:10.1007/s11854-011-0022-y

Cited by 21 publications

(36 citation statements)

References 24 publications

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“…All other operator families that are known to be localized (random potentials, strongly coupled quasi-periodic potentials or skew-shift potentials) are either not known to be uniformly localized or known to be not uniformly localized. Motivated by [71] and extending [135], Han [125] showed that phase uniformity is a general phenomenon in the context of uniform localization.…”

Section: 3mentioning

confidence: 99%

Schrödinger operators with dynamically defined potentials

Damanik¹

2016

Ergod. Th. Dynam. Sys.

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135

110

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Abstract. In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schrödinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the cases of random and almost periodic potentials.

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Section: 3mentioning

confidence: 99%

Schrödinger operators with dynamically defined potentials

Damanik¹

2016

Ergod. Th. Dynam. Sys.

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135

110

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“…It is however possible to prove pure point spectrum uniformly across the hull in certain scenarios (which are not dense). This follows from a combination of works of Pöschel [53] and Damanik-Gan [17].…”

Section: Pure Point Spectrummentioning

confidence: 99%

Spectral properties of limit-periodic Schrödinger operators

Guan¹

2010

CPAA

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We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schrödinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum Schrödinger operators and multi-dimensional Schrödinger operators, are discussed as well.We explain that each basic spectral type occurs, and it does so for a dense set of limit-periodic potentials. The spectrum has a strong tendency to be a Cantor set, but there are also cases where the spectrum has no gaps at all. The possible regularity properties of the integrated density of states range from extremely irregular to extremely regular. Additionally, we present background about periodic Schrödinger operators and almost-periodic sequences.In many cases we outline the proofs of the results we present.

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“…In this paper we will answer this question which arose naturally from the known results, and which was asked in print on several occasions (e.g., [14,18]). Theorem 1.1.…”

Section: Introductionmentioning

confidence: 94%

An extension of the Kunz–Souillard approach to localization in one dimension and applications to almost-periodic Schrödinger operators

Damanik

Gorodetski

2016

Advances in Mathematics

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We generalize the approach to localization in one dimension introduced by Kunz-Souillard, and refined by Delyon-Kunz-Souillard and Simon, in the early 1980's in such a way that certain correlations are allowed. Several applications of this generalized Kunz-Souillard method to almost periodic Schrödinger operators are presented.On the one hand, we show that the Schrödinger operators on ℓ 2 (Z) with limit-periodic potential that have pure point spectrum form a dense subset in the space of all limit-periodic Schrödinger operators on ℓ 2 (Z). More generally, for any bounded potential, one can find an arbitrarily small limit-periodic perturbation so that the resulting operator has pure point spectrum. Our result complements the known denseness of absolutely continuous spectrum and the known genericity of singular continuous spectrum in the space of all limit-periodic Schrödinger operators on ℓ 2 (Z).On the other hand, we show that Schrödinger operators on ℓ 2 (Z) with arbitrarily small one-frequency quasi-periodic potential may have pure point spectrum for some phases. This was previously known only for one-frequency quasiperiodic potentials with · ∞ norm exceeding 2, namely the super-critical almost Mathieu operator with a typical frequency and phase. Moreover, this phenomenon can occur for any frequency, whereas no previous quasi-periodic potential with Liouville frequency was known that may admit eigenvalues for any phase.

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Limit-periodic Schrödinger operators with uniformly localized eigenfunctions

Cited by 21 publications

References 24 publications

Schrödinger operators with dynamically defined potentials

Schrödinger operators with dynamically defined potentials

Spectral properties of limit-periodic Schrödinger operators

An extension of the Kunz–Souillard approach to localization in one dimension and applications to almost-periodic Schrödinger operators

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