Absence of localization in a class of Schr�dinger operators with quasiperiodic potential

Delyon, F.; Petritis, Dimitri

doi:10.1007/bf01211759

Cited by 108 publications

(77 citation statements)

References 6 publications

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“…It was stated and proved in this form by Delyon and Petritis [DP86]. Lemma 3.1 was proved by Sütő in [Sü87].…”

Section: Pointwise Methods and Variants Of The Gordon Criterionmentioning

confidence: 82%

“…These two classes of MEFA families have been studied extensively in the past; see [BIST,BIT,D98a,DKL,DL99a,DL99b,DP86,IT,IRT,HKS,J91,Ka,Ra,Sü87,Sü89] and [Be,BBG91,BBG92,BG93,D98b,D98c,D99b,DP91,H,HKS] for some important contributions in the case of circle map models and substitution models, respectively. The results comprise in particular singular continuity of spectral measures, zero Lebesgue measure of the spectrum, Gap labelling via K-theory, opening of the gaps at low coupling, continuity of gap boundaries with respect to the rotation number, and uniform existence of the Lyapunov exponent for large subclasses.…”

Section: Models Generated By Circle Maps and Primitive Substitutionsmentioning

confidence: 99%

“…In [DP86], the following theorem for circle map models has been proved. Theorem 6.6 (Delyon-Petritis).…”

Section: Absence Of Eigenvalues: Locating Squares and Cubesmentioning

confidence: 99%

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Gordon-type arguments in the spectral theory of one-dimensional quasi-crystals

Baake¹,

Moody²

2000

Directions in Mathematical Quasicrystals

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Abstract. We review the recent developments in the spectral theory of discrete one-dimensional Schrödinger operators with potentials generated by substitutions and circle maps. We discuss how occurrences of local repetitive structures allow for estimates of generalized eigenfunctions. Among the recent applications of this general approach are almost sure and uniform results on the absence of eigenvalues as well as continuity of the spectral measures with respect to Hausdorff measures.

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“…It was stated and proved in this form by Delyon and Petritis [DP86]. Lemma 3.1 was proved by Sütő in [Sü87].…”

Section: Pointwise Methods and Variants Of The Gordon Criterionmentioning

confidence: 82%

Section: Models Generated By Circle Maps and Primitive Substitutionsmentioning

confidence: 99%

See 1 more Smart Citation

Gordon-type arguments in the spectral theory of one-dimensional quasi-crystals

Baake¹,

Moody²

2000

Directions in Mathematical Quasicrystals

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show abstract

“…The first major application of this observation was in the context of the almost Mathieu operator: For super-critical coupling and Liouville frequencies, there is purely singular continuous spectrum for all phases, as shown by Avron and Simon in 1982 [6]. 4 The first application of direct relevance to this survey was obtained by Delyon and Petritis [58] in 1986 who proved absence of eigenvalues for certain codings of rotations, including most Sturmian models. Further applications will be mentioned below.…”

Section: Absence Of Point Spectrummentioning

confidence: 84%

Strictly ergodic subshifts and associated operators

Damanik¹

2007

Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday

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Dedicated to Barry Simon on the occasion of his 60th birthday.Abstract. We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twenty years. We review the mechanisms leading to these results and briefly discuss analogues for CMV matrices.

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“…With a similar method one can prove a result analogous to Theorem 5. For more details, we refer the reader to the paper of Delyon and Petritis [11] (treating the one-dimensional case).…”

Section: Then the Operator H With Potential V As Defined In (22) Has mentioning

confidence: 99%